Grade 7 / Quadrilaterals 111 exercises (including 110 corrected)

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ABCD ABCDEFGHIJKLMNOP (d(d(d==(d (t(t Jesais(AB==(MNet(CD==(MNJ’uti-liseJ’endé-duis(AB==(CDa Jesais(EF(GHet(GH(IJJ’uti-liseJ’endé-duis(EF==(IJb ABCD3cmI 1. Reminders E.6620 Consider the quadrilateral ABCD below : 1 What does the segment [ DC ] represent for this quadri-lateral? 2 What does the segment [ BD ] represent for the quadrilat-eral ABCD ? 3 What do the pair of segments [ AD ] and [ BC ] represent for ABCD ? 4 Name a pair of consecutive sides. E.6621 Consider the four quadrilaterals shown below. Each of these quadrilaterals is a particular quadrilateral: 1 Give the nature of each of these quadrilaterals. 2 For each of these quadrilaterals, name the properties, if any, related to , opposite sides, adjacent sides, angles, and their diagonals. E.6627 1 With the information encoded in the figure opposite, which proposition al-lows you to state that the straight lines ( d ) and (Δ) are perpendicular? 2 With the information coded on the figure opposite, which proposition al-lows you to state that the straight lines ( t ) and ( t ) are parallel? E.11724 Compléter les deux chaînons déductifs : 2. Properties of the parallelogram E.2067 Proposition: If a quadrilateral is a parallelogram then its opposite sides are parallel. If a quadrilateral is a parallelogram then its opposite sides are of the same measure. If a quadrilateral is a parallelogram then its opposite an-gles have the same measure. If a quadrilateral is a parallelogram then its diagonals intersect at their middles. Consider the par-allelogram ABCD opposite. I is the middle of the diag-onal [ AC ] . 1 This parallelogram can also be named DCBA . Name the eight different ways of naming this quadrilateral. 2 What can be said about the middle of the diagonal [ DB ] ? Quote the proposition that affirms this property. 3 What is the measure of side [ BC ] ? Quote the proposition that affirms this property. 4 What can be said about the angles DAC and ACB ? Quote the proposition by which this property can be as-serted. https://chingmath.fr chapExoCorrec/6620 sacados/6620 ABCD chapExoCorrec/6621 sacados/6621 ABCDEFGHIJKLMNOP chapExoCorrec/6627 sacados/6627 (d(d(d==(d (t(t chapExoCorrec/11724 sacados/11724 Jesais(AB==(MNet(CD==(MNJ’uti-liseJ’endé-duis(AB==(CDa Jesais(EF(GHet(GH(IJJ’uti-liseJ’endé-duis(EF==(IJb chapExoCorrec/2067 sacados/2067 ABCD3cmI
ABCD3cmI TrapèzeParallé-logrammeLosangeRectangleCarréLescôtésopposéssontparallèlesLes côtés opposés sont demême longueurLescôtésconsécutifs sontperpendiculairesLescôtésconsécutifs sontde même longueur TrapèzeParallé-logrammeLosangeRectangleCarréLes diagonales se coupenten leurs milieuxLesdiagonalesontlesmêmes longueursLesdiagonalessontper-pendiculaires ABCDEFG JesaisABCDest un rectangleJ’uti-liseSi un quadrilatère est un...........................Alors ses côtés consécutifs sont.....................J’endé-duis(AD(DC JesaisEFGCest un rectangle.J’uti-lise..........................................................................................................J’endé-duis(DG(GF Jesais(AD(DGet(DG(GFJ’uti-lise..........................................................................................................J’endé-duis(AD==(FG E.6626 Consider the parallelogram ABCD shown below : 1 The segment [ BC ] measures 3 cm . Which proposition allows you to state this? 2 The segment [ AC ] admits the point I as its middle. Which proposition allows you to state this? 3 The lines ( AB ) and ( DC ) are parallel. Which proposi-tion allows you to state this? E.11027 For each of the proposed quadrilat-erals, tick the properties they possess : E.11028 For each of the proposed quadrilat-erals, tick the properties they possess : 3. Properties and deductive links E.11029 In the plane, consider the figure opposite, which consists of two rectangles ABCD and EFGC . Complete the deductive links be-low : Theorem: If two straight lines are parallel to the same third straight line then they are parallel to each other. If two straight lines are perpendicular to the same third straight line then they are parallel to each other. If two straight lines are parallel to each other and a third straight line is perpendicular to one then it is perpendicular to the other. https://chingmath.fr chapExoCorrec/6626 sacados/6626 ABCD3cmI chapExoCorrec/11027 sacados/11027 TrapèzeParallé-logrammeLosangeRectangleCarréLescôtésopposéssontparallèlesLes côtés opposés sont demême longueurLescôtésconsécutifs sontperpendiculairesLescôtésconsécutifs sontde même longueur chapExoCorrec/11028 sacados/11028 TrapèzeParallé-logrammeLosangeRectangleCarréLes diagonales se coupenten leurs milieuxLesdiagonalesontlesmêmes longueursLesdiagonalessontper-pendiculaires chapExoCorrec/11029 sacados/11029 ABCDEFG JesaisABCDest un rectangleJ’uti-liseSi un quadrilatère est un...........................Alors ses côtés consécutifs sont.....................J’endé-duis(AD(DC JesaisEFGCest un rectangle.J’uti-lise..........................................................................................................J’endé-duis(DG(GF Jesais(AD(DGet(DG(GFJ’uti-lise..........................................................................................................J’endé-duis(AD==(FG
ABCDEF JesaisABEFest un parallélogrammeJ’uti-liseSi un quadrilatère est un...........................Alors ses côtés opposés sont........................J’endé-duis(FA==(EB JesaisEBCDest un parallélogramme.J’uti-lise..........................................................................................................J’endé-duis(EB==(DC Jesais(AF==(BEet(BE==(CDJ’uti-lise..........................................................................................................J’endé-duis(FA==(CD ABCDMNIJ JesaisABCDest un rectangle etAC10cmJ’uti-liseSi un quadrilatère est un rectangle alors ses diagonalesaet ses diagonalesbJ’endé-duisDIDB2cabc JesaisDMINest un rectangle etDI5cmJ’uti-liseSi un quadrilatère est un rectangle alorsdJ’endé-duisMJede E.11030 Consider the two parallelograms ABEF and BCDE shown below : Complete the deductive links below : Theorem: If two straight lines are parallel to the same third straight line then they are parallel to each other. If two straight lines are perpendicular to the same third straight line then they are parallel to each other. If two straight lines are parallel to each other and a third straight line is perpendicular to one then it is perpendicular to the other. E.11032 1 Make the plot program on your copy: a Draw a rectangle ABCD such as : AB = 9 cm ; AC = 10 cm b Name I the point of intersection of the intersections of the diagonals of the rectangle ABCD . c Construct the rectangle DMIN where : M ∈ [ AD ] ; N ∈ [ CD ] You should get a figure similar to : 2 On the properties of the figure below, we have con-structed the deductive links below : Copy, on your copy, the parts a , b , c , d , e missing in this reasoning. 4. Characterization of parallelograms E.2068 Proposition: (characterizing properties) If a quadrilateral has its opposite sides of the same length then that quadrilateral is a parallelogram. If a quadrilateral has its opposite angles of the same mea-sure then this quadrilateral is a parallelogram. If a quadrilateral has its opposite sides parallel then this quadrilateral is a parallelogram. If a quadrilateral has its diagonals intersecting in their middles then this quadrilateral is a parallelogram. If a quadrilateral has two of its opposite sides parallel and of the same length then this quadrilateral is a parallel-ogram. https://chingmath.fr chapExoCorrec/11030 sacados/11030 ABCDEF JesaisABEFest un parallélogrammeJ’uti-liseSi un quadrilatère est un...........................Alors ses côtés opposés sont........................J’endé-duis(FA==(EB JesaisEBCDest un parallélogramme.J’uti-lise..........................................................................................................J’endé-duis(EB==(DC Jesais(AF==(BEet(BE==(CDJ’uti-lise..........................................................................................................J’endé-duis(FA==(CD chapExoCorrec/11032 sacados/11032 ABCDMNIJ JesaisABCDest un rectangle etAC10cmJ’uti-liseSi un quadrilatère est un rectangle alors ses diagonalesaet ses diagonalesbJ’endé-duisDIDB2cabc JesaisDMINest un rectangle etDI5cmJ’uti-liseSi un quadrilatère est un rectangle alorsdJ’endé-duisMJede chapExoCorrec/2068 sacados/2068
ABCDI1ABCD2CDAB(AB==(CD(AD==(BC3BADC4 ABCDEFO(BC==(FE ABCDOCC ABCD(d(DC==(AB JesaisJ’utiliseJ’endéduisSi deux droites sont perpendiculaires à une mêmedroite alors elles sont parallèles entre elles JesaisJ’utiliseJ’endéduisABCDest un parallélogramme ABCDEFO In each case, justify, citing the property used, that the quadri-lateral ABCD is a parallelogram. E.5625 The configuration below consists of the two quadrilaterals ABCD and BCEF . 1 Justify that the quadrilateral ABCD is a parallelogram. 2 Justify that the quadrilateral BCEF is a parallelogram. 3 Without justification, what can be said about segments [ AD ] and [ EF ] . E.5624 The figure below shows two circles C and C with center O . The segment [ BD ] is a diameter of the circle C and the segment [ AC ] is a diameter of the circle C . Show that the quadrilateral ABCD is a parallelogram. 5. Characterization of the Parallelogram and Deductive Links E.4438 Consider the quadrilateral shown below : Complete the following deductive links: E.2084 Consider the configuration below composed of the two quadrilaterals ABCD and CEFB : 1 What is the nature of the quadrilateral ABCD ? Justify your answer. 2 What is the nature of the quadrilateral CEFB ? Justify your answer. 3 Justify that the lines ( AD ) and ( EF ) are parallel. https://chingmath.fr ABCDI1ABCD2CDAB(AB==(CD(AD==(BC3BADC4 chapExoCorrec/5625 sacados/5625 ABCDEFO(BC==(FE chapExoCorrec/5624 sacados/5624 ABCDOCC chapExoCorrec/4438 sacados/4438 ABCD(d(DC==(AB JesaisJ’utiliseJ’endéduisSi deux droites sont perpendiculaires à une mêmedroite alors elles sont parallèles entre elles JesaisJ’utiliseJ’endéduisABCDest un parallélogramme chapExoCorrec/2084 sacados/2084 ABCDEFO
ABCD(dM(AD==(BC JesaisJ’utiliseJ’endéduis(DCet(ABab JesaisJ’utiliseJ’endéduisABCDest un parallélogramme.cd JesaisJ’utiliseJ’endéduis(d(BCet(AD==(BCef JesaisJ’utiliseJ’endéduisAMBest un triangle rectanglegh Jesais(AB==(DCet(AD==(BCJ’uti-liseJ’endé-duisABCDest un parallélogramme.a JesaisMNQPetMQNPJ’uti-liseJ’endé-duisMNPQest un parallélogrammeb JesaisLes diagonales deRSTUs’interceptennt enV.RVTVetSVUVJ’uti-liseJ’endé-duisRSTUest un parallélogrammec JesaisABBCCDDAJ’uti-liseSiunquadrilatèreasesquatrescôtésconsécutifsdemême longueur alors ce quadrilatère est un losangeJ’endé-duis..................................................... JesaisEGFH;EGetFHse coupent en leur mi-lieuxJ’uti-lise..........................................................................................................J’endé-duisEFGHest un rectangle Jesais..........................................................................................................J’uti-liseSi un quadrilatère a ses diagonales perpendiculaire, demême longueur et se coupant en leurs milieux alors cequadrilatère est un carréJ’endé-duisIJKLest un carré E.4465 Consider the quadrilateral ABCD and the two straight lines ( d ) and (Δ) shown below : Complete the following two reasonings : 1 2 E.11725 Compléter les chaînons déductifs : 6. Characterization of Specific Quadrilaterals and Deductive Chains E.11036 Complete the deductive chain be-low : https://chingmath.fr chapExoCorrec/4465 sacados/4465 ABCD(dM(AD==(BC JesaisJ’utiliseJ’endéduis(DCet(ABab JesaisJ’utiliseJ’endéduisABCDest un parallélogramme.cd JesaisJ’utiliseJ’endéduis(d(BCet(AD==(BCef JesaisJ’utiliseJ’endéduisAMBest un triangle rectanglegh chapExoCorrec/11725 sacados/11725 Jesais(AB==(DCet(AD==(BCJ’uti-liseJ’endé-duisABCDest un parallélogramme.a JesaisMNQPetMQNPJ’uti-liseJ’endé-duisMNPQest un parallélogrammeb JesaisLes diagonales deRSTUs’interceptennt enV.RVTVetSVUVJ’uti-liseJ’endé-duisRSTUest un parallélogrammec chapExoCorrec/11036 sacados/11036 JesaisABBCCDDAJ’uti-liseSiunquadrilatèreasesquatrescôtésconsécutifsdemême longueur alors ce quadrilatère est un losangeJ’endé-duis..................................................... JesaisEGFH;EGetFHse coupent en leur mi-lieuxJ’uti-lise..........................................................................................................J’endé-duisEFGHest un rectangle Jesais..........................................................................................................J’uti-liseSi un quadrilatère a ses diagonales perpendiculaire, demême longueur et se coupant en leurs milieux alors cequadrilatère est un carréJ’endé-duisIJKLest un carré
ABCDO JesaisJ’uti-liseJ’endéduisSiunquadrilatèreasesdiagonalesdemêmelongueur et se coupent en leurs milieurs alors cequadrilatère est un rectangle JesaisJ’uti-liseJ’endéduisABC90o ABCDOE JesaisJ’utiliseJ’endéduisABCDest un rectangle JesaisJ’utiliseJ’endéduisODOA JesaisJ’utiliseJ’endéduis(ADest la médiatrice du segmentEO JesaisJ’utiliseJ’endéduisDEDO JesaisJ’utiliseJ’endéduisAEAO JesaisJ’utiliseJ’endéduisAODEest un losange E.4437 Consider the quadrilateral shown below : E.4436 Consider the figure below, where point O is the intersection of the diagonals of quadrilateral ABCD ; the coding provides properties between the different elements of this figure : Complete the following deductive links: https://chingmath.fr chapExoCorrec/4437 sacados/4437 ABCDO JesaisJ’uti-liseJ’endéduisSiunquadrilatèreasesdiagonalesdemêmelongueur et se coupent en leurs milieurs alors cequadrilatère est un rectangle JesaisJ’uti-liseJ’endéduisABC90o chapExoCorrec/4436 sacados/4436 ABCDOE JesaisJ’utiliseJ’endéduisABCDest un rectangle JesaisJ’utiliseJ’endéduisODOA JesaisJ’utiliseJ’endéduis(ADest la médiatrice du segmentEO JesaisJ’utiliseJ’endéduisDEDO JesaisJ’utiliseJ’endéduisAEAO JesaisJ’utiliseJ’endéduisAODEest un losange
ABCDO(d JesaisJ’utiliseJ’endéduisABCDest un losange JesaisJ’utiliseJ’endéduisSi un quadrilatère est un losange alors ses diago-nales sont perpendiculaires JesaisJ’utiliseJ’endéduis(d(AO JesaisJ’utiliseJ’endéduis(d==(DB LosangeRectangleCarréSiun parallélogramme a ses diagonales perpendic-ulairesAlorsce quadrilatère est ...Siun parallélogramme a ses côtés consécutifs per-pendiculairesAlorsce quadrilatère est ...SiunrectangleasescôtésconsécutifsdemêmelongueurAlorsce quadrilatère est ...Siunlosangeasescôtésconsécut∏sperpendicu-lairesAlorsce quadrilatère est ... LosangeRectangleCarréSiunrectangleasescôtésconsécutifsdemêmelongueurAlorsce quadrilatère est ...Siun parallélogramme a ses diagonales perpendic-ulairesAlorsce quadrilatère est ...Siun parallélogramme a ses côtés consécutifs per-pendiculairesAlorsce quadrilatère est ...Siunlosange asesdiagonales demême longueurAlorsce quadrilatère est ... E.6180 The figure opposite shows a quadri-lateral ABCD in which all sides are equal in length : AB = BC = CD = DA The line ( d ) is the perpendicular bi-sector of the segment [ AO ] . Complete the following deductive chains : 7. Construction program E.11034 For each of the proposed quadrilat-erals, tick the properties they possess : E.11035 E.11737 Recopier et compléter les phrases suiv-antes : 1 Si un parallélogramme a ses diagonales perpendiculaire alors ce quadrilatère est un . . . 2 Si un parallélogramme a deux de ses côtés consécutifs de même longueur alors ce quadrilatère est un . . . 3 Si un rectangle a ses diagonales perpendiculaires alors ce quadrilatère est un . . . 4 Si un losange possède un angle droit alors ce quadrilatère est un . . . E.11739 Pour chacune des affirmations suiv-antes, donnez les deux propriétés permettant de compléter les pointillés afin que l’affirmation soit vraie. 1 Si un losange . . . alors ce quadrilatère est un carré. 2 Si un parallélogramme . . . alors ce quadrilatère est un rectangle. 8. Classifications of Parallelograms and Deductive Chains https://chingmath.fr chapExoCorrec/6180 sacados/6180 ABCDO(d JesaisJ’utiliseJ’endéduisABCDest un losange JesaisJ’utiliseJ’endéduisSi un quadrilatère est un losange alors ses diago-nales sont perpendiculaires JesaisJ’utiliseJ’endéduis(d(AO JesaisJ’utiliseJ’endéduis(d==(DB chapExoCorrec/11034 sacados/11034 LosangeRectangleCarréSiun parallélogramme a ses diagonales perpendic-ulairesAlorsce quadrilatère est ...Siun parallélogramme a ses côtés consécutifs per-pendiculairesAlorsce quadrilatère est ...SiunrectangleasescôtésconsécutifsdemêmelongueurAlorsce quadrilatère est ...Siunlosangeasescôtésconsécut∏sperpendicu-lairesAlorsce quadrilatère est ... chapExoCorrec/11035 sacados/11035 LosangeRectangleCarréSiunrectangleasescôtésconsécutifsdemêmelongueurAlorsce quadrilatère est ...Siun parallélogramme a ses diagonales perpendic-ulairesAlorsce quadrilatère est ...Siun parallélogramme a ses côtés consécutifs per-pendiculairesAlorsce quadrilatère est ...Siunlosange asesdiagonales demême longueurAlorsce quadrilatère est ... chapExoCorrec/11737 sacados/11737 sacados/11739
IABCDEFG JesaisABDCetADBCJ’uti-liseSi un quadrilatère a ses côtés............................................................................alors ce quadrilatère est un rectangleJ’endé-duisABCDest un parallélogramme JesaisABCDest un rectangle etDABest un angle droit.J’uti-liseSi un parallélogramme a ......alors ce quadrilatère estun rectangle.J’endé-duisABCDest un rectangle JesaisICIEIFIGJ’uti-liseSi un quadrilatère a ses diagonales.......................................................................alors ce quadrilatère est un rectangleJ’endé-duisCEFGest un rectangle JesaisABDCetADBCJ’uti-liseSi un quadrilatère a ses côtés............................................................................alors ce quadrilatère est un parallélogrammeJ’endé-duisABCDest un parallélogramme JesaisICIEIFIGJ’uti-liseSi un parallélogramme a ses diagonales..................................................................alors ce quadrilatère est un rectangleJ’endé-duisCEFG JesaisJ’uti-liseJ’endé-duis(CE(EF Jesais(AB==(CEet(CE(EFJ’uti-liseJ’endé-duis(AB(EF ABCDIJLK E.11031 Consider the figure below, where information has been encoded, consisting of a right triangle BCE rectangle at C and two quadrilaterals: 1 Complete the two deductive links to determine the na-ture of these two quadrilaterals: For quadrilateral ABCD : For quadrilateral CEFG : 2 a Complete the following deductive links to justify that lines ( AB ) and ( CE ) are parallel: b Complete the following deductive steps to justify that lines ( AB ) and ( EF ) are perpendicular : Hint: you can use the following properties : If two lines are parallel to each other and a third line is perpendicular to one of them, then it is perpendicular to the other. If two lines are perpendicular to the same third line, then they are parallel to each other. If a quadrilateral is a rectangle, then its consecutive sides are perpendicular. If a quadrilateral has diagonals of equal length that in-tersect at their midpoints, then the quadrilateral is a rectangle. If a quadrilateral has opposite sides of equal length and has a right angle, then the quadrilateral is a rectangle. E.2599 Consider the figure below : The point K is obtained by intersection of the segment [ DC ] with the arc of a circle with center D and passing through the point J . 1 Justifying, give the nature of each of the quadrilaterals below : a ABCD b CIDJ c DJLK 2 a Justify that the straight lines ( KL ) and ( DJ ) are parallel. b Justify that the straight lines ( IC ) and ( KL ) are par-allel. c Deduce the nature of the quadrilateral ICLK . https://chingmath.fr chapExoCorrec/11031 sacados/11031 IABCDEFG JesaisABDCetADBCJ’uti-liseSi un quadrilatère a ses côtés............................................................................alors ce quadrilatère est un rectangleJ’endé-duisABCDest un parallélogramme JesaisABCDest un rectangle etDABest un angle droit.J’uti-liseSi un parallélogramme a ......alors ce quadrilatère estun rectangle.J’endé-duisABCDest un rectangle JesaisICIEIFIGJ’uti-liseSi un quadrilatère a ses diagonales.......................................................................alors ce quadrilatère est un rectangleJ’endé-duisCEFGest un rectangle JesaisABDCetADBCJ’uti-liseSi un quadrilatère a ses côtés............................................................................alors ce quadrilatère est un parallélogrammeJ’endé-duisABCDest un parallélogramme JesaisICIEIFIGJ’uti-liseSi un parallélogramme a ses diagonales..................................................................alors ce quadrilatère est un rectangleJ’endé-duisCEFG JesaisJ’uti-liseJ’endé-duis(CE(EF Jesais(AB==(CEet(CE(EFJ’uti-liseJ’endé-duis(AB(EF chapExoCorrec/2599 sacados/2599 ABCDIJLK
JesaisABCDest un losange etACBDJ’uti-liseSi un losange a ses diagonales de même longueur Alorsce quadrilatère est un carréJ’endé-duis..................................................... JesaisDEFGest un parallélogramme et(DF(EGJ’uti-lise..........................................................................................................J’endé-duisDEFGest un losange Jesais..........................................................................................................J’uti-liseSiunparallélogrammeasesdiagonalesdemêmelongueur Alors ce quadrilatère est un rectangleJ’endé-duisHIJKest un rectangle ABCD JesaisJ’utiliseJ’endéduisSiunquadrilatèreasesquatrecôtésdemêmelongueur alors ce quadrilatère est un losange JesaisJ’utiliseJ’endéduisABCDest un losange etABC90oABCDest un carré ABCD55o5cm3cmEFJI105o4cm7cm GHP45o2;5cm3;5cmLTRSV4cm5;5cm6cm 5cm5cm110o5cm8cmABCDEFG ABCDEFGHO40oOGcmOHcm4;5cm E.11037 Complete the deductive links: E.6178 The figure opposite represents a quadrilat-eral such that its vertex B defines a right angle and such that the following lengths are equal: AB = BC = CD = DA Complete the following deductive chains : 9. Parallelogram drawing E.2076 Reproduce the parallelograms below, respecting the indications on the figures : Hint: to draw the parallelogram, start by drawing a trian-gle constructed from two of its sides and one of the diagonals and use the center of symmetry of the parallelogram. E.4126 The figure below is composed of three parallelograms: ABCD ; ABEF ; ADGF Reproduce this figure, respecting the measurements shown. 10. Traces of particular quadrilaterals E.2082 Consider the two rhombuses ABCD and EFGH : https://chingmath.fr chapExoCorrec/11037 sacados/11037 JesaisABCDest un losange etACBDJ’uti-liseSi un losange a ses diagonales de même longueur Alorsce quadrilatère est un carréJ’endé-duis..................................................... JesaisDEFGest un parallélogramme et(DF(EGJ’uti-lise..........................................................................................................J’endé-duisDEFGest un losange Jesais..........................................................................................................J’uti-liseSiunparallélogrammeasesdiagonalesdemêmelongueur Alors ce quadrilatère est un rectangleJ’endé-duisHIJKest un rectangle chapExoCorrec/6178 sacados/6178 ABCD JesaisJ’utiliseJ’endéduisSiunquadrilatèreasesquatrecôtésdemêmelongueur alors ce quadrilatère est un losange JesaisJ’utiliseJ’endéduisABCDest un losange etABC90oABCDest un carré chapExoCorrec/2076 sacados/2076 ABCD55o5cm3cmEFJI105o4cm7cm GHP45o2;5cm3;5cmLTRSV4cm5;5cm6cm chapExoCorrec/4126 sacados/4126 5cm5cm110o5cm8cmABCDEFG chapExoCorrec/2082 sacados/2082 ABCDEFGHO40oOGcmOHcm4;5cm
GHIJK35o4;5cm 9cm3cm7cmABCDEFGH8cm9cmIJKL9cm4cm 5cm9cmNPMO IJLK5cm37o URSTO5cm58o 3;5cmLMNVK48o Construct, at full size, these two rhombuses. E.2083 Consider the rectangle GHIJ below. Construct this rectangle to full size.: E.2873 Consider the four quadrilaterals shown below : Draw the required lines, taking into account the suggested indications : 1 Draw the parallelogram ABCD . (the parallelogram is composed of two triangles) 2 Draw the rectangle EFGH . (the rectangle has four right angles) 3 Draw the rhombus IJKL . (the diagonals of the rhombus intersect in their middles and are perpendicular) E.10172 Consider the four quadrilaterals shown below : Draw the required lines, taking into account the proposed indications : Draw the rhombus MNOP . (the rhombus is composed of four right-angled triangles) E.2598 Carry out the following layouts in accordance with the instructions : 1 Draw a parallelogram ABCD such that : CAB = 70 o ; ABC = 40 o ; AB = 5 cm 2 Draw a rhombus EFGH with the following measures : EF = 5 cm ; FEH = 60 o E.2085 1 Draw a parallelogram ABCD such that : AB = 8 cm ; AD = 4 cm ; ADC = 75 o 2 Draw the rectangle IJKL whose diagonals intercept at the point M verifying: IK = 10 cm ; IMJ = 120 o E.2100 Draw the following quadrilaterals: 1 ABCD a rectangle of center O such that : BD = 6 cm ; ∠ DOC = 140 o 2 IJKL a parallelogram such that : LI = 4 cm ; ∠ LIK = 100 o ; IK = 6 cm E.10170 Perform the following plots accord-ing to the instructions : 1 Draw the rectangle IJKL such that : LK = 5 cm ; ∠ IJL = 60 o 2 Draw a diamond EFGH such that : EG = 8 cm ; ∠ FEG = 40 o E.10173 Below are two quadrilaterals: IJKL is a rhombus, RSTU is a rectangle. Reproduce these two quadrilaterals in their actual sizes. E.10174 Reproduce the rectangle below by respecting the indications on the figure : E.10169 Draw the rhombus EFGH having the measure : HF = 8 cm ; ∠ HGF = 114 o https://chingmath.fr chapExoCorrec/2083 sacados/2083 GHIJK35o4;5cm chapExoCorrec/2873 sacados/2873 9cm3cm7cmABCDEFGH8cm9cmIJKL9cm4cm chapExoCorrec/10172 sacados/10172 5cm9cmNPMO chapExoCorrec/2598 sacados/2598 chapExoCorrec/2085 sacados/2085 chapExoCorrec/2100 sacados/2100 chapExoCorrec/10170 sacados/10170 chapExoCorrec/10173 sacados/10173 IJLK5cm37o URSTO5cm58o chapExoCorrec/10174 sacados/10174 3;5cmLMNVK48o chapExoCorrec/10169 sacados/10169
5cm25oABCD6cm2;5cm4;5cmEFGH ABCDO55o3cm4;5cm EGFHP5cm144o E.11740 On considère le losange ABCD et le parallélogramme EFGH représentés ci-dessous : Tracer en vraie grandeur ces deux quadrilatères. 11. Characterization and construction of quadrilaterals E.1545 In the plane, consider the quadrilat-eral ABCD below : 1 What is the nature of the quadrilateral ABCD ? Justify your answer. 2 Draw the quadrilateral ABDC to full size. E.10171 In the plane, consider the two quadrilaterals ABCD and EFGH below : 1 What is the nature of the quadrilateral EFGH ? Justify your answer. 2 Perform a full-scale plot of the quadrilateral EFGH . E.1459 Consider the three parallelograms below : ABCD such that : AB = AD =4.5 cm ; BAD =35 o . EFGH such that : EG =9 cm ; EF =5 cm ; ( EG ) ⊥ ( HF ) IJKL such that : IJ =3 cm ; IK = JL =6 cm 1 Specify the nature of each of these parallelograms. 2 Draw each of these quadrilaterals. 12. Tracing triangles and quadrilaterals E.4177 Consider the figure below où : AHBC is a parallelogram; ACGF is a rhombus ; BCDE is a rectangle. https://chingmath.fr chapExoCorrec/11740 sacados/11740 5cm25oABCD6cm2;5cm4;5cmEFGH chapExoCorrec/1545 sacados/1545 ABCDO55o3cm4;5cm chapExoCorrec/10171 sacados/10171 EGFHP5cm144o chapExoCorrec/1459 sacados/1459 chapExoCorrec/4177 sacados/4177
6cm5cm4cm3cmABCFG5cmDEH ABCDEFGHIJ2cm4cm6cm30o40o ABCD 76oABCDO Reproduce, full size, this figure. E.6628 Consider the figure below where the quadrilateral ABEH is a parallelogram. 1 a What is the nature of the quadrilateral CDEF ? Jus-tify by citing the property used. b What is the nature of the quadrilateral FGIJ ? Justify by citing the property used. c What is the nature of the quadrilateral EFGH ? Jus-tify by citing the property used. 2 Reproduce the figure at full size. 13. Parallelogram and corresponding angles E.2069 Consider a quadri-lateral ABCD such that : DAC = ACB CDB = DBA 1 a What can be said about the pair of angles DAC and ACB ? b Deduce that : ( AD ) == ( BC ) . 2 Establish that : ( DC ) == ( AB ) . 3 Demonstrate that the quadrilateral ABCD is a parallel-ogram. E.5623 Consider the parallelogram ABCD shown below où : AD = AO ; ∠ DOA =76 o Answers to the following questions must be justified : 1 a Determine the measure of the angle ∠ ODA . b Determine the measure of the angle ∠ OAD . 2 Deduce the measure of the angle ∠ OCB . Justify your approach. https://chingmath.fr 6cm5cm4cm3cmABCFG5cmDEH chapExoCorrec/6628 sacados/6628 ABCDEFGHIJ2cm4cm6cm30o40o chapExoCorrec/2069 sacados/2069 ABCD chapExoCorrec/5623 sacados/5623 76oABCDO
ABCDO35o70o ABCDxxyyM JesaisJ’utiliseJ’endéduisABCDest un quadrilatèreLasommedesanglesdetoutquadrilatèrevaut360o.xy:::::: JesaisJ’utiliseJ’endéduisLes anglesMDAetADCsont adjacents et sup-plémentairesMDA180−x:::::: JesaisJ’utiliseJ’endéduisRelativement aux droites(ADet(BC,les an-glesMDAetDCBsontcorrespondantsetdemême mesure(AD==(BC JesaisJ’utiliseJ’endéduisSiuncoupled’anglesalternes-internesamêmemesure alors les droites dé∏nissant ces angles sontparallèles.(AB==(DC JesaisJ’utiliseJ’endéduis(AB==(DCet(AD==(BCABCDest un parallélogramme. MNOPTIJKLSEFGHRABCDQ E.2103 Consider the figure below where ABCD is a parallelogram. 1 Give the measure of the angles below, briefly justifying your approach : a AOD b DAO c DOC d DBC 2 Give the measure of the angle CBO . Justify your ap-proach. E.2072 Consider an uncrossed quadrilateral ABCD with the following properties : ADC = ABC = x DAB = DCB = y We note I the midpoint of segment [ AC ] . Using the deductive chain below, we will show that the image of the point D by symmetry of center I is the point B ; thus, ABCD admits a center of symmetry: ABCD is a parallelo-gram. For this, we admit the following property: The sum of the angles in any quadrilateral is 360 o . 14. Particular quadrilaterals: properties of diagonals E.10441 Consider the four quadrilaterals below : 1 Give the name and nature of each of these quadrilaterals. 2 For each of these quadrilaterals, name the geometric properties related to their diagonals. E.10443 Answer the following questions with true or false : a The diagonals of a quadrilateral intersect at their mid-points. b The diagonals of a rectangle are perpendicular. c The diagonals of the square are perpendicular. d The diagonals of the rhombus have the same length. https://chingmath.fr chapExoCorrec/2103 sacados/2103 ABCDO35o70o chapExoCorrec/2072 sacados/2072 ABCDxxyyM JesaisJ’utiliseJ’endéduisABCDest un quadrilatèreLasommedesanglesdetoutquadrilatèrevaut360o.xy:::::: JesaisJ’utiliseJ’endéduisLes anglesMDAetADCsont adjacents et sup-plémentairesMDA180−x:::::: JesaisJ’utiliseJ’endéduisRelativement aux droites(ADet(BC,les an-glesMDAetDCBsontcorrespondantsetdemême mesure(AD==(BC JesaisJ’utiliseJ’endéduisSiuncoupled’anglesalternes-internesamêmemesure alors les droites dé∏nissant ces angles sontparallèles.(AB==(DC JesaisJ’utiliseJ’endéduis(AB==(DCet(AD==(BCABCDest un parallélogramme. chapExoCorrec/10441 sacados/10441 MNOPTIJKLSEFGHRABCDQ chapExoCorrec/10443 sacados/10443
ABCDI1ABCD2CDAB(AB==(CD(AD==(BC3BADC4 TrapèzeParallé-logrammeLosangeRectangleCarréLescôtésopposéssontparallèlesLes côtés opposés sont demême longueurLescôtésconsécutifs sontperpendiculairesLescôtésconsécutifs sontde même longueur E.11009 For each of the questions below and without justification, if the statement below is true or false : 1 If a quadrilateral is a rhombus then its diagonals are of the same length and parallel. 2 If a quadrilateral is a rectangle then its diagonals are perpendicular. 3 If a quadrilateral is a rectangle then its opposite sides are of the same length. 4 If a quadrilateral is a rectangle then the diagonals are the same length. 5 If a quadrilateral is a rhombus then its diagonals are of the same length. 6 If a quadrilateral is a rhombus then its diagonals intersect in the middle. 7 If a quadrilateral is a rhombus then its diagonals are per-pendicular. 8 If a quadrilateral is a parallelogram then its diagonals are of the same length. E.11010 For each of the questions below and without justification, if the statement below is true or false : 1 If a quadrilateral is a parallelogram then its opposite sides are always parallel. 2 If a quadrilateral is a rectangle then its diagonals are perpendicular. 3 If a quadrilateral is a rhombus then its opposite sides have the same length. 4 If a quadrilateral is a square then its diagonals are the same length and intersect in the middle. 5 If a quadrilateral is a parallelogram then its consecutive sides are of the same length. 6 If a quadritateral is a rectangle then its opposite sides are parallel. E.10442 Among the rhombus, rectangle, and square : a Which quadrilaterals have perpendicular diagonals? b Which quadrilaterals have parallel opposite sides? c Which quadrilaterals have diagonals of equal length? d Which quadrilaterals have diagonals that intersect at their midpoints? E.2890 1 Draw any quadrilateral that has its two diagonals of equal length. 2 Draw any quadrilateral that has its diagonals perpendic-ular. 15. Parallelogram E.10448 Proposition: (characterizing properties) If a quadrilateral has its opposite sides parallel then that quadrilateral is a parallelogram. If a quadrilateral has its opposite sides of the same length then this quadrilateral is a parallelogram. If a quadrilateral has its opposite angles of the same mea-sure then this quadrilateral is a parallelogram. If a quadrilateral has its diagonals intersecting at their midpoints then this quadrilateral is a parallelogram. If a quadrilateral has two of its opposite sides parallel and of the same length then this quadrilateral is a parallelogram. In each case, justify, citing the property used, that the quadri-lateral ABCD is a parallelogram. E.10469 For each of the proposed quadrilat-erals, check the properties they possess : https://chingmath.fr chapExoCorrec/11009 sacados/11009 chapExoCorrec/11010 sacados/11010 chapExoCorrec/10442 sacados/10442 chapExoCorrec/2890 sacados/2890 chapExoCorrec/10448 sacados/10448 ABCDI1ABCD2CDAB(AB==(CD(AD==(BC3BADC4 chapExoCorrec/10469 sacados/10469 TrapèzeParallé-logrammeLosangeRectangleCarréLescôtésopposéssontparallèlesLes côtés opposés sont demême longueurLescôtésconsécutifs sontperpendiculairesLescôtésconsécutifs sontde même longueur
TrapèzeParallé-logrammeLosangeRectangleCarréLes diagonales se coupenten leurs milieuxLesdiagonalesontlesmêmes longueursLesdiagonalessontper-pendiculaires ABCDEFH4cm2cm ABCD55o5cm3cmEFJI105o4cm7cm E.10470 For each of the proposed quadrilat-erals, check the properties they possess : 16. Parallelogram tracing with the compass E.10446 Consider the quadrilateral ABCD shown below : 1 What properties of the quadrilateral ABCD allow us to say that it is a parallelogram? 2 Using a compass, check that the measurements are equal: AB = EF ; EH = AD 3 Using a compass and an unmarked ruler, place point G so that quadrilateral EFGH is a parallelogram. E.10447 Reproduce the parallelo-grams below, respecting the indications on the figures : 17. Use of the properties of particular quadrilaterals E.1549 1 Consider two points O and O of the plane such that the circle C of center O of radius 4 cm and the circle C of center O of diameter 7 cm intersect at two points E and F . a Draw such a configuration. b What is the nature of the triangle OEF ? 2 a Make a configuration identical to the previous ques-tion, but such that the circles C and C have 5 cm for radius. b What is the nature of the quadrilateral OEO F ? https://chingmath.fr chapExoCorrec/10470 sacados/10470 TrapèzeParallé-logrammeLosangeRectangleCarréLes diagonales se coupenten leurs milieuxLesdiagonalesontlesmêmes longueursLesdiagonalessontper-pendiculaires chapExoCorrec/10446 sacados/10446 ABCDEFH4cm2cm chapExoCorrec/10447 sacados/10447 Avec utilisation des angles ABCD55o5cm3cmEFJI105o4cm7cm chapExoCorrec/1549 sacados/1549
ABCDEFIJ COABCDE3cm EFGH5cmIK7cmJL4cm E.2635 Consider the figure below : where : The quadrilateral ABCD is a rhombus of center I such that : AC =6 cm ; BD =3 cm Let’s note J the middle of segment [ AB ] . The points E and F are such that the quadrilateral AEBF is a square. 1 a What are the segments [ AC ] and [ BD ] called for the rhombus ABCD ? b What can we say about the straight lines ( AC ) and ( BD ) ? c Note I the point of intersection of the straight lines ( BD ) and ( AC ) . Give the measure of the segment [ IC ] ? 2 a What are the segments [ AB ] and [ EF ] called for the square AFBE ? b What does the point J represent for the square AFBE ? c What does the line ( FE ) represent for the segment [ AB ] ? 3 The aim of this question is to reproduce the whole of this figure : a Draw two straight lines ( d ) and ( d ) perpendicular ; name I the point of intersection of these two straight lines. b Place the points A , B , C , D to make the rhombus ABCD with the required dimensions. c Using the compass, draw the perpendicular bisector of segment [ AB ] ; name J the midpoint of segment [ AB ] . d Place the points E and F on this bisector in order to draw the square AEDF with the required dimensions. E.2637 Let C be a circle of center O and radius 3 cm ; the points A , B , C , D , E , belonging to the circle C verify: segments [ BD ] and [ AE ] are diameters of C ; the quadrilateral OABC is a rhombus ; the quadrilateral ABED is a rectangle. 1 a What property makes it possible to state that the segments [ OB ] and [ OC ] are of the same length? b Which property makes it possible to assert that seg-ments [ AB ] and [ BC ] are of the same length? c Compare the lengths of segment [ OB ] and segment [ BD ] . Name the property used. d What is the segment [ BD ] called for the circle C ? What is the name of the segment [ BD ] for the rect-angle ADEB ? 2 The aim of this question is to reproduce, in true sizes, this figure : a Draw the circle with center O and radius O ; b Place a point A on the circle C , then draw the rhombus OABC ; c Draw the diagonals of the rectangle ABED , then trace this quadrilateral. 18. Tracing quadrilaterals E.961 Consider the four quadrilaterals shown below : a Draw the square EFGH verifying FH =5 cm b Draw rhombus IJKL verifying: LJ = 7 cm ; IK = 4 cm Note: the construction lines used to produce these figures must not be erased. https://chingmath.fr chapExoCorrec/2635 sacados/2635 ABCDEFIJ chapExoCorrec/2637 sacados/2637 COABCDE3cm chapExoCorrec/961 sacados/961 EFGH5cmIK7cmJL4cm
6cm5cmABCD4cm7;5cmNPMO 9cm3cm7cmABCDEFGH8cm9cmIJKL9cm4cm 8;5cm7;5cm9cm7cm3cm10cm5;5cmABCDEFGHIJKL E.10009 Consider the four quadrilaterals shown below : a Draw the rectangle ABCD verifying: AC = 6 cm ; CD = 5 cm b Draw the rhombus MNOP verifying: MO = 7.5 cm ; MN = 4 cm The construction lines used to produce these figures must not be erased. E.10957 Consider the four quadrilaterals shown below : Carry out the requested tracings, taking into account the sug-gested indications : 1 Trace the parallelogram ABCD . (the parallelogram is made up of two triangles) 2 Draw the rectangle EFGH . (the rectangle has four right angles) 3 Draw the rhombus IJKL . (the diagonals of the rhombus intersect in their middles and are perpendicular) E.11006 Below are shown : the rhombus ABCD , the rectangle EFGH , the parallelogram IJKL : Draw these three quadrilaterals full-scale. 19. Tracing quadrilaterals E.1556 Consider the rectangle DJEU . 1 What do the segments [ DJ ] and [ JU ] represent for the rectangle DJEU ? 2 Draw a rectangle DJEU such that : DJ =3 cm and JU =8 cm E.10010 Consider the rhombus IJKL veri-fying : KI = 2 cm ; JL = 8 cm 1 In the IJKL quadrilateral, what do the segments [ KI ] and [ JL ] represent? 2 What can we say about the straight lines ( KI ) and ( JL ) in the quadrilateral IJKL ? 3 Draw the rhombus IJKL . E.10012 Consider the rhombus MNOP ver-ifying: MO = 8 cm ; MN = 4.5 cm 1 What do the segments [ MO ] and [ MN ] represent for the diamond MNOP ? 2 We note I the middle of segment [ MO ] . What is the nature of the triangle MIN ? 3 Draw the diamond MNOP . E.1554 For each question, construct the rectangle ABCD by respecting the indications data : a AB =5 cm ; AD =6 cm b AB = 4 cm ; BD = 8 cm E.10011 Draw the rectangle EFGH such that : EF =5 cm ; FH =6 cm E.10014 Draw the rectangle ABCD such that : AB =4 cm ; AC =7 cm E.2655 Draw the rhombus EFGH such that : EG =7 cm ; EF =4 cm E.2883 Draw a rectangle ABCD is a rect-angle such that : AC = 5 cm 20. Quadrilateral layout and layout program https://chingmath.fr chapExoCorrec/10009 sacados/10009 6cm5cmABCD4cm7;5cmNPMO chapExoCorrec/10957 sacados/10957 9cm3cm7cmABCDEFGH8cm9cmIJKL9cm4cm chapExoCorrec/11006 sacados/11006 8;5cm7;5cm9cm7cm3cm10cm5;5cmABCDEFGHIJKL chapExoCorrec/1556 sacados/1556 chapExoCorrec/10010 sacados/10010 chapExoCorrec/10012 sacados/10012 chapExoCorrec/1554 sacados/1554 chapExoCorrec/10011 sacados/10011 chapExoCorrec/10014 sacados/10014 chapExoCorrec/2655 sacados/2655 chapExoCorrec/2883 sacados/2883
ab4cm6cm4cm5cm 6cm7cmABCDE5cmF ABCD7cm8cm3cmEFGH 5cm6cm4cm6;5cm8cmABCDEFG E.1551 Draw the rhombus DOUX verify-ing : UX = 6 cm ; OX = 3 cm then, complete the program below : 1 Draw the segment [ OX ] verifying . . . 2 Draw the line ( d ) . . . of segment [ OX ] . 3 Draw the circle C such that . . . 4 Name D and U the points of intersection . . . 5 Draw the rhombus DOUX . E.10013 Draw the rectangle AMER verify-ing : AM = 3.5 cm ; AE = 6 cm and then complete the rectangle plotting program AMER : 1 Draw the segment [ AM ] such that . . . 2 Draw the line ( d ) . . . through the point M . 3 Draw the circle C with center A and radius . . . 4 Name E the point of intersection . . . 5 Draw the line (Δ) perpendicular to the line . . . passing through the point E . 6 Draw the line (Δ ) perpendicular to the line . . . passing through the point A . 7 Name R the point of intersection . . . 8 draw the rectangle AMER . 21. Reproduce a figure E.2889 Carry out the following plot pro-gram: 1 Draw the triangle ABC verifying the following measures : AB = 7 cm ; AC = 4 cm ; BC = 8.5 cm 2 Draw, on the previous figure, the rectangle CAFG such that AG =6 cm . 3 Complete the drawing by tracing the square ADBE . E.1552 The following figures have been drawn freehand. Reproduce them following the instructions on the figures and using your drawing instruments (ruler, square, compass) The rounded part of the figure a is a half circle E.6332 Reproduce the figure below in full size : E.6346 We consider the following configura-tion : 1 Give the nature of each of the quadrilaterals ABCD , EFBC , HFEG . 2 Use a ruler and a compass to reproduce this configura-tion. E.11007 Consider the figure below, which is composed of the rhombus BCDE and the rectangle ACFG : Reproduce this figure in full size. 22. Carry out a construction program https://chingmath.fr chapExoCorrec/1551 sacados/1551 chapExoCorrec/10013 sacados/10013 chapExoCorrec/2889 sacados/2889 chapExoCorrec/1552 sacados/1552 ab4cm6cm4cm5cm chapExoCorrec/6332 sacados/6332 6cm7cmABCDE5cmF chapExoCorrec/6346 sacados/6346 ABCD7cm8cm3cmEFGH chapExoCorrec/11007 sacados/11007 5cm6cm4cm6;5cm8cmABCDEFG
ABCDEFG 2508cmABCD9cm300EFGH E.2656 Carry out the plot program below : 1 Draw the rhombus ABCD having the following mea-sures : AC = 8 cm ; BD = 5 cm 2 a Name O the point of intersection of the diagonals. b Place the point E such that OCED is a rectangle. 3 Place the points F and G so that AFBG is a square. 23. Construction program E.2888 Consider the figure below : 1 Give the nature of the triangle ABC and the quadrilat-eral CBED . Justify your answers. 2 a Justify that the two segments [ FC ] and [ CD ] are of equal length. b Specify the nature of the triangle FCD . 3 Justify that the triangle CEG is isosceles at C . 24. Tracing quadrilaterals and angles E.2970 Consider, in the figure below, the two quadrilaterals ABCD and EFGH : 1 a What is the nature of the quadrilateral ABCD ? Jus-tify. b Reproduce, full size, the quadrilateral ABCD . 2 a What is the nature of the quadrilateral EFGH ? Jus-tify. b What does the line ( FH ) represent for the segment [ EG ] ? Justify your answer. c Reproduce, full size, the quadrilateral EFGH . 25. Unclassified exercises E.1750 Copy and complete each of the tables below representing a deductive link: 1 Let ( d ) and ( d ) be two straight lines : On sait On utilise On déduit ( d ) == ( d ) ( d ) ⊥ ( d  ) . . . . . . ( d ) ⊥ ( d  ) 2 Let ABCD be a quadrilateral and O the intersection of its diagonals. https://chingmath.fr chapExoCorrec/2656 sacados/2656 chapExoCorrec/2888 sacados/2888 ABCDEFG chapExoCorrec/2970 sacados/2970 2508cmABCD9cm300EFGH chapExoCorrec/1750 sacados/1750
ABCDEF JesaisABEFest un parallélogrammeJ’uti-liseSi un quadrilatère est un...........................Alors ses côtés opposés sont........................J’endé-duis(FA==(EB JesaisEBCDest un parallélogramme.J’uti-lise..........................................................................................................J’endé-duis(EB==(DC Jesais(AF==(BEet(BE==(CDJ’uti-lise..........................................................................................................J’endé-duis(FA==(CD On sait On utilise On déduit . . . . . . If a quadrilateral has its di-agonals intersecting at their midpoints then this quadri-lateral is a parallelogram ABCD is a parallélo-gramme E.1752 The deductive link below contains an error. Find the error: On sait On utilise On en déduit ABCD is a quadri-lateral having its diagonals intersect-ing in their middles If a quadrilateral is a parallelogram then its diagonals intersect in their middles ABCD is a paral-lelogram E.1749 1 Cite the conclusions of each of the following propositions : a If a quadrilateral is a parallelogram then its diagonals intersect at their middles. b If an integer is a multiple of 4 then that integer is even. 2 Name the conditions of use for each of the following propositions : a If a quadrilateral has its diagonals perpendicular then it’s a square. b If the sum of the digits of an integer is a multiple of 3 then that integer is divisible by 3. 3 Of the mathematical statements above, name the ones that are true. E.394 1 For each of the following statements, explicitly state the conclusions : a If a quadrilateral is a rectangle, then its diagonals are of equal length. b If an integer is a multiple of 9 , then the sum of its digits is a multiple of 9 . 2 For each of the following statements, explicitly state the conditions : a If a quadrilateral has all sides of equal length, then it is a rhombus. b If one integer is even and another is odd, then their sum is odd. E.1460 Complete the following deductive links: 1 . . . . . . . . . . . . . . . . . . . . . . . . . If a parallelogram has two consecutive sides of the same length Then it’s a rhombus. ABCD is a rhombus. 2 ( AB ) ⊥ ( CD ) ; [ AB ] and [ CD ] have O as their midpoint. If a quadrilateral has its diagonals perpendicular and in-tersecting in their middles Then it’s a rhombus. . . . . . . . . . . . . . . . . . . . . . . . . . 3 EF = GH ; [ EF ] and [ GH ] have the same midpoint. . . . . . . . . . . . . . . . . . . . . . . . . . The quadrilateral EGFH is a rectangle E.2102 1 a Draw a rectangle ABCD . b Draw the parallel to the line ( BD ) passing through the point C . This line intercepts the line ( AD ) at F . 2 What is the nature of the quadrilateral BDFC ? Justify your statement. 3 Justify each of the following statements : a ˇ D is the middle of the segment [ AF ] ı b ˇ ( DC ) is the perpendicular bisector of the segment [ AF ] ı c ˇ The triangle ACF is isosceles at A ı E.10444 Consider the two parallelograms ABEF and BCDE shown below : Complete the deductive chains below : Theorem: If two lines are parallel to the same third line, then they are parallel to each other. If two lines are perpendicular to the same third line, then they are parallel to each other. If two lines are parallel to each other and a third line is perpendicular to one of them, then it is perpendicular to the other. https://chingmath.fr chapExoCorrec/1752 sacados/1752 chapExoCorrec/1749 sacados/1749 chapExoCorrec/394 sacados/394 chapExoCorrec/1460 sacados/1460 chapExoCorrec/2102 sacados/2102 chapExoCorrec/10444 sacados/10444 ABCDEF JesaisABEFest un parallélogrammeJ’uti-liseSi un quadrilatère est un...........................Alors ses côtés opposés sont........................J’endé-duis(FA==(EB JesaisEBCDest un parallélogramme.J’uti-lise..........................................................................................................J’endé-duis(EB==(DC Jesais(AF==(BEet(BE==(CDJ’uti-lise..........................................................................................................J’endé-duis(FA==(CD
ABD JesaisJ’uti-liseJ’endéduis(AD==(BC JesaisJ’uti-liseJ’endéduis(AD==(BE JesaisJ’uti-liseJ’endéduis(AD==(BCet(AD==(BE(BC==(BE JesaisJ’uti-liseJ’endéduisSi deux droites sont parallèles et si elles possèdentun point en commun alors elles sont parallèles.Les droites(BCet(BEsont confondues. IABCDEFG JesaisABDC;ADBC;DAB90oJ’uti-liseSi un quadrilatère a..........................................................................................................................................alors ce quadrilatère est un rectangleJ’endé-duisABCDest un rectangle JesaisICIEIFIGJ’uti-liseSi un quadrilatère a ses diagonales.......................................................................alors ce quadrilatère est un rectangleJ’endé-duisCEFGest un rectangle JesaisABDC;ADBC;(AB(ADJ’uti-liseSi un quadrilatère a ses côtés............................................................................alors ce quadrilatère est un rectangleJ’endé-duisABCDest un retangle JesaisICIEIFIGJ’uti-liseSi un quadrilatère a ses diagonales.......................................................................alors ce quadrilatère est un rectangleJ’endé-duisCEFG JesaisJ’uti-liseJ’endé-duis(CE(EF E.10944 Consider the three points A , B , C shown below : 1 Using the compass : a Place the point C such that ABCD is a parallelogram. b Place the point E such that AEBD is a parallelogram. 2 Complete the deductive link below to state that the points B , C , E are aligned. E.10945 Consider the figure below, where information has been encoded, consisting of a right triangle BCE rectangle at C and two quadrilaterals: 1 Complete the two deductive links to determine the na-ture of these two quadrilaterals: For quadrilateral ABCD : For quadrilateral CEFG : 2 a Complete the following deductive links to justify that lines ( AB ) and ( CE ) are parallel: b Complete the following deductive steps to justify that lines ( AB ) and ( EF ) are perpendicular : https://chingmath.fr chapExoCorrec/10944 sacados/10944 ABD JesaisJ’uti-liseJ’endéduis(AD==(BC JesaisJ’uti-liseJ’endéduis(AD==(BE JesaisJ’uti-liseJ’endéduis(AD==(BCet(AD==(BE(BC==(BE JesaisJ’uti-liseJ’endéduisSi deux droites sont parallèles et si elles possèdentun point en commun alors elles sont parallèles.Les droites(BCet(BEsont confondues. chapExoCorrec/10945 sacados/10945 IABCDEFG JesaisABDC;ADBC;DAB90oJ’uti-liseSi un quadrilatère a..........................................................................................................................................alors ce quadrilatère est un rectangleJ’endé-duisABCDest un rectangle JesaisICIEIFIGJ’uti-liseSi un quadrilatère a ses diagonales.......................................................................alors ce quadrilatère est un rectangleJ’endé-duisCEFGest un rectangle JesaisABDC;ADBC;(AB(ADJ’uti-liseSi un quadrilatère a ses côtés............................................................................alors ce quadrilatère est un rectangleJ’endé-duisABCDest un retangle JesaisICIEIFIGJ’uti-liseSi un quadrilatère a ses diagonales.......................................................................alors ce quadrilatère est un rectangleJ’endé-duisCEFG JesaisJ’uti-liseJ’endé-duis(CE(EF
Jesais(AB==(CEet(CE(EFJ’uti-liseJ’endé-duis(AB(EF ABCDMNIJ JesaisABCDest un rectangle etAC10cmJ’uti-liseSi un quadrilatère est un rectangle alors ses diagonalesaet ses diagonalesbJ’endé-duisDIDB2cabc JesaisDMINest un rectangle etDI5cmJ’uti-liseSi un quadrilatère est un rectangle alorsdJ’endé-duisMJede xyBD ABCDOE JesaisABCDest un rectangleJ’uti-liseJ’endé-duis(DC(DAa JesaisAODEest un losangeJ’uti-liseJ’endé-duis(DA(EOb Jesais(DC(DAetJ’uti-liseJ’endé-duis(DC==(EOc Hint: you can use the following properties : If two lines are parallel to each other and a third line is perpendicular to one of them, then it is perpendicular to the other. If two lines are perpendicular to the same third line, then they are parallel to each other. If a quadrilateral is a rectangle, then its consecutive sides are perpendicular. If a quadrilateral has diagonals of equal length that in-tersect at their midpoints, then the quadrilateral is a rectangle. If a quadrilateral has opposite sides of equal length and has a right angle, then the quadrilateral is a rectangle. E.11008 1 Make the plot program on your copy: a Draw a rectangle ABCD such as : AB = 9 cm ; AC = 10 cm b Name I the point of intersection of the intersections of the diagonals of the rectangle ABCD . c Construct the rectangle DMIN where : M ∈ [ AD ] ; N ∈ [ CD ] You should get a figure similar to : 2 On the properties of the figure below, we have con-structed the deductive links below : Copy, on your copy, the parts a , b , c , d , e missing in this reasoning. E.10766 All construction lines must be made ex-clusively with the compass and non graduated rule and must be preserved. Let [ Dy ) and [ Bx ) be two half-lines intercepting at A . Complete the figure below to obtain the parallelogram ABCD . E.11024 Consider the figure below, where quadrilateral ABCD is a rectangle and quadrilateral AODE is a rhombus. Complete the deductive chain : Fill in the missing information on your copy ( a , b , c , d ) . https://chingmath.fr Jesais(AB==(CEet(CE(EFJ’uti-liseJ’endé-duis(AB(EF chapExoCorrec/11008 sacados/11008 ABCDMNIJ JesaisABCDest un rectangle etAC10cmJ’uti-liseSi un quadrilatère est un rectangle alors ses diagonalesaet ses diagonalesbJ’endé-duisDIDB2cabc JesaisDMINest un rectangle etDI5cmJ’uti-liseSi un quadrilatère est un rectangle alorsdJ’endé-duisMJede chapExoCorrec/10766 sacados/10766 xyBD chapExoCorrec/11024 sacados/11024 ABCDOE JesaisABCDest un rectangleJ’uti-liseJ’endé-duis(DC(DAa JesaisAODEest un losangeJ’uti-liseJ’endé-duis(DA(EOb Jesais(DC(DAetJ’uti-liseJ’endé-duis(DC==(EOc
ABCDEFIJ6cm JesaisABCDest un losange etBC6cmJ’uti-liseJ’endé-duisAB6cma JesaisAEBFest un carré etAB6cmJ’uti-liseJ’endé-duisFE6cmb JesaisFE6cmJ’uti-liseJ’endé-duisFJ3cmc ABCDEF27o62o ABCD(dIKLM(d(AI== ABCD2;7cmI Instructions: Some of these properties will be used : If a quadrilateral is a rectangle, then its consecutive sides are perpendicular. If a quadrilateral is a rectangle, then its opposite sides are parallel and of equal length. If a quadrilateral is a rectangle, then its diagonals are of equal length and intersect at their midpoints. If a quadrilateral is a rhombus, then its consecutive sides are of equal length. If a quadrilateral is a rhombus, then its opposite sides are parallel and of equal length. If a quadrilateral is a rhombus, then its diagonals are perpendicular and intersect at their midpoints. If two lines are parallel to a third line, then they are parallel to each other. If two lines are perpendicular to the same third line, then they are parallel to each other. If two lines are parallel to each other and a third line is perpendicular to one of them, then it is perpendicular to the other. E.11025 Consider the figure : where ABCD is a rhombus and AEBF is a square. The point J is the middle of [ AB ] . Complete the following deductive links: Hint: the following properties can be used : If a quadrilateral is a square then its opposite sides are parallel and of the same length. If a quadrilateral is a square then its consecutive sides are perpendicular and of the same length. If a quadrilateral is a square then its diagonals are per-pendicular, of the same length and intersect at their midpoints. E.1658 Consider the figure below formed by a square ABCD and two triangles CDF and BCE such that : ∠ DCF = 27 o ; ∠ BCE = 62 o . Justify that the points F , C , E are not aligned. E.6237 We consider the following configura-tion : Write the plot program for this configuration using once the word ˇ median ı and starting the following two points : Draw a rectangle ABCD . Place the point I in the middle of [ CD ] . E.2073 Consider the parallelogram below. An-swer the following questions with true or false : 1 The segment [ BC ] has a length of 2.7 cm . 2 [ BD ) is the bisector of the angle ∠ ABC . 3 I is the midpoint of segment [ AC ] . 4 The diagonals of ABCD are perpendicular. 5 diagonals have the same length. https://chingmath.fr chapExoCorrec/11025 sacados/11025 ABCDEFIJ6cm JesaisABCDest un losange etBC6cmJ’uti-liseJ’endé-duisAB6cma JesaisAEBFest un carré etAB6cmJ’uti-liseJ’endé-duisFE6cmb JesaisFE6cmJ’uti-liseJ’endé-duisFJ3cmc chapExoCorrec/1658 sacados/1658 ABCDEF27o62o chapExoCorrec/6237 sacados/6237 ABCD(dIKLM(d(AI== chapExoCorrec/2073 sacados/2073 ABCD2;7cmI
AB(d ABCD5cmM36o18o ABCD -6-4-201246810-20124xy E.5576 Consider the figure below où the point A belongs to the line ( d ) : 1 Draw the quadrilateral ABCD such that ABCD is a rhombus accepting the line ( d ) as its axis of symmetry. 2 What does the straight line ( d ) represent for the angle ∠ BAC . E.2981 Consider a rectangle ABCD such that ∠ DCA =36 o ; M is a point on the segment [ BC ] such that ∠ BAM =18 o 1 What does the half line [ AM ) represent for the angle ∠ CAB ? Justify. 2 a Justify that the angles ∠ DCA and ∠ ACB are adja-cent angles. b Give, by presenting your calculation, the measure of the angle ∠ ACB . 3 Reproduce this figure in full size. E.2314 The box below shows the parallelo-gram ABCD : Perform the following tracing program using the ungraduated ruler and compass : 1 Draw the segment [ AC ] . 2 Draw the perpendicular bisector of segment [ AC ] . 3 Name I the midpoint of segment [ AC ] and J the point of intersection of the perpendicular bisector of [ AC ] with segment [ AB ] . 4 Draw the perpendicular bisector of segment [ AJ ] 5 Name K the midpoint of segment [ AJ ] . E.11187 1 In the marker below, place the following points : A ( − 7 ; − 2) ; B ( − 5 ; 3) ; C ( − 2 ; 0) D (0 ; 2) ; E (6 ; 4) ; F (10 ; 2) ; G (4 ; 0) 2 a Connect the points A , B , C and color the triangle ABC blue. What is its nature? b Connect the points D , E , F , G and color the quadri-lateral DEFG in red. What is its nature? https://chingmath.fr chapExoCorrec/5576 sacados/5576 AB(d chapExoCorrec/2981 sacados/2981 ABCD5cmM36o18o chapExoCorrec/2314 sacados/2314 ABCD chapExoCorrec/11187 sacados/11187 -6-4-201246810-20124xy
ABCDEFGH E.6478 In the plane, consider the 8 points below : 1 a Draw the segment [ BE ] and the half-line [ AF ) . b Name P the point of intersection of segment [ BE ] and half-line [ AF ) . c Draw the half-lines [ AC ) and [ BD ) . d Name M the point of intersection of the half-lines [ AC ) and [ BD ) . e Draw the quadrangle APBM . f What is the nature of the quadrilateral APBM ? 2 a Draw the straight lines ( GM ) and ( AH ) . b Name N the point of intersection of the straight lines ( AH ) and ( GM ) . c Draw the triangle AMN . d What is the nature of the triangle AMN ? E.2604 1 Perform the following plot program : a Draw an isosceles triangle at B such that : AB = 5 cm ; ∠ ABC = 50 o b Draw the bisector of the segment [ AC ] with a com-pass and a straight edge. Note I the midpoint of the segment [ AC ] . c Draw the circle with center I and radius [ IB ] . It in-tersects the line ( IB ) a second time at D . d Draw the quadrilateral ABCD . 2 What is the nature of the quadrilateral ABCD ? Justify your answer. https://chingmath.fr chapExoCorrec/6478 sacados/6478 ABCDEFGH chapExoCorrec/2604 sacados/2604