Lesson 3

Congruent Triangles, Part 1

Problem 1

Triangle $$ABC$$ is congruent to triangle $$EDF$$. So, Kiran knows that there is a sequence of rigid motions that takes $$ABC$$ to $$EDF$$.

Select all true statements after the transformations:

A:

Angle $$A$$ coincides with angle $$F$$.

B:

Angle $$B$$ coincides with angle $$D$$.

C:

Segment $$AC$$ coincides with segment $$EF$$.

D:

Segment $$BC$$ coincides with segment $$ED$$.

E:

Segment $$AB$$ coincides with segment $$ED$$.

Solution

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Problem 2

A rotation by angle $$ACE$$ using point $$C$$ as the center takes triangle $$CBA$$ onto triangle $$CDE$$.

1. Explain why the image of ray $$CA$$ lines up with ray $$CE$$.
2. Explain why the image of $$A$$ coincides with $$E$$.
3. Is triangle $$CBA$$ congruent to triangle $$CDE$$? Explain your reasoning.

Solution

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Problem 3

The triangles are congruent. Which sequence of rigid motions will take triangle $$XYZ$$ onto triangle $$BCA$$?

A:

Translate $$XYZ$$ using directed line segment $$YC$$. Rotate $$X’Y’Z’$$ using $$C$$ as the center so that $$X’$$ coincides with $$B$$. Reflect $$X’’Y’’Z’’$$ across line $$CB$$

B:

Translate $$XYZ$$ using directed line segment $$YC$$. Rotate $$X’Y’Z’$$ using $$C$$ as the center so that $$X’$$ coincides with $$B$$. Reflect $$X’’Y’’Z’’$$ across line $$AC$$.

C:

Translate $$XYZ$$ using directed line segment $$YC$$. Rotate $$X’Y’Z’$$ using $$C$$ as the center so that $$X’$$ coincides with $$A$$. Reflect $$X’’Y’’Z’’$$ across line $$CB$$.

D:

Translate $$XYZ$$ using directed line segment $$YC$$. Rotate $$X’Y’Z’$$ using $$C$$ as the center so that $$X’$$ coincides with $$A$$. Reflect $$X’’Y’’Z’’$$ across line $$AC$$.

Solution

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Problem 4

Triangle $$HEF$$ is the image of triangle $$FGH$$ after a 180 degree rotation around point $$K$$. Select all statements that must be true.

A:

Triangle $$HGF$$ is congruent to triangle $$FEH$$.

B:

Triangle $$GFH$$ is congruent to triangle $$EFH$$.

C:

Angle $$KHE$$ is congruent to angle $$KHG$$.

D:

Angle $$GHK$$ is congruent to angle $$EFK$$.

E:

Segment $$EH$$ is congruent to segment $$GH$$.

F:

Segment $$HG$$ is congruent to segment $$FE$$.

G:

Segment $$FH$$ is congruent to segment $$HF$$.

Solution

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(From Unit 2, Lesson 2.)

Problem 5

Line $$SD$$ is a line of symmetry for figure $$ASMHZDPX$$. Tyler says that $$ASDPX$$ is congruent to $$SMDZH$$ because sides $$AS$$ and $$MS$$ are corresponding.

1. Why is Tyler's congruence statement incorrect?
2. Write a correct congruence statement for the pentagons.

Solution

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(From Unit 2, Lesson 2.)

Problem 6

Triangle $$ABC$$ is congruent to triangle $$DEF$$.  Select all the statements that are a result of corresponding parts of congruent triangles being congruent.

A:

Segment $$AC$$ is congruent to segment $$EF$$.

B:

Segment $$BC$$ is congruent to segment $$EF$$.

C:

Angle $$BAC$$ is congruent to angle $$EDF$$.

D:

Angle $$BCA$$ is congruent to angle $$EDF$$.

E:

Angle $$CBA$$ is congruent to angle $$FED$$.

Solution

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(From Unit 2, Lesson 1.)

Problem 7

When triangle $$ABC$$ is reflected across line $$AB$$, the image is triangle $$ABD$$. Why is angle $$ACD$$ congruent to angle $$ADB$$?

A:

Corresponding parts of congruent figures are congruent.

B:

Congruent parts of congruent figures are corresponding.

C:

Segment $$AB$$ is a perpendicular bisector of segment $$DC$$.

D:

An isosceles triangle has a pair of congruent angles.

Solution

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(From Unit 2, Lesson 1.)

Problem 8

Line $$DE$$ is parallel to line $$BC$$.

1. What is the measure of angle $$EAC$$?
2. What is the measure of angle $$DAB$$?

Solution

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(From Unit 1, Lesson 21.)