Skill: Apply properties of triangles to determine unknown measures

Use the facts given in the problem to write equations. Do not rely on how the triangle looks in a picture.

• The interior angles of a triangle add to $180^\circ$.
• Equal sides have equal opposite angles.
• An exterior angle equals the sum of the two remote interior angles.

Example

In $\triangle ABC$, $AB=AC$. Side $BC$ is extended past $C$ to point $D$. If $\angle B=50^\circ$, find $\angle A$, $\angle C$, and the exterior angle $\angle ACD$.

Step 1: Use the equal sides

Since $AB=AC$, the opposite angles are equal:

$$\angle B=\angle C$$

Because $\angle B=50^\circ$,

$$\angle C=50^\circ$$

Step 2: Use the triangle angle sum

The three interior angles add to $180^\circ$:

$$\angle A+\angle B+\angle C=180^\circ$$

Substitute the known measures:

$$\angle A+50^\circ+50^\circ=180^\circ$$ $$\angle A=80^\circ$$

Step 3: Find the exterior angle

The exterior angle $\angle ACD$ has remote interior angles $\angle A$ and $\angle B$. Therefore,

$$\angle ACD=\angle A+\angle B$$ $$\angle ACD=80^\circ+50^\circ=130^\circ$$

So the unknown measures are

$$\boxed{\angle A=80^\circ,\quad \angle C=50^\circ,\quad \angle ACD=130^\circ}$$