Side-Angle-Side Triangle Congruence

ID: kogik-kanuh
Created by Illustrative MathIllustrative Math
Subject: Geometry
Grade: 9-12

Side-Angle-Side Triangle Congruence

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Problem 1

1) Triangle DACDAC is isosceles with congruent sides ADAD and ACAC. Which additional given information is sufficient for showing that triangle DBCDBC is isosceles?

Select all that apply. Write each corresponding letter in the answer box and separate letters with commas.

a) Line ABAB is an angle bisector of DACDAC. \quad\quad\quad b) Angle BADBAD is congruent to angle ABCABC. \quad\quad\quad c) Angle BDCBDC is congruent to angle BCDBCD. \quad\quad d) Angle ABDABD is congruent to angle ABCABC. \quad\quad e) Triangle DABDAB is congruent to triangle CABCAB.

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Problem 2

2) Tyler has written an incorrect proof to show that quadrilateral ABCDABCD is a parallelogram. He knows segments ABAB and DCDC are congruent. He also knows angles ABCABC and ADCADC are congruent. Find the mistake in his proof:

Segment ACAC is congruent to itself, so triangle ABCABC is congruent to triangle ADCADC by Side-Angle-Side Triangle Congruence Theorem. Since the triangles are congruent, so are the corresponding parts, and so angle DACDAC is congruent to ACBACB. In quadrilateral ABCDABCD, ABAB is congruent to CDCD and ADAD is parallel to CBCB. Since ADAD is parallel to CBCB, alternate interior angles DACDAC and BCABCA are congruent. Since alternate interior angles are congruent, ABAB must be parallel to CDCD. Quadrilateral ABCDABCD must be a parallelogram since both pairs of opposite sides are parallel.

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Problem 3

3) Triangles ACDACD and BCDBCD are isosceles. Angle BACBAC has a measure of 18 degrees and angle BDCBDC has a measure of 48 degrees. Find the measure of angle ABDABD .


ADAC\overline{AD} \cong \overline{AC}

BDBC\overline{BD} \cong \overline{BC}

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Problem 4

4) Here are some statements about 2 zigzags. Put them in order to prove figure ABCABC is congruent to figure DEFDEF.

1. If necessary, reflect the image of figure ABCABC across DEDE to be sure the image of CC, which we will call CC', is on the same side of DEDE as FF.

2. CC' must be on ray EFEF since both CC' and FF are on the same side of DEDE and make the same angle with it at EE.

3. Segments ABAB and DEDE are the same length so they are congruent. Therefore, there is a rigid motion that takes ABAB to DEDE. Apply that rigid motion to figure ABCABC.

4. Since points CC' and FF are the same distance along the same ray from EE they have to be in the same place.

5. Therefore, figure ABCABC is congruent to figure DEFDEF.

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

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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.

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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 ABCABC is congruent to triangle EDFEDF. So, Priya 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) Segment ABAB coincides with segment EFEF. \quad\quad b) Segment BCBC coincides with segment DFDF. \quad\quad c) Segment ACAC coincides with segment EDED. \quad\quad d) Angle AA coincides with angle EE. \quad\quad e) Angle CC coincides with angle FF.

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