Exercises: Prove Theorems About Triangles

Lines $\ell$ and $m$ are parallel, cut by a transversal $t$. Which theorem guarantees that the alternate interior angles are congruent?

Triangle $ABC$ has $AB = AC$. An angle bisector is drawn from $A$ to point $D$ on $\overline{BC}$. Which congruence criterion proves $\triangle ABD \cong \triangle ACD$?

Point $D$ is the midpoint of $\overline{AB}$, where $A = (0, 0)$ and $B = (8, 0)$. What are the coordinates of $D$?

In triangle $PQR$, $m\angle P = 47^\circ$ and $m\angle Q = 68^\circ$. Find $m\angle R$.

Which theorem from HSG.CO.C.9 is the essential step in the proof of the Triangle Angle Sum Theorem?

Isosceles triangle $ABC$ has $AB = AC$. The vertex angle measures $\angle A = 50^\circ$. Find $m\angle B$.

In equilateral triangle $XYZ$, all three sides are equal. What is $m\angle X$?

In triangle $ABC$, $D$ is the midpoint of $\overline{AB}$ and $E$ is the midpoint of $\overline{AC}$. If $BC = 24$, find the length of midsegment $\overline{DE}$.