XYZP is a rhombus. Explain why triangle XZY is congruent to triangle ZXP. (Hint: You must prove the triangle congruent with a triangle congruence theorem and explain the criteria required including naming the congruent sides/angles
To prove that triangle XYZ is congruent to triangle ZXP, we need to establish a congruence theorem and show that the necessary criteria are satisfied.
The congruence theorem that can be applied here is the Side-Angle-Side (SAS) congruence criterion. According to this criterion, if two triangles have two pairs of corresponding sides that are congruent, and the included angles formed by these sides are congruent as well, then the two triangles are congruent.
Now let's identify the corresponding sides and angles:
1. Side XY is congruent to side XZ (both sides of the rhombus are equal in length).
2. Side XZ is congruent to side ZP (both sides of the rhombus are equal in length).
3. Angle XZY is congruent to angle ZXP (these angles are formed by the congruent sides XZ and ZP, which are opposite sides of the rhombus).
Therefore, we have a pair of congruent sides (XY and ZP), followed by the included congruent angle (angle XZY = angle ZXP). This satisfies the criteria of the SAS congruence theorem.
Hence, triangle XZY is congruent to triangle ZXP.