Congruent Triangles, Part 1

ID: basur-mopor
Created by Illustrative MathIllustrative Math
Subject: Geometry
Grade: 9-12

Congruent Triangles, Part 1

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Problem 1

1) Triangle ABCABC is congruent to triangle EDFEDF. So, Kiran knows that there is a sequence of rigid motions that takes ABCABC to EDFEDF.

Select all true statements after the transformations. Write each corresponding letter in the answer box and separate letters with commas.

a) Angle AA coincides with angle FF. \quad\quad b) Angle BB coincides with angle DD. \quad\quad c) Segment ACAC coincides with segment EFEF. \quad\quad d) Segment BCBC coincides with segment EDED. \quad\quad e) Segment ABAB coincides with segment EDED.

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Problem 2

A rotation by angle ACEACE using point CC as the center takes triangle CBACBA onto triangle CDECDE.

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2) Explain why the image of ray CACA lines up with ray CECE.

3) Explain why the image of AA coincides with EE.

4) Is triangle CBACBA congruent to triangle CDECDE?

True or false? Write below.

5) Explain your reasoning.

Problem 3

6) The triangles are congruent. Which sequence of rigid motions will take XYZXYZ onto BCABCA?

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a) Translate XYZ using directed line segment YC. Rotate XYZ using C as the center so that X coincides with B Reflect XYZ across line CB\text{Translate } XYZ \text{ using directed line segment } YC \text{. Rotate } X'Y'Z' \text{ using } C \text{ as the center so that } X' \text{ coincides with } B \text{. } \newline \text{ Reflect } X''Y''Z'' \text{ across line } CB \text{. }b) Translate XYZ using directed line segment YC. Rotate XYZ using C as the center so that X coincides with B Reflect XYZ across line AC\text{Translate } XYZ \text{ using directed line segment } YC \text{. Rotate } X'Y'Z' \text{ using } C \text{ as the center so that } X' \text{ coincides with } B \text{. } \newline \text{ Reflect } X''Y''Z'' \text{ across line } AC \text{. }c) Translate XYZ using directed line segment YC. Rotate XYZ using C as the center so that X coincides with A Reflect XYZ across line CB\text{Translate } XYZ \text{ using directed line segment } YC \text{. Rotate } X'Y'Z' \text{ using } C \text{ as the center so that } X' \text{ coincides with } A \text{. } \newline \text{ Reflect } X''Y''Z'' \text{ across line } CB \text{. }d) Translate XYZ using directed line segment YC. Rotate XYZ using C as the center so that X coincides with A Reflect XYZ across line AC\text{Translate } XYZ \text{ using directed line segment } YC \text{. Rotate } X'Y'Z' \text{ using } C \text{ as the center so that } X' \text{ coincides with } A \text{. } \newline \text{ Reflect } X''Y''Z'' \text{ across line } AC \text{. }
Problem 4

7) Triangle HEFHEF is the image of triangle FGHFGH after a 180 degree rotation around point KK. Select all statements that must be true. Write each corresponding letter in the answer box and separate letters with commas.

a) Triangle HGFHGF is congruent to triangle FEHFEH.\quad\quad b) Triangle GFHGFH is congruent to triangle EFHEFH. \quad\quad c) Angle KHEKHE is congruent to angle KHGKHG. \quad\quad d) Angle GHKGHK is congruent to angle EFKEFK. \quad\quad e) Segment EFEF is congruent to segment GHGH. \quad\quad f) Segment HGHG is congruent to segment FEFE. \quad\quad g) Segment FHFH is congruent to segment HFHF.

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Problem 5

Line SDSD is a line of symmetry for figure ASMHZDPXASMHZDPX. Tyler says that ASDPXASDPX is congruent to SMDZHSMDZH because sides ASAS and MSMS are corresponding.

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8) Why is Tyler's congruence statement incorrect?

9) Which pentagon is congruent to pentagon ASDPXASDPX?

Problem 6

10) Triangle ABCABC is congruent to triangle DEFDEF. Select all the statements that are a result of corresponding parts of congruent triangles being congruent. Write each corresponding letter in the answer box and separate letters with commas.

a) Segment ACAC is congruent to segment EFEF. \quad\quad b) Segment BCBC is congruent to segment EFEF. \quad\quad c) Angle BACBAC is congruent to angle EDFEDF. \quad\quad d) Angle BCABCA is congruent to angle EDFEDF. \quad\quad e) Angle CBACBA is congruent to angle DEFDEF.

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Problem 7

11) When triangle ABCABC is reflected across line ABAB, the image is triangle ABDABD. Why is angle ACDACD congruent to angle ADBADB?

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a) Corresponding parts of congruent figures are congruent.\text{Corresponding parts of congruent figures are congruent.}b) Congruent parts of congruent figures are corresponding.\text{Congruent parts of congruent figures are corresponding.}c) Segment AB is a perpendicular bisector of segment DC.\text{Segment } AB \text{ is a perpendicular bisector of segment } DC \text{.}d) An isosceles triangle has a pair of congruent angles.\text{An isosceles triangle has a pair of congruent angles.}
Problem 8

Line DEDE is parallel to line BCBC.

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12) What is the measure of angle EACEAC?

13) What is the measure of angle DABDAB?