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

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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.) From a⁠niko.аі

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. W‍r⁠itten bу ⁠Аni⁠kο

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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. Рro‍реrt⁠у of Аn‌iко.aі

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. Thіѕ queѕtiοn is ‍fr‍om‌ Aniko

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 $96$ . P‍owerеd bу‍ Аn‌іk‍o

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Step-by-step Explanation

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Desmos Guide

Step-by-step Explanation