Question 324·Easy·Math

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In $\triangle LMN$ , $\angle L$ is twice the measure of $\angle M$ , and $\angle N$ is $60^\circ$ greater than $\angle M$ . What is the measure of $\angle L$ ?

The cue is that one triangle’s angles are described in terms of another angle. Use the triangle angle sum: name one angle $x$ , translate the others, and enter their $180^\circ$ equation as a Desmos regression. Keep track of which angle the question asks for. The value of $x$ may be only an intermediate measure.

Hints

Hint 1
Let $x$ be the measure of $\angle M$ . Twice that measure gives $\angle L$ ; $60^\circ$ greater gives $\angle N$ . Write both in terms of $x$ before using the triangle’s angle sum.
Hint 2
A triangle’s interior angles, the angles inside it, total $180^\circ$ . Put all three angle expressions into that equation. Once you find $x$ , which expression gives the angle the question asks for?

Step-by-step

Use the triangle angle sum
Step 1
Write each angle in terms of one unknown
Let $x$ be the measure of $\angle M$ in degrees. Here $m\angle M$ means the measure of that angle. Since $\angle L$ is twice $\angle M$ and $\angle N$ is $60^\circ$ greater than $\angle M$ ,
$m\angle M=x^\circ,\qquad m\angle L=2x^\circ,\qquad m\angle N=(x+60)^\circ.$
Step 2
Make an equation from the triangle
The triangle angle sum says the three angles inside a triangle add to $180^\circ$ , so: $2x+x+(x+60)=180$ .Use the whole expression $x+60$ for $\angle N$ , not $60$ alone.
Step 3
Find the measure of angle M
Type $2x_1+x_1+(x_1+60)\sim180$ in Desmos. Use $x_1$ for the unknown and $\sim$ to ask Desmos to match the two sides. Under PARAMETERS, it reports $x_1=30$ . That is $\angle M$ ’s measure, not yet the requested angle.
Step 4
Find the requested angle
Because $\angle L$ is twice $\angle M$ , add $2x_1$ below the regression. Desmos prints $60$ , so the measure of $\angle L$ is $60^\circ$ . Choice C.

Lessons that teach this

SAT Geometry and TrigonometryIntermediateCoreUse triangle angle theorems

Question 324·Easy·Math

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