# Similar Triangles

Triangles that are **congruent** are exactly the same. If $\triangle ABC \cong \triangle DEF$ (this is the notation that shows that triangles are congruent), then all of the sides and angles in $\triangle ABC$ are equal to all of the corresponding sides and angles in $\triangle DEF$.

Triangles that are **similar** have corresponding angles with the same angle measures and corresponding sides that are proportional. If $\triangle ABC \sim \triangle DEF$ (this is the notation that shows that triangles are similar), then all of the angles in $\triangle ABC$ are equal to all of the corresponding angles in $\triangle DEF$, and all of the sides in $\triangle ABC$ are proportional to all of the corresponding sides in $\triangle DEF$.

Consequently, when you know that triangles are similar, it's easy to deduce their angle measures (just make sure that you match up the correct angles when you set up your proportions!).

So if you know that in the figure below, $\triangle ABC \sim \triangle DEF$, then you know that:

$\measuredangle B= \measuredangle E$ and $\measuredangle A= \measuredangle D$ and $\measuredangle C= \measuredangle F$.

So, if $\measuredangle A=30$, then $\measuredangle E=30$.

We also know that the corresponding sides of similar triangles are proportional.

$\dfrac{AB}{AC}=\dfrac{DE}{DF}$

So, if side $AB=12$ and side $AC=9$, and side $DE=4$, then you can use a proportion to find the length of side $DF$.

$$\eqalign{\dfrac{AB}{AC}=\dfrac{DE}{DF}\\\dfrac{12}{9}=\dfrac{4}{x}\\12x=36\\x=3\\}$$

So, $DF$ is 3 units long.

Often, tests won't tell you that triangles are similar. You're supposed to remember a geometry rule that says that when a triangle is intersected by a line that is parallel to one of its sides, then the two triangles that are formed (the entire triangle and the smaller one made by the parallel line) are similar.

In the to figure to the right, if $\overline{DE} \parallel \overline{AC}$, then $\triangle ABC \sim \triangle DBE$.

So, you know that the corresponding angles of $\triangle ABC$ equal the corresponding angles of $\triangle DBE$ and the corresponding sides of $\triangle ABC$ are proportional to the corresponding sides of $\triangle DBE$

Note: a line does not have to be parallel to the base in order to create similar triangles. Any line that is parallel to any of a triangle's sides will form similar triangles. In the lower figure to the right, if $\overline{DE} \parallel \overline{AC}$, then $\triangle ABC \sim \triangle DBE$ and corresponding angles are equal and corresponding sides are proportional.

#### Practice Problems:

## Similar Triangles

In the figure above, $\triangle ABC \sim \triangle DEF$. $\overline{AB}=30$, $\overline{BC}=25$, $\overline{AC}=27$, $\overline{DE}=20$, $\measuredangle{A}=65$, and $\measuredangle{F}=70$.

- What is $\measuredangle C$?
- What is $\measuredangle D$?
- What is $\measuredangle E$?
- What is $\measuredangle B$?
- What is the length of $\overline{EF}$?
- What is the length of $\overline{DF}$?

In the figure above, $\overline{DE} \parallel \overline{AC}$. $\overline{AB}=10$, $\overline{BC}=12$, $\overline{AC}=8$, $\overline{DE}=6$, $\measuredangle{A}=90$, and $\measuredangle{C}=40$.

- What is $\measuredangle BED$?
- What is $\measuredangle BDE$?
- What is $\measuredangle B$?
- What is the length of $\overline{BE}$?
- What is the length of $\overline{BD}$?

In the figure above, $\overline{DE} \parallel \overline{AB}$. $\overline{AB}=42$, $\overline{BC}=30$, $\overline{AC}=27$, $\overline{DE}=36$, $\measuredangle{A}=50$, and $\measuredangle{E}=70$.

- What is $\measuredangle C$?
- What is $\measuredangle EDC$?
- What is $\measuredangle B$?
- What is the perimeter of $\triangle DEC$?

#### Answer Key: