Grade 6 / angles: complements 31 exercises (including 30 corrected)

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abcAxyztBCxyzDxyz ABCDE ABCDO 15o35oAxyz 28o47oBxyz 1. Definition of adjacent angles E.2974 Definition: Two angles are said to be adjacent if they satisfy the following three conditions : They have the same vertex. They share a common side. they are located on either side of this common side. In each of the three cases below, justify that the coded pairs of angles are not a pair of adjacent angles : E.2975 Consider a quadrilateral ABCD and a point E inside it. For each of the angles coded in the figure, name at least one associated adjacent angle. E.10858 Consider the quadrilateral ABCD shown below. We name O the intersection of its diagonals : 1 Name the angle adjacent to the angle OBC 2 Name the angle adjacent to the angle ODA 3 Name angle adjacent to angle DOC 2. Measurement and adjacent angles E.11476 1 Consider the figure below composed of point A and rays [ Ax ) , [ Ay ) , [ Az ) : a Name a pair of adjacent angles. b Deduce the measure of angle xAz 2 Consider the figure below composed of point B and rays [ Bx ) , [ By ) , and [ Bz ) : a Name a pair of adjacent angles. b Deduce the measure of angle y Bz https://chingmath.fr chapExoCorrec/2974 sacados/2974 abcAxyztBCxyzDxyz chapExoCorrec/2975 sacados/2975 ABCDE chapExoCorrec/10858 sacados/10858 ABCDO chapExoCorrec/11476 sacados/11476 15o35oAxyz 28o47oBxyz
17o38oAxyz25o42oBxyz Omnpxy25o67o 28o65oABCD 01020304050607080901001101201301401501601701801801701601501401301201101009080706050403020100xyztuvO Oxyztuv E.5621 We consider the two configurations below : 1 Determine the measure of the angle ∠ xAz . Justify your approach. 2 Determine the measure of the angle ∠ y Bz . Justify your approach. E.2553 The figure below is composed of five half-lines with origin the point O . The angle ∠ yOx is a flat angle and the angle ∠ nOx is a right angle. Note: the figure was not constructed with true dimensions. 1 a Justify that the angles ∠ xOm and ∠ mOn are adja-cent. b Determine the measure of the angle ∠ mOx . 2 a Noting that the angles ∠ mOn and ∠ nOp are adja-cent, justify, by explanation and calculation, that the measure of the angle ∠ pOn =42 o . b Noting that the angles ∠ yOp and ∠ pOn are adjacent angles, determine the measure of the angle ∠ yOp E.10859 Consider the three half-lines [ AB ) , [ AC ) , [ AD ) shown below : We have the measurements : BAD = 65 o ; CAD = 28 o 1 Justify that the angles BAC and CAD are adjacent. 2 Give the measure of the angle BAC . Justify your ap-proach. E.2457 Consider the protractor below on which some half-lines have been drawn : 1 a Are the angles xOy and yOz adjacent? Justify your answer. b By reading on the protractor, give the measures of the angles xOy and xOz . c Deduce the measure of the angle yOz . 2 Are the angles tOy and zOt adjacent? Justify your an-swer. 3 a Are the angles zOu and uOv adjacent? b Deduce the measure of the angle zOu . 3. Definition and complementary, supplementary and opposite angles through the vertex E.10800 Definition: Two angles are said to be opposite at the vertex if they satisfy the following three properties : they have the same vertex, their sides are carried by two lines, their sides are extensions of each other. Consider the three lines ( xy ) , ( zt ) , ( uv ) that intersect at point O . For each question, name the angle opposite the vertex of the angle mentioned : a vOy b xOt c zOu https://chingmath.fr chapExoCorrec/5621 sacados/5621 17o38oAxyz25o42oBxyz chapExoCorrec/2553 sacados/2553 Omnpxy25o67o chapExoCorrec/10859 sacados/10859 28o65oABCD chapExoCorrec/2457 sacados/2457 01020304050607080901001101201301401501601701801801701601501401301201101009080706050403020100xyztuvO chapExoCorrec/10800 sacados/10800 Oxyztuv
Oxxyyzz ABCIJ yuvz95oO xyztuv25o70oO xxyyzz42o75oO E.5622 Definition: Two adjacent angles are said to be complementary if the sum of their measures is 90 o . Two adjacent angles are said to be supplementary if the sum of their measures is 180 o . Consider the configuration below, consisting of three lines that intersect at point O . 1 a Name a pair of adjacent and complementary angles that includes angle zOy . b Name another pair of adjacent and complementary an-gles in this configuration. 2 a Name a pair of adjacent supplementary angles that include angle x Oz . b Name another pair of adjacent supplementary angles present in this configuration. E.6670 Consider the figure below resting on a right triangle ABC at A . Points I and J are the midpoints of segments [ AB ] and [ BC ] , respectively. 1 List all pairs of adjacent and complementary angles. 2 List all pairs of adjacent and supplementary angles. 4. Measurement and complementary, supplementary and opposite angles through the vertex E.1403 Consider the two lines ( yu ) and ( vz ) intersecting at the point O and such that ∠ uOv =95 o Determine the measure of the angle ∠ zOu . Justify your an-swer. E.2063 The figure below is com-posed of three straight lines intercepting at O . 1 Give the measure of the angle yOu . Justify. 2 Determine the measure of the angle yOz . Justify. E.11477 Consider the three lines ( xx ) , ( yy ) , ( zz ) intersecting at point O : The measurements are given : xOz = 42 o ; zOy = 75 o 1 Give the measurement of angle yOz . 2 Justify that angles xOz and z Oy form a pair of adjacent angles. 3 Deduce the measure of angle x Oy . https://chingmath.fr chapExoCorrec/5622 sacados/5622 Oxxyyzz chapExoCorrec/6670 sacados/6670 ABCIJ chapExoCorrec/1403 sacados/1403 yuvz95oO chapExoCorrec/2063 sacados/2063 xyztuv25o70oO chapExoCorrec/11477 sacados/11477 xxyyzz42o75oO
xxyy36oAB22oCDO OABCDE28o 115o44oOxxyyzztt ABC5;5cm4;5cm8cm(dMN AxyzAxyzAxyz E.6669 Consider the figure below, composed of four lines intersecting at point O , and more specifically, lines ( xx ) and ( yy ) , which are perpendicular to each other : Without justification, give the measure of the following an-gles : a BOy b y OD c DOA d DOB E.10860 Consider the three straight lines ( AB ) , ( CD ) , ( EO ) intercepting at the point O and such that ( EO ) ⊥ ( AB ) : 1 Give the measure of the angle COA . Justify. 2 Give the measure of the angle COE . Justify. E.2982 In the plane we consider the four straight lines ( xx ) , ( yy ) , ( zz ) , ( tt ) intersecting at the same point O : The protractor must not be used in this exercise. 1 Using adjacent angles and noting the calculations made, determine the measure of the angles below : a zOy b tOx c xOy 2 a Determine the measure of the angle x Ot . b Deduce the measure of angle zOx . E.1405 Complete the table below : Mesure de angle A Mesure de angle B Les angles A et B sont 37 o complementary 67.5 o 112.5 o 34 o supplémentaires 27.19 o 62.81 o 5. Angle bisector: definition E.5643 Consider the following configuration : 1 Specify the nature of the straight line ( AM ) relative to the triangle ABC ? 2 a Reproduce this figure in true dimension. b Write a plotting program for this figure. E.5645 In each of the three cases below, spec-ify which of the half-lines [ A;z ) la (s) quelle (s) represent the bisector of the angle xAy : 6. Angle bisector: drawing with a protractor https://chingmath.fr chapExoCorrec/6669 sacados/6669 xxyy36oAB22oCDO chapExoCorrec/10860 sacados/10860 OABCDE28o chapExoCorrec/2982 sacados/2982 115o44oOxxyyzztt chapExoCorrec/1405 sacados/1405 chapExoCorrec/5643 sacados/5643 ABC5;5cm4;5cm8cm(dMN chapExoCorrec/5645 sacados/5645 AxyzAxyzAxyz
BrsCtuAxy BrsCtuAxy AxyBztCrs 8cm6cmABCLM42o E.2458 The box below shows three angles xAy , rBs and tCu . Using a protractor and ruler, construct the bisectors of these three angles. E.6445 The box below shows three angles xAy , rBs and tCu . Using a protractor and ruler, construct the bisectors of these three angles. E.3920 Consider the three angles xAy , sBr and uCt below : 1 a Using rapporteur , give the measure of the angle xAy . b Using protractor and ruler, draw the bisector of the angle xAy . 2 Using the compass and ruler, trace the bisector of the angle tBz and the angle sCr . E.1223 (We’ll leave the construction traces) 1 a Draw the triangle ABC such that : AB = 7.5 cm ; BC = 4 cm ; AC = 8 cm b What can we say about the nature of the triangle ABC ? 2 a Draw the triangle MNP such that : MN = 5 cm ; MP = 8 cm ; NMP = 75 o b Draw the bisector of the angle MNP . 7. Tracing program with protractor E.1666 In the plan, we consider the config-uration below : 1 Give the program of plot of the figure above. 2 a Draw the triangle ABC at using a rule graduated and a reporter. b Finish to reproduce the figure by using the rule and the compass. https://chingmath.fr chapExoCorrec/2458 sacados/2458 BrsCtuAxy chapExoCorrec/6445 sacados/6445 BrsCtuAxy chapExoCorrec/3920 sacados/3920 AxyBztCrs chapExoCorrec/1223 sacados/1223 chapExoCorrec/1666 sacados/1666 8cm6cmABCLM42o
ABC ABCD ABC36oM E.2479 Consider the triangle ABC shown be-low : 1 a Using the protractor and ruler, draw the bisectors arising from the three vertices of this triangle. b Name O the point of concurrence of the three bisectors. 2 a Draw a circle with center O entirely contained within the triangle ABC and whose radius is as large as possible. b What is special about this circle? Note : this circle is called the circle inscribed in the triangle ABC . E.6637 Consider the quadrilateral ABCD shown below : 1 Draw the line ( d ) perpendicular to the line ( BC ) and through the point D . 2 Draw the line ( d ) parallel to the line ( CD ) through the point B . 3 Using the protractor, draw the bisector of the angle ∠ DAB . 4 Draw the perpendicular bisector of segment [ CD ] . E.2481 Carry out the drawing program below. Use protractor and square. 1 Draw the triangle ABC verifying the following measures : AB =8 cm ; BC =5.5 cm ; AC =7 cm 2 Draw the perpendicular bisector of segment [ AB ] . 3 Draw the bisector of the angle CBA 4 Draw the perpendicular to the line ( BC ) passing through A . 8. Reasoning About the Bisector E.5644 Consider the configuration below where ABC is an isosceles triangle and the angle of the main ver-tex measures 36 o . The line ( CM ) is the bisector of the angle ACB 1 Determine the measures of the three angles of the trian-gle BCM . 2 Deduce the nature of the triangle BCM . https://chingmath.fr chapExoCorrec/2479 sacados/2479 ABC chapExoCorrec/6637 sacados/6637 ABCD chapExoCorrec/2481 sacados/2481 chapExoCorrec/5644 sacados/5644 ABC36oM
MRUNS32o E.1659 Consider the figure be-low : Without using a pro-tractor, determine the measures of the follow-ing angles : 1 SMN 2 UMN 3 UMS 9. An Approach to the Sum of the Angles in a Triangle E.11765 L’exercice n’existe pas. https://chingmath.fr chapExoCorrec/1659 sacados/1659 MRUNS32o sacados/11765