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.

Kite W X Y Z. Sides Z W and W X have single tick marks. Sides X Y and Y Z have double tick marks.

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 \(ZYX\)?

Triangle WYZ and WYX. WZ is congruent to WX and YZ is congruent to YX. The triangles share side WY.

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 \(ZYW\).

Kite WXYZ. WX is congruent to WZ and XY is congruent to ZY.


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.

(From Unit 2, Lesson 8.)

Problem 5

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

\(DC \parallel AB\)

Line DC is parallel to and above line AB and cut by transversal DB. Angles A and C are right angles.
(From Unit 2, Lesson 7.)

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 \(BAC\).

\(\overline{AD} \cong \overline{AC}\) and \(\overline{BD} \cong \overline{BC}\)

Triangle A C D, isosceles, point B inside. Side A D and A C have one tick mark. Segments B D, B C, and BA are drawn. Triangle B C D is isosceles. Sides B D and B C have two tick marks.
(From Unit 2, Lesson 6.)

Problem 7

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

Triangle ABC with angle ABC marked as a right angle.


(From Unit 2, Lesson 1.)