Angle-Angle Similarity Theorem & Side-Side-Side Similarity Theorem Explained

In this lesson some conditions will be developed that guarantee the similarity of triangles.

Catch-Up and Review

Here are a few recommended readings before getting started with this lesson.

The concept of similarity.
Conditions for similarity of polygons.
The concept of congruence.
Conditions that guarantee the congruence of triangles.

Challenge

Using Similarity of Triangles to Solve Problems

Find the ratio of the length of a diagonal and a side of a regular pentagon.

Regular pentagon ABCDE with all five diagonals drawn

Explore

Investigating Triangles With Two Pairs of Congruent Angles

In the applet, move points $Q$ and $R .$ The applet copies $∠ B$ and $∠ C$ on $\overline{Q R}$ and labels the intersection of the corresponding rays by $P .$

What do you notice about the angles and side lengths of the two triangles?
Are the two triangles similar?

Discussion

Angle-Angle Similarity Theorem

Two polygons are similar if corresponding angles are congruent and corresponding sides are proportional. For triangles, the congruence of two angles already implies similarity.

If two angles of a triangle are congruent to two angles of another triangle, then the triangles are similar.

Triangles ABC and DEF with congruent angles marked at A and D and B and E.

If $∠ A ≅ ∠ D$ and $∠ B ≅ ∠ E,$ then $△ A B C \sim △ D E F .$

Proof

Angle-Angle Similarity Theorem

Consider two triangles $△ A B C$ and $△ D E F,$ whose two corresponding angles are congruent.

These triangles can be proven to be similar by identifying a similarity transformation that maps one triangle onto the other. First, $△ D E F$ can be dilated with the scale factor $k = \frac{A B}{D E}$ about $D,$ forming the new triangle $△ D E^{'} F^{'} .$

Since a dilation is a similarity transformation, it can be concluded that

△ D E^{'} F^{'}

and

△ D E F

are similar triangles. Next, it has to be proven that a rigid motion that maps

△ D E^{'} F^{'}

onto

△ A B C

exists. The corresponding angles of similar figures are congruent, so

∠ E^{'}

and

∠ E

are congruent angles.

∠ E^{'} ≅ ∠ E

Additionally, since

∠ E

is congruent to

∠ B,

by the Transitive Property of Congruence,

∠ E^{'}

is congruent to

∠ B .

∠ E^{'} ≅ ∠ B

The ratios of the corresponding side lengths of similar polygons are the same and equal to the scale factor.

\frac{D E ^{'}}{D E} = k

In this case, the scale factor

k

\frac{A B}{D E} .

Applying the Transitive Property of Equality, an equation can be formed and simplified.

\frac{D E ^{'}}{D E} = \frac{A B}{D E} ⇕ D E^{'} = A B

It has been obtained that the two angles and the included side of

△ D E^{'} F^{'}

are congruent to the corresponding two angles and the included side of

△ A B C .

Therefore, by the Angle-Side-Angle (ASA) Congruence Theorem, the two triangles are congruent.

△ A B C ≅ △ D E^{'} F^{'}

Since congruent figures can be transformed into each other using rigid motions, and

△ A B C

and

△ D E^{'} F^{'}

are congruent triangles, there is a rigid motion placing

△ D E^{'} F^{'}

onto

△ A B C .

The combination of this rigid motion and the dilation performed earlier forms a similarity transformation that maps

△ D E F

onto

△ A B C .

Therefore, it can be concluded that $△ A B C$ and $△ D E F$ are similar triangles.

$△ A B C \sim △ D E F$

The proof is now complete.

Example

Solving Problems Using Angle-Angle-Similarity Theorem

The Grim Reaper, who is $5$ feet tall, stands $16$ feet away from a street lamp at night. The Grim Reaper's shadow cast by the streetlamp light is $8$ feet long. How tall is the street lamp?

Hint

Both the lamp post and the Grim Reaper stand vertically on horizontal ground.

Solution

A sketch of the situation is helpful for finding the solution. Under the assumption that the lamp post and the Grim Reaper make right angles in relation to the ground, two right triangles can be drawn. The unknown height of the lamp post is labeled as $x .$

As these triangles both have a right angle and share the angle on the right-hand side, they are similar by the Angle-Angle (AA) Similarity Theorem. Notice that the base of the larger triangle measures to be $24$ feet.

Since the triangles are similar, the ratios between corresponding side lengths are the same.

\frac{x}{5} = \frac{2 4}{8}

The solution of this equation defines the value of

x

— the height of the street lamp.

\frac{x}{5} = \frac{2 4}{8}

Solve for

x

CalcQuot

Calculate quotient

\frac{x}{5} = 3

MultEqn

$LHS \cdot 5 = RHS \cdot 5$

x = 15

The street lamp at $15$ feet high towers over The Grimp Reaper.

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