# Lesson 9

Side-Side-Side Triangle Congruence

- Let’s see if we can prove one more set of conditions that guarantee triangles are congruent, and apply theorems.

### Problem 1

A kite is a quadrilateral which has 2 sides next to each other that are congruent and where the other 2 sides are also congruent. Given kite \(WXYZ\), show that at least one of the diagonals of a kite decomposes the kite into 2 congruent triangles.

### Problem 2

Mai has proven that triangle \(WYZ\) is congruent to triangle \(WYX\) using the Side-Side-Side Triangle Congruence Theorem. Why can she now conclude that diagonal \(WY\) bisects angles \(ZWX\) and \(ZYX\)?

### Problem 3

\(WXYZ\) is a kite. Angle \(WXY\) has a measure of 133 degrees and angle \(ZWX\) has a measure of 60 degrees. Find the measure of angle \(ZYW\).

### Problem 4

Each statement is always true. Select **all** statements for which the converse is also always true.

Statement: If 2 angles form a straight angle, then they are supplementary. Converse: If 2 angles are supplementary, then they form a straight angle.

Statement: In an isosceles triangle, the base angles are congruent. Converse: If the base angles of a triangle are congruent, then the triangle is isosceles.

Statement: If a point is equidistant from the 2 endpoints of a segment, then it lies on the perpendicular bisector of the segment. Converse: If a point lies on the perpendicular bisector of a segment, then it is equidistant from the 2 endpoints of the segment.

Statement: If 2 angles are vertical, then they are congruent. Converse: If 2 angles are congruent, then they are vertical.

Statement: If 2 lines are perpendicular, then they intersect to form 4 right angles. Converse: If 2 lines intersect to form 4 right angles, then they are perpendicular.

### Problem 5

Prove triangle \(ABD\) is congruent to triangle \(CDB\).

### Problem 6

Triangles \(ACD\) and \(BCD\) are isosceles. Angle \(DBC\) has a measure of 84 degrees and angle \(BDA\) has a measure of 24 degrees. Find the measure of angle \(BAC\).

### Problem 7

Reflect right triangle \(ABC\) across line \(AB\). Classify triangle \(CAC’\) according to its side lengths. Explain how you know.