In isosceles triangle , . If , what is , in degrees? (Disregard the degree symbol when entering your answer.)

Solution available with step-by-step explanation

SAT Lines, Angles & Triangles Question 35 | Free SAT Prep | Aniko

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Question 35·Medium·Lines, Angles, and Triangles

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In isosceles triangle $PQR$ , $\overline{PQ} \cong \overline{PR}$ . If $m\angle Q = 42^\circ$ , what is $m\angle P$ , in degrees? (Disregard the degree symbol when entering your answer.)

Your answer

(Express the answer as an integer)

For isosceles triangle questions, first identify which sides are congruent and then match them to the angles they are opposite; those opposite angles are equal (the base angles). Use any given angle to find its congruent partner, then apply the triangle angle sum rule ( $180^\circ$ total) to solve for the remaining angle with a quick subtraction. This avoids unnecessary algebra and lets you solve in just a couple of steps.

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Think about which angles are equal

In an isosceles triangle, the angles opposite the congruent sides are equal. Look at which sides are marked congruent ( $\overline{PQ}$ and $\overline{PR}$ ) and decide which angles must match.

Use the given angle measure

Once you know which angle is equal to $\angle Q$ , assign it the same measure of $42^\circ$ .

Apply the triangle angle sum

The three interior angles of any triangle add up to $180^\circ$ . Add the two known angles and subtract from 180 to find the remaining angle at $P$ .

Desmos Guide

Verify the angle calculation

In Desmos, type the expression 180 - 42 - 42 into the input line. The value that Desmos outputs is the measure of $\angle P$ in degrees.

Step-by-step Explanation

Identify which sides and angles are equal

You are told that $\overline{PQ} \cong \overline{PR}$ . When two sides in a triangle are congruent, the angles opposite those sides are also congruent.

Side $PQ$ is opposite angle $R$ .
Side $PR$ is opposite angle $Q$ .

So $\angle Q$ and $\angle R$ are congruent (they have the same measure). These are called the base angles of the isosceles triangle, and $\angle P$ is the vertex angle.

Use the given angle measure to find the other base angle

You are given that $m\angle Q = 42^\circ$ .

Since $\angle Q$ and $\angle R$ are congruent in this isosceles triangle, it follows that

m\angle R = 42^\circ.

Use the triangle angle sum to find angle P

The sum of the interior angles in any triangle is $180^\circ$ .

So for triangle $PQR$ :

m\angle P + m\angle Q + m\angle R = 180^\circ \\[0.7em] m\angle P + 42^\circ + 42^\circ = 180^\circ

Combine the two known angles:

42^\circ + 42^\circ = 84^\circ

m\angle P + 84^\circ = 180^\circ.

Subtract $84^\circ$ from both sides:

m\angle P = 180^\circ - 84^\circ = 96^\circ.

Therefore, the measure of $\angle P$ is $96$ .

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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