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)
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To prove that triangle XZY is congruent to triangle ZXP, we need to use the congruence property of rhombus XYZP.
A rhombus is a parallelogram with all sides congruent. Therefore, in rhombus XYZP, all sides XY, YZ, ZP, and PX are congruent.
Using this property, we can prove congruence of triangle XZY and triangle ZXP by the Side-Side-Side (SSS) congruence theorem.
According to the SSS congruence theorem, two triangles are congruent if all corresponding sides of the triangles are congruent.
In this case, we have:
1. Side XY = Side XP (both equal to a side of the rhombus)
2. Side YZ = Side ZP (both equal to a side of the rhombus)
3. Side ZX is common to both triangles (a side of rhombus XYZP)
Thus, all three sides of triangle XZY, namely XY, YZ, and ZX, are congruent to the corresponding sides of triangle ZXP, namely XP, ZP, and ZX, respectively.
Therefore, by the SSS congruence theorem, triangle XZY is congruent to triangle ZXP.