# Side-Angle-Side Triangle Congruence

# Side-Angle-Side Triangle Congruence

##### Problem 1

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. Write each corresponding letter in the answer box and separate letters with commas.

a) Line $AB$ is an angle bisector of $DAC$. $\quad\quad\quad$ b) Angle $BAD$ is congruent to angle $ABC$. $\quad\quad\quad$ c) Angle $BDC$ is congruent to angle $BCD$. $\quad\quad$ d) Angle $ABD$ is congruent to angle $ABC$. $\quad\quad$ e) Triangle $DAB$ is congruent to triangle $CAB$.

##### Problem 2

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.

##### Problem 3

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 $ABD$ .

$\overline{AD} \cong \overline{AC}$

$\overline{BD} \cong \overline{BC}$

##### Problem 4

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 $DEF$.

Write your answer as a list of statement numbers in correct order for the proof, separated by commas.

##### Problem 5

Match each statement below to one of the pairs of congruent triangles. Write the number of the correct image in the answer box.

5) The 2 angles and the included side of one triangle are congruent to 2 angles and the included side of another triangle.

6) In the 2 triangles there are 3 pairs of congruent sides.

7) The 2 sides and the included angle of one triangle are congruent to 2 sides and the included angle of another triangle.

##### Problem 6

8) 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. Write each corresponding letter in the answer box and separate letters with commas.

a) Segment $AB$ coincides with segment $EF$. $\quad\quad$ b) Segment $BC$ coincides with segment $DF$. $\quad\quad$ c) Segment $AC$ coincides with segment $ED$. $\quad\quad$ d) Angle $A$ coincides with angle $E$. $\quad\quad$ e) Angle $C$ coincides with angle $F$.