A triangular sign is supported by two braces of equal length. In the figure, is isosceles with . Segment is extended through to point . If , which choice is the measure of ?

Solution available with step-by-step explanation

SAT Lines, Angles & Triangles Question 32 | Free SAT Prep | Aniko

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Question 32·Medium·Lines, Angles, and Triangles

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A triangular sign is supported by two braces of equal length. In the figure, $\triangle PQR$ is isosceles with $PQ = PR$ . Segment $\overline{PQ}$ is extended through $Q$ to point $S$ . If $\angle SQR = 146^\circ$ , which choice is the measure of $\angle QPR$ ?

When a side of a triangle is extended, the given exterior angle and the adjacent interior angle form a linear pair (sum to $180^\circ$ ). If the triangle is isosceles, set the two base angles equal, then use the triangle angle sum $180^\circ$ to get the remaining angle.

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Relate the exterior and interior angles at $Q$

Because $\overline{PQ}$ is extended to $S$ , the angle labeled $146^\circ$ and the interior angle at $Q$ add to $180^\circ$ .

Use the isosceles triangle fact

If $PQ=PR$ , then the angles at $Q$ and $R$ (the base angles) are equal.

Finish with the triangle angle sum

After you know the angles at $Q$ and $R$ , subtract them from $180^\circ$ to get the angle at $P$ .

Desmos Guide

Compute the interior angle at $Q$

In Desmos, define

q = 180 - 146

Express the vertex angle at $P$

Because the base angles are both $q$ , enter

p = 180 - 2q

Read the value of $p$

Desmos will evaluate $p$ , which is the measure of $\angle QPR$ . The correct choice matches this value.

Step-by-step Explanation

Find the interior angle at $Q$

Because $\overline{PQ}$ is extended through $Q$ to $S$ , $\angle SQR$ and $\angle PQR$ form a linear pair (they are supplementary).

So,

\angle PQR = 180^\circ - 146^\circ = 34^\circ.

Use the isosceles triangle angle relationship

Given $PQ = PR$ , $\triangle PQR$ is isosceles with base $\overline{QR}$ . Therefore, the base angles are equal:

\angle PQR = \angle PRQ = 34^\circ.

Use the triangle angle sum to find $\angle QPR$

The angles of a triangle sum to $180^\circ$ :

\angle QPR = 180^\circ - 34^\circ - 34^\circ = 112^\circ.

So the correct choice is $112^\circ$ .

Strategy

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Desmos Guide

Step-by-step Explanation

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Strategy

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Desmos Guide

Step-by-step Explanation