Grade 7 / Angles 49 exercises corrected

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ChingQuizz : 5 exercises available for Quizz assessment : 1. Measurement and reproduction of an angle E.5620 Give the names of the coded angles in the figure below and their measurements : E.10802 Definition: we categorize angles into 5 groups : Consider the plane provided with the two straight lines and points shown below : Give the nature of each of the angles below : a ABC b DEG c DEF d FEG e JHI f HJI g JIH E.10804 The figure opposite was created freehand. Using geometry instruments, re-produce this figure. 2. Introduction to alternate/internal and corresponding angles E.1397 1 Reproduce the figure above. 2 On the figure and using the protractor, measure and in- dicate the measure of each of the angles formed by the straight lines ( d ) and ( d ) with the intersection of (Δ) . E.1392 Consider the configuration below consisting of the three straight lines ( d ) , ( d ) and (Δ) where the two straight lines ( d ) and ( d ) are parallel. Name three angles of equal measure to the angle CAD . https://chingmath.fr quizzChap/1019 chapExoCorrec/5620 sacados/5620 ABCDO chapExoCorrec/10802 sacados/10802 AnglenulAngleaiguAngledroitAngleobtusAngleplat ABCDEFGIJH chapExoCorrec/10804 sacados/10804 7cmABC70o45o chapExoCorrec/1397 sacados/1397 ABCDEFGH(d(d(d==(d chapExoCorrec/1392 sacados/1392 ABCDEFGH(d(d(d==(d
3. Definition and alternate/internal and corresponding angles E.1406 Definition: In a configuration of two lines and a third line intersecting the first two: two angles are said to be alternate interior angles if they satisfy the following three properties : their vertices are the points of intersection of the secant line with the two lines, they are located on either side of the secant line, they are located between the two lines. Two angles are said to be corresponding if they sat-isfy the following three properties : Their vertices are the points of intersection of the secant with the two lines. they are located on the same side of the secant, one is located between the two lines, the other out-side. Below, we consider the configuration consisting of three lines : Complete, if possible , the following table : A B Angles A and B are: a 1 a 3 a 4 b 3 a 2 b 3 a 1 b 1 a 3 alternate interior angles b 4 opposite the vertex E.1388 In the figure below, AECD is a trapezoid. We call: the point B the intersection of line ( AE ) with the line passing through point C and parallel to line ( AD ) . the point F intersection of the line ( AE ) with the line passing through the point D and parallel to the line ( CE ) . point G intersection of lines ( DF ) and ( CB ) . 1 Reproduce the above figure freehand. 2 Lines ( AE ) and ( DC ) define pairs of corresponding an-gles of equal measure. Color one of these pairs and label them. 3 Lines ( AD ) and ( CG ) define pairs of alternate interior angles of equal measure. Color one of these pairs and name them. 4 In the figure, color a pair of angles opposite at the vertex in red and name them. E.10863 Consider the tiling of the plane below : 1 Give two alternate interior angles with respective vertices I and S . 2 Give two corresponding angles with respective vertices G and N . 4. Measurement and alternate/internal and corresponding angles E.1396 Proposition: In a configuration of two lines and a third intersecting line. Si The two lines are parallel. Then The pairs of alternate interior angles are equal in measure, and the pairs of corre-sponding angles are equal in measure. https://chingmath.fr chapExoCorrec/1406 sacados/1406 a1a2a3a4b1b2b3b4 chapExoCorrec/1388 sacados/1388 ABCDEFG chapExoCorrec/10863 sacados/10863 ABCDEFGHIJKLMNOPQRST chapExoCorrec/1396 sacados/1396
In the figure opposite, lines ( xy ) and ( uv ) are parallel: 1 List all angles equal to angle xAt . 2 Give the measure of angle vBs . Justify your answer. E.6671 Consider the figure below : 1 a Name a pair of corresponding angles. b Name a pair of alternate interior angles. 2 What property is missing from this figure for angle CAE to measure 118 o ? E.10866 Consider the four straight lines ( ux ) , ( ty ) , ( sv ) , ( wz ) shown below : In addition, we have the information : vBx = 155 o ; wDt = 25 o ; ( ux ) == ( ty ) ; ( sv ) == ( zw ) 1 Give the measure of the angle wCx . Justify your result. 2 Give the measure of the angle zCx . Justify your result. E.11478 Consider the figure below where : ( xx ) == ( yy ) ; ( zz ) == ( tt ) 1 Justify that angle DBA measures 22 o . 2 Justify that angle ABt measures 116 o . 3 Deduce the measure of angle DBC . 5. Alternate/internal and corresponding angles: reciprocal properties E.10801 Proposition: Consider two lines intersected by a secant. Si Two alternate interior angles in this configuration have the same measure. Therefore, these two lines are parallel. Consider the configuration below : Justify that lines ( xy ) and ( uv ) are parallel. E.10803 Proposition: Consider two lines intersected by a secant. Si Two corresponding angles in this configuration have the same measure. Therefore, these two lines are parallel. Consider the configuration below : Justify that lines ( xy ) and ( uv ) are parallel. https://chingmath.fr ABxyvust62o chapExoCorrec/6671 sacados/6671 ABCDEFGH(d(d62o chapExoCorrec/10866 sacados/10866 ABCDwzsvtyxu155o25o chapExoCorrec/11478 sacados/11478 ABCDxxyyzztt22o116o chapExoCorrec/10801 sacados/10801 ABxyvust55o55o chapExoCorrec/10803 sacados/10803 ABxyvust109o109o
E.11506 Consider the configuration below, drawn freehand : We have the following information : x Mt =143 o ; y Nt = 37 o 1 Justify that : t Ny = 143 o 2 What can be said about lines ( x x ) and ( y y ) ? Justify your answer. E.10864 Below has been reproduced, free-hand, a figure including the three straight lines [ AB ] , [ CD ] , [ EF ] cut by the secant [ GH ] with for information : BIG = 75 o ; JKE = 105 o ; ( AB ) == ( CD ) 1 Give the measure of the angle IJD . Justify your reason-ing. 2 a Give the measure of the angle DJK . Justify your reasoning. b Justify that the straight lines ( CD ) and ( EF ) are par-allel. Indication : each answer must be justified. E.10861 Below is a trapezoid ABCD and its diagonals : 1 Give two alternate interior angles of equal measure with respective vertices D and B . 2 Give two alternate interior angles of different measures with respective vertices A and C . 6. Sum of the angles in a triangle E.1394 Consider the two triangles ABC and DEF shown below : 1 Determine the measure of the angle ∠ ACB . Justify your approach. 2 Determine the measure of the angle ∠ DFE . Justify your approach. E.1407 Determine the measure of the angle CBD . Justify your approach. E.1387 1 Draw a figure similar to the figure opposite whose angles have the following value : AEB = 65 o ; ACB = 30 o BDE = 60 o 2 Prove that the straight lines ( AC ) and ( DE ) are perpendic-ular. E.1391 In each case, say whether it is pos-sible to construct the following triangles : a AB = 6 cm ; BC = 5 cm ; AC = 12 cm b ABC = 32 o ; BAC = 98.5 o ; ACB = 49.5 o c ZC = 3 cm ; CT = 3 cm ; TZ = 6 cm d XY Z = 36 o ; Y XZ = 47 o ; XZY = 98 o e AB = 5 cm ; CAB = 50 o ; ABC = 130 o https://chingmath.fr chapExoCorrec/11506 sacados/11506 xxyyttMN143o37o chapExoCorrec/10864 sacados/10864 ABCDEFGHIJK75o105o(AB==(CD chapExoCorrec/10861 sacados/10861 ABCD(AB==(CD chapExoCorrec/1394 sacados/1394 ABC37o46oDEF43o62o chapExoCorrec/1407 sacados/1407 72o86;5o33o122oABCD chapExoCorrec/1387 sacados/1387 ABCDE chapExoCorrec/1391 sacados/1391
E.11507 Consider the quadrilateral ABCD drawn freehand. Let O be the point of intersection of the diagonals : We have the following information : DOC = 103 o ; DAC = 76 o ; ABD = 22 o 1 Give the measure of angle AOB . 2 In triangle ABO , determine the measure of angle OAB . Justify your approach. 3 a Justify that angles DAO and OAB are adjacent. b Give the measure of angle DAB . E.1400 In the figure opposite, ABC is any triangle and the point D belongs to the segment [ AC ] 1 Calculate the measure of the angle BDA . 2 Deduce the measure of the angle BAD It is assumed for the next question that the triangle ABC is isosceles in A : 3 Deduce the measure of the angle ACB E.6672 The configuration below consists of a right triangle ABC at C such that points D , B , and A are aligned. Determine the measure of angle DCB . The wording of your answers and the presence of the steps in your reasoning will be taken into account in the evaluation. E.1404 Consider the triangle ABC shown below, with points E and F such that ( AB ) == ( EF ) . 1 Determine the measure of angle CBA . Justify your an-swer. 2 a Determine the measure of angle CEF . Justify your answer. b Deduce the measure of angle CFE . Justify your an-swer. E.1399 Consider the figure below, consisting of : the triangle ABC , which is a rectangle at C , such that : BAC =46 o the points E and F , lying on [ AB ] and [ BC ] respectively, such that : ( BC ) ⊥ ( EF ) point D , which lies on [ AC ] , satisfies the following con-dition : AED =43 o 1 What can be said about lines ( CD ) and ( FE ) ? 2 Find the measure of angle BEF . 3 What type of quadrilateral is CDEF ? E.10865 Consider the figure below composed of the four straight lines ( AB ) , ( CD ) , ( AD ) , ( BC ) below : We have the additional data : MCN = 34 o ; IAB = 27 o ; ( AB ) == ( CD ) Determine the measure of the angle CID . Hint: this question should be dealt with in several steps. Each step should be justified. 7. Sum of angles and particular triangles https://chingmath.fr chapExoCorrec/11507 sacados/11507 ABCDO103o76o22o chapExoCorrec/1400 sacados/1400 ABCD32o83o chapExoCorrec/6672 sacados/6672 34o41oABCD chapExoCorrec/1404 sacados/1404 ABCEF83o66o chapExoCorrec/1399 sacados/1399 ABCDEF46o43o chapExoCorrec/10865 sacados/10865 ABCDI27oMN34o(AB==(CD
E.1398 Determine, if possible for each ques-tion, the measure of the angle ABC E.1402 Each line represents a different isosceles triangle ABC . Using the marked information, find the (s) information (s) that is missing (s) : Main vertex A B C A 30 o B 30 o C 45 o 35 o 110 o 35 o 50 o 50 o 80 o E.11479 Each line represents a different isosceles triangle ABC . Using the marked information, find the (s) information (s) that is missing (s) : Main vertex A B C a A 35 o b B 70 o c 120 o E.1395 Consider the two triangles shown below : 1 a What is the nature of the triangle ABC ? b What is the measure of the angle ∠ ABC ? 2 a What is the nature of the triangle DEF ? b What is the measure of the angle ∠ EDF ? E.1408 Consider the figure below : 1 a Show that the triangle DCB is an equilateral trian-gle. Justify your approach. b What is the nature of the triangle ABD ? 2 What is the measure of the angle ABD ? Justify your approach? 3 Deduce the value of the angle DAB . Justify your ap-proach. E.6673 Consider the configuration below where triangle ABC is isoscele at A . Point D belongs to segment [ AB ] . Determine the measure of angle DCB . The wording of your answers and the presence of the steps in your reasoning will be taken into account in the evaluation. E.1389 ABCD is a rectangle. Points M and N are the points of intersection of circle C with segments [ BC ] and [ AD ] , respectively. 1 What is the nature of triangle AMN ? 2 What is the value of angle MAN ? 3 Calculate the value of angle ANM . 4 What is the value of angle CMN ? Justify your approach. https://chingmath.fr chapExoCorrec/1398 sacados/1398 ABCa25oABCb72oABCc36o chapExoCorrec/1402 sacados/1402 chapExoCorrec/11479 sacados/11479 chapExoCorrec/1395 sacados/1395 ABCDEF70o chapExoCorrec/1408 sacados/1408 ABCD60o chapExoCorrec/6673 sacados/6673 ABC75o120oD chapExoCorrec/1389 sacados/1389 ABCDMNC47o
E.10691 Consider the triangle ABC below and the point M belonging to the segment [ AC ] . What is the nature of the triangle BCD ? Justify your asser- tion. E.5755 Consider a square ABCD in-side which an equilateral trian-gle ABE is drawn. A repre-sentation of this configuration is shown opposite. Determine the measure of the angles represented by the sym-bol ˇ?ı and justify each of your results. 8. Deductive reasoning E.1401 The figure below consists of the straight lines ( d ) and ( d ) parallel to each other. The straight line (Δ) intercepting the straight lines ( d ) and ( d ) A and B respectively. We note I the middle of segment [ AB ] . In this exercise, we want to show that the alternating-internal angles IAx and IBy are equal. To do this, we place a point C on the half-line [ Ax ) , trace C its symmetrical with respect to I . The aim is to show that the point C is a point on the half-line [ By ) : thus, the half-right [ AC ) will have as its image [ BC ] , making it possible to affirm that the angles CAI and C BI are symmetrical and they will be of the same measurement. Complete the table below representing the deductive link in the demonstration : https://chingmath.fr chapExoCorrec/10691 sacados/10691 4cmABCD40o25o chapExoCorrec/5755 sacados/5755 ???????????EABCD chapExoCorrec/1401 sacados/1401 xy(d)(d)(d)//(d)ABICC JesaisJ’uti-liseJ’endéduisIest le milieu du segmentABBest l’image deApar la symétrie de centreI JesaisJ’uti-liseJ’endéduisSideuxdemi-droitessontparallèlesetd’orientation opposéalorsellessontsymétriquesde centre de symétrie le milieu des originesLa demi-droiteAxa pour imageBypar rap-port àI JesaisJ’uti-liseJ’endéduisCAx,l’imagedeAxparrapportàIestByet celle deCestCL’appartenance est conservée par la symétrie cen-traleC::: JesaisJ’uti-liseJ’endéduisL’angleIACapourimagel’angleIBCparlasymétrie de centreI
E.2064 The figure below shows the triangle ABD isosceles at D and a point C of the segment [ BD ] such that ABC is isosceles at A . The aim of the exercise is to determine the measure of the angle BAC . In your notebook, copy the missing sentences in the deductive links below : E.2054 Consider the configuration shown opposite. Correctly complete each of the de-ductive links below in order to answer each of the questions cor-rectly: 1 Calculate the measure of the angle BDA . 2 Deduce the measure of the angle BAD It is assumed for the next question that the triangle ABC is isosceles in A : 3 Deduce the measure of the angle ACB https://chingmath.fr chapExoCorrec/2064 sacados/2064 ABCD72;5o JesaisJ’uti-liseJ’endéduisSi un triangle est isocèle alors la mesure des deuxangles de la base principale sont égalesBCA JesaisJ’uti-liseJ’endéduisLespointsB,CetDsontalignésetformentl’angle platBCDDeux angles adjacents formant un angle plat sontsupplémentairesACD JesaisJ’uti-liseJ’endéduisLe triangleABDest isocèle enDSi un triangle est isocèle alors les deux angles dela base principale sont de mesures égales. JesaisJ’uti-liseJ’endéduisDansletriangleABD,onaABD72;5oetBAD72;5oBDA JesaisJ’uti-liseJ’endéduisACD107;5oetCDA35oCAD chapExoCorrec/2054 sacados/2054 ABCD32o83o JesaisJ’uti-liseJ’endéduisTrois points alignés forment un angle platLes anglesADBetBDCsont adjacents et sup-plémentaires JesaisJ’uti-liseJ’endéduisDeux angles sont dits supplémentaires si la sommede leur mesure vaut180oBDA JesaisJ’uti-liseJ’endéduisDans le triangleBDA, on aB32oetD97oBAD JesaisJ’uti-liseJ’endéduisDans un triangle isocèle, les angles de la base prin-cipale ont même mesure
E.1390 Let ABCD be a rectangle and C a circle with center A intersecting segments [ BC ] and [ AD ] at M and N , respectively. 1 Copy the missing phrase from each deductive chain into your notebook: 2 a Complete the following deductive chain : b Give the measure of angle BMA . 3 We want to determine the value of angle CMN : a Determine the measure of angle MAN . b Deduce the measure of angle AMN . c Give the measure of angle NMC . 9. Us-math E.9628 American notation: losrque deux angles sont ˇ de mesures égales ı, aux Etats-Unis, on dit que les angles sont congru-ents. For example, below the two angles ∠ BOA and ∠ EOD are of the same measure, we note : ∠ AOB ∼ = ∠ DOF Complete the following blanks : a ∠ BOC ∼ = ∠ : : : ; b ∠ AOB ∼ = ∠ : : : c m ∠ COD + m ∠ DOE + m ∠ DOE = : : : E.9629 We consider the configuration be-low : where : ←→ DE and ←→ CF intercepts in A ; ←→ GH and ←→ CF intercepts in B ; ←→ DE ←→ GH Complete the blanks : 1 ∠ CAD ∼ = ∠ : : : ∼ = ∠ : : : ∼ = ∠ : : : 2 ∠ ABH ∼ = ∠ : : : ∼ = ∠ : : : ∼ = ∠ : : : 10. Unclassified exercises https://chingmath.fr chapExoCorrec/1390 sacados/1390 ABCDMNC47o JesaisJ’uti-liseJ’endéduisLes pointsMetNsont deux points du cercleOn en déduit:AMAN JesaisJ’uti-liseJ’endéduisAMetANsontdeuxsegmentsdemêmelongueurLe triangleAMNest isocèle enA. JesaisJ’uti-liseJ’endéduisLe triangleAMNest isocèle de sommet principalAAMNANM JesaisJ’uti-liseJ’endéduisSi un quadrilatère est un rectangle alors ses anglessont des angles droitsOn en déduit:BADABC90o chapExoCorrec/9628 sacados/9628 OADBEFC chapExoCorrec/9629 sacados/9629 ABCDEFGH
E.1745 Consider the figure below composed of 6 points : 1 Here are the properties to use in this question : If two straight lines are parallel to the same straight line, then they are parallel to each other If two straight lines are perpendicular to the same straight line, then they are parallel If two straight lines are parallel to each other and si a third straight line is perpendicular to one, then it is perpendicular to autre If a quadrilateral has four right angles then this quadrilateral is a rectangle. a How do you justify that the straight lines ( CD ) and ( FE ) are parallel? b Justify that the angle DEF is a right angle. c What is the nature of the quadrilateral CDEF ? 2 a Are the angles DAE and AED complementary? b Are the angles ADE and EDC supplementary? c Are the points A , D and C aligned? E.1461 Let ABCD be a rectangle and C a circle of center A intercepting segments [ BC ] and [ AD ] at M and N respectively. 1 Copy the missing sentence from each deductive link onto your notebook: a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . If a quadrilateral is a rectangle Then its angles are right angles. We deduce : BAD = ABC =90 o . b The points M and N are two points on the circle C with center A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . We deduce : AM = AN . c [ AM ] and [ AN ] are two segments of equal length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The triangle AMN is isosceles at A . d triangle AMN is isosceles with principal vertex A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . We deduce : AMN = ANM . Answer the following questions, justifying each of your state-ments : 2 Give the measure of the angle BMA . 3 We wish to determine the value of the angle CMN : a Determine the measure of the angle MAN . b Deduce the measure of the angle AMN . c Give the measure of the angle NMC . E.1743 Consider the triangle ABC oppo-site such that ( AB ) == ( EF ) . Answer the questions below by writing your reasoning in the form of deductive chains : 1 Calculate the measure of the angle FBA . 2 a Determine the measure of the angle CEF . a Determine the measure of the angle CFE . https://chingmath.fr chapExoCorrec/1745 sacados/1745 ABCDEF46o43o chapExoCorrec/1461 sacados/1461 ABCDMNC47o chapExoCorrec/1743 sacados/1743 66o83oABCEF
E.1744 The figure below shows a triangle ACD and a point B belonging to the segment [ AC ] . All reasoning must be justified in the form of the deductive link. 1 a Show that the triangle DCB is an equilateral trian-gle. Justify your approach. (several deductive links are required) . b What is the nature of the triangle ABD ? 2 What is the measure of the angle ABD ? Justify your approach. 3 Deduce the value of the angle DAB . Justify your ap-proach. (several deductive chains are required) . E.1746 Let ABC be a triangle and D a point on the segment [ AC ] . Some measurements are shown in the figure below : 1 Determine the measure of angle BDA . 2 Deduce the measure of angle BAD . 3 Now suppose that triangle ABC is isosceles at A . Determine the measure of angle ACB . E.4680 Consider a triangle ABC right-angled at B and such that : ABC = 34 o Note H the foot of the height, in triangle ABC , from vertex C . 1 Determine the measure of the angle CAB . 2 Demonstrate the following equality: CBH = ACH E.2931 In the plane, consider a parallelogram ABCD ; the point I is the point of intersection of the two bisectors DAB and ABC : Justify that the triangle ABI is right-angled at I . E.6448 Let A , B and C be three non-aligned points of the plane and ( d ) the bisector of the angle ABC . 1 Place a point M on the line ( d ) . 2 a Draw the line (Δ) passing through the point M and perpendicular to the line ( BA ) . b Name I the point of intersection of the straight lines ( d ) and ( BA ) . 3 a Draw the line (Δ ) passing through the point N and perpendicular to the line ( BC ) . b Name J the point of intersection of the lines ( d ) and ( BC ) . 4 Demonstrate that the distances IM and JM are equal. https://chingmath.fr chapExoCorrec/1744 sacados/1744 ABCD60o chapExoCorrec/1746 sacados/1746 ABCD32o83o chapExoCorrec/4680 sacados/4680 Juste a verifier s il est correct ABCH34o chapExoCorrec/2931 sacados/2931 ABCDI chapExoCorrec/6448 sacados/6448 ABC(d