Question

given qt = sr, qv = su, and the diagram below, prove that triangles qut and svr are congruent. write a paragraph proof.

Explanation:

Step1: Identify right - angles

We observe that $\angle QUT=\angle SVR = 90^{\circ}$ as they are marked as right - angles in the diagram.

Step2: Recall given side - lengths

We are given that $QT = SR$ and $QV=SU$.

Step3: Apply Hypotenuse - Leg (HL) theorem

In right - triangles $\triangle QUT$ and $\triangle SVR$, the hypotenuses $QT$ and $SR$ are equal ($QT = SR$), and since $\angle QUT=\angle SVR = 90^{\circ}$, by the Hypotenuse - Leg congruence theorem for right - triangles, we can conclude that $\triangle QUT\cong\triangle SVR$.

Answer:

Triangles $\triangle QUT$ and $\triangle SVR$ are congruent by the Hypotenuse - Leg (HL) congruence theorem for right - triangles since $\angle QUT=\angle SVR = 90^{\circ}$ and $QT = SR$.