Question 388·Medium·Math

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In isosceles triangle $RST$ , the sides $RS$ and $RT$ are congruent. If $\angle SRT = 68^\circ$ , what is the measure of $\angle RST$ ?

The word isosceles signals two equal sides, so first identify the angles opposite those sides. The angle where the equal sides meet isn't one of the equal pair. Use the triangle's $180^\circ$ angle total to write an equation, then use a Desmos regression to solve it. Equal sides face equal angles.

Hints

Hint 1
In an isosceles triangle, equal sides face equal angles. Look across from $RS$ and $RT$ : which two corners must have the same angle measure?
Hint 2
The triangle angle sum says the three interior angles total $180^\circ$ . The given angle is at $R$ , so after accounting for it, how do you share what's left between the two equal angles?

Step-by-step

Match the base angles, then use the angle sum
Step 1
Identify the equal angles
An isosceles triangle has two equal sides. Equal sides face equal angles: $RS$ faces $\angle RTS$ , and $RT$ faces $\angle RST$ . So the angles at $T$ and $S$ match. Call each measure $x^\circ$ .
Step 2
Write the triangle angle equation
$\angle SRT$ is at $R$ , where the equal sides meet. It's the vertex angle, not one of the equal angles. The triangle angle sum says the three interior angles total $180^\circ$ , so $x+x+68=180$ .
Step 3
Solve for the requested angle
Type $x_1+x_1+68\sim180$ in Desmos. Use $x_1$ instead of $x$ and $\sim$ instead of $=$ so Desmos solves for the unknown rather than graphing it. Under PARAMETERS, it shows $x_1=56$ . Since $x$ is each base angle, $\angle RST$ measures $56^\circ$ . Choice D.

Lessons that teach this

SAT Geometry and TrigonometryIntermediateCoreUse triangle angle theorems

Question 388·Medium·Math

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