In triangle , the sides and are congruent. Which statement must be true?

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SAT Lines, Angles & Triangles Question 37 | Free SAT Prep | Aniko

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Question 37·Easy·Lines, Angles, and Triangles

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In triangle $PQR$ , the sides $PQ$ and $PR$ are congruent. Which statement must be true?

For problems that say two sides of a triangle are congruent, immediately recognize an isosceles triangle and recall that angles opposite congruent sides are congruent. Next, systematically match each side to its opposite angle (using the vertex in the middle of the angle name) and then choose the answer that states the congruence between the angles opposite the given congruent sides. This side–angle matching approach is fast and avoids guessing based on the diagram or on angle positions alone.

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Name the triangle type

If a triangle has two congruent sides, what special type of triangle is it called?

Recall the special angle property

In that kind of triangle, what can you say about the angles that are opposite the two congruent sides?

Match sides with angles

Which angle is opposite side $PQ$ ? Which angle is opposite side $PR$ ? Once you find those two angles, look for the choice that states they are congruent.

Desmos Guide

Create a model isosceles triangle

In Desmos (using a graph), pick a convenient point for $P$ , such as $(0,3)$ . Then plot $Q=(-2,0)$ and $R=(2,0)$ . Visually confirm that $PQ$ and $PR$ are the same length (the triangle is symmetric about the $y$ -axis).

Label and inspect angles

Use the geometry tools (or just visually inspect the graph) to focus on the angles at $Q$ and $R$ and the angle at $P$ . Observe which two angles in your picture look equal; then compare that pair of vertex letters to the answer choices to decide which statement matches what you see.

Step-by-step Explanation

Identify the type of triangle

The problem says $PQ$ and $PR$ are congruent (equal in length). A triangle with two congruent sides is called an isosceles triangle.

So triangle $PQR$ is isosceles with sides $PQ$ and $PR$ as the congruent legs.

Recall the key property of isosceles triangles

In an isosceles triangle, the angles opposite the two congruent sides are congruent. These equal angles are often called the base angles of the triangle.

So to use this property, we must find which angles are opposite sides $PQ$ and $PR$ .

Match sides to their opposite angles

In triangle notation, the angle at a vertex is opposite the side that does not include that vertex:

Angle $PQR$ is at vertex $Q$ , so it is opposite side $PR$ .
Angle $PRQ$ is at vertex $R$ , so it is opposite side $PQ$ .
Angle $QPR$ is at vertex $P$ , so it is opposite side $QR$ .

Now we know which angles correspond to which sides.

Use the isosceles property to select the correct statement

Since $PQ$ and $PR$ are congruent, their opposite angles must also be congruent. From Step 3, these opposite angles are:

Opposite $PQ$ : angle $PRQ$ .
Opposite $PR$ : angle $PQR$ .

Therefore, the statement that must be true is $\angle PQR \cong \angle PRQ$ , which matches choice D.

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Step-by-step Explanation

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