Question

select all of the diagrams that contain enough information to determine that △qrs is isosceles.

Explanation:

Step1: Recall isosceles - triangle property

An isosceles triangle has two equal angles or two equal sides. We use the angle - sum property of a triangle ($\angle Q+\angle R+\angle S = 180^{\circ}$) and linear - angle property (linear pair of angles sum to $180^{\circ}$) to find the angles of $\triangle QRS$.

Step2: Analyze option A

$\angle PQR = 180 - 100=80^{\circ}$, $\angle R = 80^{\circ}$, $\angle S=180-(80 + 80)=20^{\circ}$. Since $\angle PQR=\angle R = 80^{\circ}$, $\triangle QRS$ is isosceles.

Step3: Analyze option B

$\angle PQR = 180 - 100 = 80^{\circ}$, $\angle R = 80^{\circ}$, $\angle S=180-(80 + 80)=20^{\circ}$. Since $\angle PQR=\angle R = 80^{\circ}$, $\triangle QRS$ is isosceles.

Step4: Analyze option C

$\angle PQR = 180 - 100=80^{\circ}$, $\angle S = 40^{\circ}$, $\angle R=180-(80 + 40)=60^{\circ}$. No two angles are equal, so $\triangle QRS$ is not isosceles.

Step5: Analyze option D

$\angle PQR = 180 - 100 = 80^{\circ}$, $\angle R = 40^{\circ}$, $\angle S=180-(80 + 40)=60^{\circ}$. No two angles are equal, so $\triangle QRS$ is not isosceles.

Step6: Analyze option E

$\angle R = 40^{\circ}$, $\angle S = 80^{\circ}$, $\angle Q=180-(40 + 80)=60^{\circ}$. No two angles are equal, so $\triangle QRS$ is not isosceles.

Step7: Analyze option F

$\angle PQR = 180 - 100=80^{\circ}$, $\angle QSR=180 - 100 = 80^{\circ}$, $\angle R=180-(80 + 80)=20^{\circ}$. Since $\angle PQR=\angle QSR = 80^{\circ}$, $\triangle QRS$ is isosceles.

Answer:

A. Diagram A
B. Diagram B
C. Diagram F