1.3: Similar Triangles

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Roy Simpson
Cosumnes River College

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Note to the Instructor (click to expand): This entire section focuses solely on similar triangles and their applications.

To succeed in this section, you'll need to use some skills from previous courses. While you should already know them, this is the first time they've been required. You can review these skills in CRC's Corequisite Codex. If you have a support class, it might cover some, but not all, of these topics.

Prerequisite Skills and Support Topics (click to expand)

Arithmetic
- Fractions: Working with similar triangles often requires a strong grasp of fractions and fractional arithmetic.
Solving Equations
- Solving Rational Equations: This skill is fundamental to this section. It is required to solve the proportions that arise from similar triangles, as seen in the "Skills Refresher" and throughout the examples and homework (e.g., Example 4, where the proportion $ \frac{x}{4} = \frac{15}{6} $ is solved).

The following is a list of learning objectives for this section.

Learning Objectives (click to expand)

Determine when two triangles are congruent.
Use proportion to find a side given two similar triangles.
Solve application problems involving similar right triangles.

Congruent Triangles

We begin this section by identifying what it means for two triangles to be congruent.

Definition: Congruent Triangles

In simple terms, two triangles are considered congruent if they have the same size and shape. For example, it looks as though $ \triangle A B C $ and $ \triangle D E F $ in Figure $ \PageIndex{ 1 } $ are identical in both shape and size ($ \triangle DEF $ has been rotated from $ \triangle ABC $), so they are congruent. We write $ \triangle A B C \cong \triangle D E F $.

Screen Shot 2022-09-09 at 1.54.27 AM.png — **Figure $ \PageIndex{ 1 } $:** *Congruent triangles $ \triangle ABC $ and $ \triangle DEF $*

Example $ \PageIndex{ 1 } $

The two triangles in Figure $ \PageIndex{ 2 } $ below are congruent. List the corresponding parts, and find the angles $\theta$, $\phi$, and $\chi$, and side $z$.

Screen Shot 2022-09-09 at 1.56.27 AM.png — **Figure $ \PageIndex{ 2 } $**

Solution: In these triangles, $\angle B=\angle D$ because they are both right angles and $\angle B C A= \angle D C E$ because they are vertical angles, so $\theta=25^{\circ}$. The third angles, $\angle A$ and $\angle E$, must also be equal, so $\phi=\chi=65^{\circ}$. (Do you see why?) The sides opposite each pair of corresponding angles are equal, so $A B=D E, B C=C D$ and $A C=C E$. In particular, we find that $z=9$.

Checkpoint $\PageIndex{1}$

The two triangles in Figure $ \PageIndex{ 3 } $ below are congruent. Find the values of $\alpha, \beta$, and $\gamma$.

Screen Shot 2022-09-09 at 1.58.19 AM.png — **Figure $ \PageIndex{ 3 } $**

Answer: $ \alpha=85^{\circ}, \beta=75^{\circ}, \gamma=85^{\circ}$

Recall that the altitude of a triangle is the segment from one vertex of the triangle perpendicular to the opposite side.

Example $ \PageIndex{ 2 } $

Show that the altitude of an equilateral triangle divides it into two congruent right triangles.

Solution: Figure $ \PageIndex{ 4 } $
$Screen Shot 2022-09-09 at 2.05.01 AM.png$

Consider, for example, an equilateral triangle of side 8 inches, as shown above. The altitude is perpendicular to the base, so each half of the original triangle is a right triangle. Because each right triangle contains a $60^{\circ}$ angle, the remaining angle in each triangle must be $90^{\circ}-60^{\circ}=30^{\circ}$. Both triangles have a side of length 8 between the angles of $30^{\circ}$ and $60^{\circ}$, so they are congruent. (Consequently, the short sides of the congruent triangles are equal, so each is half the original base.)

For the following Checkpoint, you should recall the following definition from your Geometry class.

Definition: Parallelogram

Checkpoint $\PageIndex{2}$

The diagonal of a parallelogram divides it into two congruent triangles, as shown in Figure $ \PageIndex{ 5 } $ below. List the corresponding parts of the two triangles and explain why each pair is equal.

Screen Shot 2022-09-09 at 2.08.04 AM.png — **Figure $ \PageIndex{ 5 } $**

Answer: $\angle B C A=\angle C A D$ and $\angle B A C=\angle A C D$ because they are alternate interior angles. If two angles in a triangle are equal, so is the third pair, so $\angle B=\angle D$. $B C=A D$ and $A B=C D$ because they are opposite sides of a parallelogram, and $A C=A C$.

Similar Triangles

The most common comparison of triangles in both Trigonometry and Calculus will be with similar triangles.

Definition: Similar Triangles

The definition of similar triangles looks almost like the definition of congruent triangles; however, similarity between triangles is a weaker form of congruence - the triangles have the same shape but might be sized differently. This flexibility grants us much power in mathematics.

Fun Fact!

Congruent triangles are similar, but similar triangles need not be congruent.

The corresponding angles between similar triangles are equal, and the corresponding sides are proportional. We can think of one similar triangle as an enlargement or a reduction of the other (see Figure $ \PageIndex{ 6 } $ below).

Screen Shot 2022-09-09 at 2.12.03 AM.png — **Figure $ \PageIndex{ 6 } $:** *Proportional relationships between similar triangles*

To decide whether two triangles are similar, we must verify only one of the two similarity conditions (the other condition will be true automatically).

Theorem: Similarity Conditions

Example $ \PageIndex{ 3 } $

Which of the pairs of triangles shown below are similar?

Figure $ \PageIndex{ 7 } $
$Screen Shot 2022-09-09 at 2.13.32 AM.png$
Figure $ \PageIndex{ 8 } $
$Screen Shot 2022-09-09 at 2.13.46 AM.png$
Figure $ \PageIndex{ 9 } $
$Screen Shot 2022-09-09 at 2.14.06 AM.png$

Solutions

We will check whether the corresponding sides are proportional. We compute the ratios of the corresponding sides, making sure to write each ratio in the same order, $\frac{\text { larger triangle }}{\text { smaller triangle }}$. (The other order, $\frac{\text { smaller triangle }}{\text { larger triangle }}$, would also work, as long as we use the same order for all the ratios.)\[ \text{shorter legs: }\dfrac{9}{6} \quad \text{longer legs: } \dfrac{12}{8} \quad \text{hypotenuses: } \dfrac{15}{10} \nonumber \]Because all of these ratios are equal to 1.5, the triangles are similar.
The ratios of corresponding sides are not equal: the ratio of the longest sides is $\frac{6}{5}$, but the ratio of the smallest sides is $\frac{4}{3}$. The triangles are not similar.
The missing angle of the first right triangle is $48^{\circ}$, and the missing angle in the second right triangle is $42^{\circ}$, so three pairs of angles match. The triangles are similar.

Checkpoint $\PageIndex{3}$

Are the triangles below similar? Explain why or why not in each case.

Figure $ \PageIndex{ 10 } $
$Screen Shot 2022-09-09 at 2.19.31 AM.png$
Figure $ \PageIndex{ 11 } $
$Screen Shot 2022-09-09 at 2.19.55 AM.png$

Answers

The triangles are similar because $\frac{4}{6} = \frac{6}{9} = \frac{8}{12}$, so the sides are proportional.
The third angle in both triangles is $80^{\circ}$, so the triangles are similar because their corresponding angles are equal.^a

In part (b) of Checkpoint $ \PageIndex{ 3 } $, the third angle in each triangle must be $80^{\circ}$ because the sum of the angles is $180^{\circ}$. Thus, we only need to show that two pairs of angles are equal to show that two triangles are similar.

Using Proportions with Similar Triangles

Figure $ \PageIndex{ 12 } $ shows a parallelogram $A B C D$ and two triangles, $\triangle A B E$ and $\triangle F C E$. Can we find the unknown lengths $x$ and $y$ in the larger triangle?

Screen Shot 2022-09-09 at 2.29.17 AM.png — **Figure $ \PageIndex{ 12 } $**

First, note that two pairs of corresponding angles in the triangles are equal: $\angle B E A$ and $\angle F E C$ are vertical angles, and $\angle E F C$ and $\angle B A E$ are alternate interior angles. However, if two pairs of corresponding angles are equal, the third pair must also be equal. This means that the two triangles are similar, and we can use the fact that their corresponding sides are proportional to find $x$ and $y$.

Example $ \PageIndex{ 4 } $

Find the value of $x$ in Figure $ \PageIndex{ 12 } $.

Solution: We see that $x$ is the length of the shortest side in $\triangle A B E$. We know the short side in $\triangle F C E$ and the lengths of the medium sides in each triangle. If we form the ratios of the short and medium sides, we obtain the following proportion.\[\dfrac{\text { larger triangle }}{\text { smaller triangle }}: \dfrac{x}{4}=\dfrac{15}{6} \nonumber\]To solve the proportion, we multiply both sides by the LCD to get\[ \begin{array}{rrrclcl}
& & 6 x & = & 4(15) & \quad & \\
\scriptscriptstyle\mathrm{Arithmetic} & \implies & 6x & = & 60 & \quad & (\text{multiply}) \\
\scriptscriptstyle\xcancel{\mathrm{Arithmetic}} \to \mathrm{Algebra} & \implies & x & = & 10 & \quad & (\text{divide both sides by }6) \\
\end{array} \nonumber \]

What does the $ \mathrm{Arithmetic} $ and $ \xcancel{\textrm{Arithmetic}} $ mean?

To get you used to the Mathematical Mantra, every once in a while I will include the "thought processes" during solutions. Just to review, we perform Mathematics in the order we learned it - Arithmetic, Algebra, Trigonometry, $ \ldots $. At each step during a "mechanical" solution process, we should pause and ask ourselves if there is some simple Arithmetic to be done. If not, we move on to any Algebra that can be done. If there is no Algebra to be done, we then move on to Trigonometry.

Thus,

"$ \mathrm{Arithmetic} \implies$" in a solution means that there is some Arithmetic we could do to clean up the previous expression
"$ \xcancel{\mathrm{Arithmetic}} \to \mathrm{Algebra} \implies$" means there isn't any Arithmetic but there is some Algebra we could use to clean up the previous expression or equation
"$ \xcancel{\mathrm{Arithmetic}} \to \xcancel{\mathrm{Algebra}} \to \mathrm{Trigonometry} \implies$" means there isn't any Arithmetic nor is there any Algebra that can be done to clean up the previous step, so we need to start looking at doing something from Trigonometry.

Checkpoint $\PageIndex{4}$

Find the value of $y$ in the previous example.

Answer: $y=20$

Similar Right Triangles

If two right triangles have one pair of corresponding acute angles with the same measure, they are similar. We can use this fact about right triangles to make indirect measurements.

Example $ \PageIndex{ 5 } $

Ivan wants to know the height of a particular building. He asks Kim to hold up a 5-foot pole near the building and measure the length of its shadow. The shadow of the pole is 3 feet long, and the shadow of the building is 12 feet long.

Screen Shot 2022-09-09 at 2.35.05 AM.png — **Figure $ \PageIndex{ 13 } $**

Use similar triangles to write a proportion involving the height of the building.
Solve the proportion to find the height of the building.

Solutions

In Figure $ \PageIndex{ 13 } $, we see two right triangles: One triangle is formed by the building and its shadow, and the pole and its shadow form the other. Because the light rays from the sun are parallel, the two angles at the tips of the shadows are equal. Thus, the two right triangles are similar, and their corresponding sides are proportional. The ratios of heights and bases in the two triangles yield the proportion\[\dfrac{\text{larger triangle}}{\text{smaller triangle}}: \dfrac{h}{5} = \dfrac{12}{3} \nonumber \]
To solve the proportion, we multiply both sides by the LCD to get\[ \begin{array}{rrrclcl}
\scriptscriptstyle\xcancel{\mathrm{Arithmetic}} \to \mathrm{Algebra} & \implies & 3h & = & 5(12) & \quad & \left( \text{multiply both sides by the LCD of all terms in the equation} \right)\\
\scriptscriptstyle\mathrm{Arithmetic} & \implies & 3h & = & 60 & \quad & (\text{multiply}) \\
\scriptscriptstyle\xcancel{\mathrm{Arithmetic}} \to \mathrm{Algebra} & \implies & h & = & 20 & \quad & (\text{divide both sides by }3) \\
\end{array} \nonumber \]The building is 20 feet tall.

Checkpoint $\PageIndex{5}$

In Example $ \PageIndex{ 2 } $, we created a $30^{\circ} -60^{\circ} -90^{\circ} $ triangle in which the shorter leg was 4 inches and the hypotenuse was 8 inches. The hypotenuse of another $30^{\circ} -60^{\circ} -90^{\circ} $ triangle is 5 feet. What is the length of the side opposite the $30^{\circ} $ angle?

Answer: $2.5$ feet

Overlapping Triangles

In some applications, similar triangles may share a side or an angle.

Example $ \PageIndex{ 6 } $

Identify two similar triangles in Figure $ \PageIndex{ 14 } $ below, and write a proportion to find $H$.

Screen Shot 2022-09-09 at 2.40.29 AM.png — **Figure $ \PageIndex{ 14 } $**

Solution

The two triangles overlap, sharing the marked angle, as shown below. Because each triangle also has a right angle, they are similar.

Screen Shot 2022-09-09 at 2.41.16 AM.png — **Figure $ \PageIndex{ 15 } $**

Note that the base of the larger triangle is $24 + 12 = 36$. The ratio of the heights and the ratio of the bases must be equal, so we write the following proportion. \[ \begin{array}{rrrclcl}
& & \dfrac{H}{10} & = & \dfrac{36}{24} & \quad & (\text{start with the proportion}) \\
\\
\scriptscriptstyle\xcancel{\mathrm{Arithmetic}} \to \mathrm{Algebra} & \implies & 24H & = & 360 & \quad & (\text{multiply both sides by the LCD}) \\
\\
\scriptscriptstyle\xcancel{\mathrm{Arithmetic}} \to \mathrm{Algebra} & \implies & H & = & \dfrac{360}{24} & \quad & (\text{divide both sides by }24) \\
\\
\scriptscriptstyle\mathrm{Arithmetic} & \implies & H & = & 15 & \quad & (\text{division}) \\
\end{array} \nonumber \]

Checkpoint $\PageIndex{6}$

Min wants to know the height of a street lamp. She discovers that when she is 12 feet from the lamp, her shadow is 6 feet long. Find the height of the street lamp.

Screen Shot 2022-09-09 at 2.44.21 AM.png — **Figure $ \PageIndex{ 16 } $**

Answer: $15$ feet

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Congruent Triangles

Definition: Congruent Triangles

Example \( \PageIndex{ 1 } \)

Checkpoint \(\PageIndex{1}\)

Example \( \PageIndex{ 2 } \)

Definition: Parallelogram

Checkpoint \(\PageIndex{2}\)

Similar Triangles

Definition: Similar Triangles

Fun Fact!

Theorem: Similarity Conditions

Example \( \PageIndex{ 3 } \)

Checkpoint \(\PageIndex{3}\)

Using Proportions with Similar Triangles

Example \( \PageIndex{ 4 } \)

What does the \( \mathrm{Arithmetic} \) and \( \xcancel{\textrm{Arithmetic}} \) mean?

Checkpoint \(\PageIndex{4}\)

Similar Right Triangles

Example \( \PageIndex{ 5 } \)

Checkpoint \(\PageIndex{5}\)

Overlapping Triangles

Example \( \PageIndex{ 6 } \)

Checkpoint \(\PageIndex{6}\)