# Lesson 6

Side-Angle-Side Triangle Congruence

### Problem 1

Triangle $$DAC$$ is isosceles with congruent sides $$AD$$ and $$AC$$. Which additional given information is sufficient for showing that triangle $$DBC$$ is isosceles? Select all that apply.

A:

Line $$AB$$ is an angle bisector of $$DAC$$.

B:

Angle $$BAD$$ is congruent to angle $$ABC$$.

C:

Angle $$BDC$$ is congruent to angle $$BCD$$.

D:

Angle $$ABD$$ is congruent to angle $$ABC$$.

E:

Triangle $$DAB$$ is congruent to triangle $$CAB$$.

### Solution

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### Problem 2

Tyler has written an incorrect proof to show that quadrilateral $$ABCD$$ is a parallelogram. He knows segments $$AB$$ and $$DC$$ are congruent. He also knows angles $$ABC$$ and $$ADC$$ are congruent. Find the mistake in his proof.

Segment $$AC$$ is congruent to itself, so triangle $$ABC$$ is congruent to triangle $$ADC$$ by Side-Angle-Side Triangle Congruence Theorem.  Since the triangles are congruent, so are the corresponding parts, and so angle $$DAC$$ is congruent to $$ACB$$.  In quadrilateral $$ABCD$$, $$AB$$ is congruent to $$CD$$ and $$AD$$ is parallel to $$CB$$. Since $$AD$$ is parallel to $$CB$$, alternate interior angles $$DAC$$ and $$BCA$$ are congruent. Since alternate interior angles are congruent, $$AB$$ must be parallel to $$CD$$. Quadrilateral $$ABCD$$ must be a parallelogram since both pairs of opposite sides are parallel.

### Solution

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### Problem 3

Triangles $$ACD$$ and $$BCD$$ are isosceles. Angle $$BAC$$ has a measure of 18 degrees and angle $$BDC$$ has a measure of 48 degrees. Find the measure of angle $$ABD$$.

### Solution

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### Problem 4

Here are some statements about 2 zigzags. Put them in order to prove figure $$ABC$$ is congruent to figure $$DEF$$.

• 1: If necessary, reflect the image of figure $$ABC$$ across $$DE$$ to be sure the image of $$C$$, which we will call $$C'$$, is on the same side of $$DE$$ as $$F$$
• 2: $$C'$$ must be on ray $$EF$$ since both $$C'$$ and $$F$$ are on the same side of $$DE$$ and make the same angle with it at $$E$$.
• 3: Segments $$AB$$ and $$DE$$ are the same length so they are congruent. Therefore, there is a rigid motion that takes $$AB$$ to $$DE$$. Apply that rigid motion to figure $$ABC$$.
• 4: Since points $$C'$$ and $$F$$ are the same distance along the same ray from $$E$$ they have to be in the same place.
• 5: Therefore, figure $$ABC$$ is congruent to figure $$DEF$$.

### Solution

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(From Unit 2, Lesson 5.)

### Problem 5

Match each statement using only the information shown in the pairs of congruent triangles.

### Solution

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(From Unit 2, Lesson 4.)

### Problem 6

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

Select all true statements after the transformations:

A:

Segment $$AB$$ coincides with segment $$EF$$.

B:

Segment $$BC$$ coincides with segment $$DF$$.

C:

Segment $$AC$$ coincides with segment $$ED$$.

D:

Angle $$A$$ coincides with angle $$E$$.

E:

Angle $$C$$ coincides with angle $$F$$.

### Solution

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(From Unit 2, Lesson 3.)