Question 352·Medium·Math

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In $\triangle ABC$ , $\overline{AB}=\overline{AC}$ , and side $\overline{AC}$ is extended through $C$ to point $D$ . If the measure of exterior angle $\angle BCD$ is $132^\circ$ , what is $m\angle B$ ?

A side extended past a vertex creates an exterior angle, outside the triangle. That angle and the interior angle beside it form a straight $180^\circ$ angle, so subtract to find the inside angle. In an isosceles triangle, which has two equal sides, match the angles across from those sides. Halving the exterior angle would assume the wrong pair of angles is equal.

Hints

Hint 1
In an isosceles triangle, equal sides face equal angles. Side $\overline{AB}$ faces $\angle C$ , while side $\overline{AC}$ faces $\angle B$ . Which angle matches the inside angle at $C$ ?
Hint 2
An exterior angle and the interior angle beside it fill a straight $180^\circ$ angle. Subtract the outside angle to find the inside angle at $C$ . What measure do you get?

Step-by-step

Use the straight angle, then match opposite angles
Step 1
Match the equal sides to their opposite angles
No Desmos needed. These angle facts take one subtraction and a match of equal angles. The given $\overline{AB}=\overline{AC}$ makes $\triangle ABC$ isosceles, meaning it has two equal sides. $\overline{AB}$ faces $\angle C$ , and $\overline{AC}$ faces $\angle B$ . Equal sides face equal angles. Match them:
$m\angle B=m\angle BCA.$
Step 2
Form the straight-angle equation
Since $\overline{AC}$ continues through $C$ to $D$ , the exterior angle $BCD$ and the inside angle $BCA$ form a straight angle measuring $180^\circ$ . Their measures add to $180^\circ$ :
$m\angle BCA+132^\circ=180^\circ.$
Step 3
Find the inside angle at C
Subtract the given outside angle:
$m\angle BCA=180^\circ-132^\circ=48^\circ.$
The equal sides make $\angle B$ match $\angle BCA$ :
$m\angle B=m\angle BCA=48^\circ.$
So $m\angle B$ is $48^\circ$ . Choice B.

Lessons that teach this

SAT Geometry and TrigonometryIntermediateCoreUse triangle angle theorems

Question 352·Medium·Math

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Step-by-step

Lessons that teach this

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Strategy

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Step-by-step

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