Question

given: $overline{ys}congoverline{yu}$, $overline{ys}perpoverline{rt}$, $overline{yu}perpoverline{rv}$, $y$ is the mid - point of $overline{tv}$. prove: $\triangle trv$ is isosceles. 1. $overline{ys}perpoverline{rt}$, $overline{yu}perpoverline{rv}$ 1. given 2. perpendicular lines intersect to form right angles. 3. definition of a right triangle 4. $overline{ys}congoverline{yu}$, $y$ is the mid - point of $overline{tv}$. 4. given

Explanation:

Step1: Define right - angles

Since $\overline{YS}\perp\overline{RT}$ and $\overline{YU}\perp\overline{RV}$, by the definition of perpendicular lines, $\angle YSR = 90^{\circ}$ and $\angle YUV = 90^{\circ}$.

Step2: Identify right - triangles

By the definition of a right - triangle (a triangle with one right angle), $\triangle YSR$ and $\triangle YUV$ are right - triangles.

Step3: Prove triangle congruence and isosceles property

We know that $\overline{YS}\cong\overline{YU}$ and $Y$ is the mid - point of $\overline{TV}$ so $\overline{TY}\cong\overline{VY}$. By the Hypotenuse - Leg (HL) congruence criterion for right - triangles, $\triangle YSR\cong\triangle YUV$. Then, corresponding angles are equal, so $\angle T=\angle V$. In $\triangle TRV$, since $\angle T=\angle V$, the sides opposite these angles are equal, and $\triangle TRV$ is isosceles.

Answer:

1. $\angle YSR = 90^{\circ}, \angle YUV=90^{\circ}$
2. $\triangle YSR$ and $\triangle YUV$ are right - triangles
3. Prove $\triangle YSR\cong\triangle YUV$ (by Hypotenuse - Leg (HL) congruence criterion since $\overline{YS}\cong\overline{YU}$ and $\overline{TY}\cong\overline{VY}$ as $Y$ is the mid - point of $\overline{TV}$), then $\angle T=\angle V$. Since angles opposite equal sides in a triangle make the triangle isosceles, $\triangle TRV$ is isosceles.