# Side-Angle-Side Triangle Congruence

ID: kogik-kanuh
Illustrative Math
Subject: Geometry

# Side-Angle-Side Triangle Congruence

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##### 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 ﻿$F$﻿.