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)
A AAA - All angles of the triangles are equal therefore the triangles are congruent.
B They cannot be proven congruent as there is not enough information.
C SSA - Because the two outsides are equal and angle Z is congruent to angle X because the diagonals bisect the angles.
D SSS - Since all sides of a rhombus are equal and XZ is congruent because of the reflexive property.
The correct answer is D - SSS.
To prove triangle XZY congruent to triangle ZXP using the SSS (Side-Side-Side) congruence criterion, we need to show that all three sides of triangle XZY are equal to the corresponding sides of triangle ZXP.
In a rhombus, all sides are equal. Therefore, we know that segment XZ and segment ZP are equal in length.
Additionally, the diagonals of a rhombus bisect each other at right angles, meaning that angle X and angle Z are congruent.
So, we have segment XZ congruent to segment ZP (both sides are equal) and angle X congruent to angle Z (both angles are congruent).
Lastly, we can see that segment XY is equal to segment ZY (both sides of the rhombus are equal).
With all three corresponding sides of the triangles being equal, we can conclude that triangle XZY is congruent to triangle ZXP by the SSS congruence criterion.