Exercises: Apply Similarity and Congruence to Solve Problems

Two triangles share a side. You know two angles of the first triangle are congruent to two angles of the second triangle. Which congruence criterion applies?

Triangle $PQR$ is similar to triangle $XYZ$ with a scale factor of 3. If $PQ = 6$, what is $XY$?

In the diagram, $DE \parallel BC$ with $D$ on $\overline{AB}$ and $E$ on $\overline{AC}$. Which similarity criterion proves $\triangle ADE \sim \triangle ABC$?

In parallelogram $ABCD$, diagonal $\overline{AC}$ is drawn. You know $AB \parallel CD$ and $BC \parallel AD$. Which criterion proves $\triangle ABC \cong \triangle CDA$?

In rectangle $ABCD$, $\overline{AB} \cong \overline{DC}$, $\overline{BC} \cong \overline{BC}$ (reflexive), and $\angle ABC \cong \angle DCB$ (both right angles). A student concludes: "By CPCTC, $\overline{AC} \cong \overline{DB}$." Identify the missing step and state the complete proof.

In right triangle $ABC$ with the right angle at $C$, the altitude from $C$ to hypotenuse $\overline{AB}$ meets $\overline{AB}$ at point $H$. If $AH = 4$ and $HB = 9$, find the length of altitude $CH$.

In $\triangle ABC$, point $D$ is on $\overline{AB}$ and point $E$ is on $\overline{AC}$ such that $DE \parallel BC$. Given $AD = 8$, $DB = 4$, and $AE = 12$, find $EC$.

In isosceles triangle $PQR$ with $PQ \cong PR$, $M$ is the midpoint of $\overline{QR}$. Which congruence criterion proves $\triangle PQM \cong \triangle PRM$?