Exercises: Prove Theorems Using Similarity

Two triangles share the same vertex angle $A$. If a second pair of angles is also equal, which similarity criterion guarantees the triangles are similar?

Line $DE$ is parallel to line $BC$, and line $AB$ is a transversal crossing both.
Which statement correctly describes the relationship between $\angle ADE$ and $\angle ABC$?

A right triangle has legs of length 5 and 12. What is the length of the hypotenuse?

In $\triangle ABC$, segment $DE \parallel BC$ with $D$ on $\overline{AB}$ and $E$ on $\overline{AC}$.
Given $AD = 6$, $DB = 4$, and $AE = 9$, find $EC$.

In $\triangle PQR$, segment $ST \parallel QR$ with $S$ on $\overline{PQ}$ and $T$ on $\overline{PR}$.
Given $PS = 5$, $SQ = 3$, and $PT = 7.5$, find $TR$.

In $\triangle ABC$, point $D$ lies on $\overline{AB}$ and point $E$ lies on $\overline{AC}$.
Given $AD = 8$, $DB = 4$, $AE = 10$, and $EC = 5$. Is $DE \parallel BC$?

Right triangle $\triangle ABC$ has a right angle at $C$. The altitude from $C$ meets
hypotenuse $\overline{AB}$ at point $D$, with $AD = 4$ and $DB = 9$.
Using the similarity of $\triangle ACD$ and $\triangle ABC$, find $AC$.

In right $\triangle ABC$ with right angle at $C$, altitude $\overline{CD}$ meets
hypotenuse $\overline{AB}$ at $D$ with $AD = 3$ and $DB = 12$.
Using $\triangle ACD \sim \triangle CBD$, find the altitude length $CD$.