The similarity theorems

The similarity theorems say which data are enough to know that two triangles are similar, without checking all three angles and all three sides.

• They are like the congruence theorems, and even have the same names: AA, SAS, SSS.
• In congruence the sides are equal; in similarity they are in the same ratio.

The similarity theorem AA (angle, angle): if two angles of one triangle are equal to two angles of another triangle, the triangles are similar.

• The third angle is equal too: the angles of a triangle add up to $\displaystyle 180^\circ$, so it is what the other two leave.
• No side needs to be known.

The similarity theorem SAS (side, angle, side): if two sides of one triangle are in the same ratio to two sides of another triangle, and the angles between them are equal, the triangles are similar.

• An example: one triangle has sides of $\displaystyle 4$ and $\displaystyle 6$ with an angle of $\displaystyle 50^\circ$ between them; another has sides of $\displaystyle 6$ and $\displaystyle 9$ with an angle of $\displaystyle 50^\circ$ between them.
• $\displaystyle {\frac{6}{4}}={\frac{9}{6}}={1.5}$, and the angles between them are equal: the triangles are similar, with the ratio of similarity $\displaystyle 1.5$.

The similarity theorem SSS (side, side, side): if the three sides of one triangle are in the same ratio to the three sides of another triangle, the triangles are similar.

• Sort each triangle’s sides from the shortest to the longest, and divide each side by the side in its place in the other triangle.
• An example: $\displaystyle 3$, $\displaystyle 5$, $\displaystyle 6$ and $\displaystyle 6$, $\displaystyle 10$, $\displaystyle 12$: every quotient is $\displaystyle 2$, so the triangles are similar.

Are the triangles similar? If so, write by which similarity theorem.

Two angles of one triangle: $\displaystyle 40^\circ$ and $\displaystyle 75^\circ$.
Two angles of another triangle: $\displaystyle 75^\circ$ and $\displaystyle 65^\circ$.

The sides of one triangle: $\displaystyle 4$, $\displaystyle 5$, $\displaystyle 7$.
The sides of another triangle: $\displaystyle 12$, $\displaystyle 15$, $\displaystyle 21$.

One triangle has sides of $\displaystyle 3$ and $\displaystyle 5$, with an angle of $\displaystyle 60^\circ$ between them.
Another triangle has sides of $\displaystyle 6$ and $\displaystyle 10$, with an angle of $\displaystyle 60^\circ$ between them.

The sides of one triangle: $\displaystyle 2$, $\displaystyle 3$, $\displaystyle 4$.
The sides of another triangle: $\displaystyle 4$, $\displaystyle 6$, $\displaystyle 9$.

A segment parallel to a side of a triangle cuts off a small triangle similar to it.

In triangle $\displaystyle ABC$, the point $\displaystyle D$ is on the side $\displaystyle AB$, the point $\displaystyle E$ is on the side $\displaystyle AC$, and $\displaystyle DE\parallel BC$.
Prove: $\displaystyle \triangle ADE\sim\triangle ABC$.

The proof below shows that two angles of triangle $\displaystyle ADE$ are equal to two angles of triangle $\displaystyle ABC$, and uses the similarity theorem AA:

In the same drawing, given: $\displaystyle {AD}={3}$, $\displaystyle {AB}={9}$, $\displaystyle {BC}={12}$.
How long is $\displaystyle DE$?

• By the order of the letters in $\displaystyle \triangle ADE\sim\triangle ABC$, the side $\displaystyle AD$ corresponds to $\displaystyle AB$, and $\displaystyle DE$ to $\displaystyle BC$.
• A proportion: $\displaystyle {\frac{DE}{12}}={\frac{3}{9}}$.
• Cross-multiplying: $\displaystyle {9\cdot DE}={36}$, so $\displaystyle {DE}={4}$.

The segments $\displaystyle AB$ and $\displaystyle CD$ cross at the point $\displaystyle O$.
Given: $\displaystyle AC\parallel BD$.

a. Prove: $\displaystyle \triangle AOC\sim\triangle BOD$.

Also given: $\displaystyle {AO}={4}$, $\displaystyle {OB}={6}$, $\displaystyle {AC}={5}$.
Find $\displaystyle BD$.

A tree’s height can be measured without climbing it: measure its shadow, and the shadow of a stick whose height is known.

A stick $\displaystyle 1.5$ metres long stands upright on the ground, and its shadow is $\displaystyle 2$ metres long.
At the same hour, the shadow of a tree nearby is $\displaystyle 12$ metres long.
How tall is the tree?

• The sun is very far away, so its rays reaching the stick and the tree are parallel. The angle between a sun ray and the ground ($\displaystyle \alpha$) is the same at the stick and at the tree.
• The stick and the tree are upright, so the angles at the ground below them are equal too: both are right.
• By the similarity theorem AA the two triangles are similar, so $\displaystyle {\frac{h}{1.5}}={\frac{12}{2}}$.
• $\displaystyle {\frac{12}{2}}={6}$, so $\displaystyle {h}={1.5\cdot6}={9}$: the tree is $\displaystyle 9$ metres tall.

Noa, who is $\displaystyle 1.6$ metres tall, stands near a building.
Her shadow is $\displaystyle 2.4$ metres long, and at the same hour the building’s shadow is $\displaystyle 30$ metres long.
How tall is the building? Write a proportion and solve it.

In triangle $\displaystyle ABC$, the point $\displaystyle D$ is on the side $\displaystyle AB$, the point $\displaystyle E$ is on the side $\displaystyle AC$, and $\displaystyle DE\parallel BC$.
Given: $\displaystyle {AD}={4}$, $\displaystyle {DB}={6}$, $\displaystyle {DE}={5}$.
Find $\displaystyle BC$.