Similar Triangles and Proportions

Similar triangles are triangles that have the same shape but may be different sizes. Their corresponding angles are equal, and their corresponding sides are in proportion. This property is the foundation of scaling, indirect measurement, and blueprint reading.

What Makes Triangles Similar?

Two triangles are similar (written $\triangle ABC \sim \triangle DEF$) when:

1. All three pairs of corresponding angles are equal, and
2. All three pairs of corresponding sides are proportional

If the angles match, the sides are automatically proportional — and vice versa. You only need to verify one of these conditions.

AA Similarity (Angle-Angle)

The most common way to prove two triangles are similar is the AA criterion: if two angles of one triangle are equal to two angles of another triangle, the triangles are similar.

Why do you only need two angles? Because the angles in any triangle add up to $180°$. If two angles match, the third must also match:

$\text{Third angle} = 180° - \text{first angle} - \text{second angle}$

Example 1: Are these triangles similar?

Triangle 1 has angles of $40°$, $60°$, and $80°$. Triangle 2 has angles of $60°$, $80°$, and $40°$.

Both triangles have the same three angle measures ($40°$, $60°$, $80°$), just listed in a different order. By AA similarity, the triangles are similar.

Corresponding Sides and Proportions

When triangles are similar, their corresponding sides form equal ratios. If $\triangle ABC \sim \triangle DEF$, then:

$\frac{AB}{DE} = \frac{BC}{EF} = \frac{AC}{DF}$

This common ratio is called the scale factor.

Example 2: Two similar triangles have sides as follows. Find the unknown side $x$.

Triangle 1 has sides 6, 8, and 10. Triangle 2 has sides 9, 12, and $x$.

Step 1 — Find the scale factor using a pair of known corresponding sides:

$\text{Scale factor} = \frac{9}{6} = 1.5$

Step 2 — Apply the scale factor to find $x$:

$x = 10 \times 1.5 = 15$

Verify with the other pair: $\frac{12}{8} = 1.5$ . The ratio is consistent.

Answer: The unknown side is 15.

Example 3: Solving with a proportion

In two similar triangles, the sides of the first triangle are 4 and 7 (with the third side unknown), and the corresponding sides of the second triangle are 12 and $x$. Find $x$.

Set up a proportion with corresponding sides:

$\frac{4}{12} = \frac{7}{x}$

Cross-multiply:

$4x = 12 \times 7$

$4x = 84$

$x = 21$

Answer: The unknown side is 21.

Scale Factor

The scale factor is the ratio between corresponding sides of two similar figures. It tells you how much larger or smaller one figure is compared to the other.

• A scale factor greater than 1 means the second figure is larger (scaling up)
• A scale factor less than 1 means the second figure is smaller (scaling down)
• A scale factor of exactly 1 means the figures are the same size (congruent)

$\text{Scale factor} = \frac{\text{side of new figure}}{\text{side of original figure}}$

If Triangle 1 has a side of 5 and the corresponding side of Triangle 2 is 20, the scale factor is $\frac{20}{5} = 4$. Every side of Triangle 2 is four times the corresponding side of Triangle 1.

Indirect Measurement

One of the most practical uses of similar triangles is indirect measurement — finding a length you can’t measure directly by using proportions with a length you can measure.

Example 4: Finding the height of a tree

A 6-foot-tall person stands so that the tip of their shadow and the tip of a tree’s shadow line up at the same point. The person’s shadow is 4 feet long, and the tree’s shadow is 20 feet long. How tall is the tree?

The person and their shadow form a right triangle. The tree and its shadow form a larger right triangle. Because the sun’s rays hit at the same angle, these triangles are similar (by AA — both have a right angle and share the angle of the sun’s rays).

Set up the proportion:

$\frac{\text{person's height}}{\text{person's shadow}} = \frac{\text{tree's height}}{\text{tree's shadow}}$

$\frac{6}{4} = \frac{h}{20}$

Cross-multiply:

$4h = 6 \times 20 = 120$

$h = 30$

Answer: The tree is 30 feet tall.

Real-World Application: Carpentry — Reading a Scaled Blueprint

A carpenter is working from a blueprint drawn at a scale of $\frac{1}{4}$ inch = 1 foot. On the blueprint, a wall measures $3\frac{1}{2}$ inches long, and a window opening measures $\frac{3}{4}$ inch wide. What are the actual dimensions?

Step 1 — Identify the scale factor. The scale is $\frac{1}{4}$ inch = 1 foot, which means:

$\text{Scale factor} = \frac{1 \text{ foot}}{\frac{1}{4} \text{ inch}} = 4 \text{ feet per inch}$

Every inch on the blueprint represents 4 feet in reality.

Step 2 — Find the actual wall length:

$3.5 \text{ in} \times 4 \text{ ft/in} = 14 \text{ ft}$

Step 3 — Find the actual window width:

$0.75 \text{ in} \times 4 \text{ ft/in} = 3 \text{ ft}$

Answer: The wall is 14 feet long and the window opening is 3 feet wide.

Alternatively, you can set this up as a proportion:

$\frac{0.25 \text{ in}}{1 \text{ ft}} = \frac{3.5 \text{ in}}{x \text{ ft}}$

$0.25x = 3.5$

$x = 14 \text{ ft}$

Both approaches give the same result. Blueprints rely on the same principle as similar triangles — every measurement on the drawing is proportional to the actual measurement by a consistent scale factor.

Practice Problems

Test your understanding with these problems. Click to reveal each answer.

Key Takeaways

• Similar triangles have equal angles and proportional sides
• AA similarity: if two angles match, the triangles are similar
• Set up proportions with corresponding sides to find unknown lengths: $\frac{a}{d} = \frac{b}{e} = \frac{c}{f}$
• The scale factor is the constant ratio between corresponding sides
• Indirect measurement uses similar triangles to find heights or distances you can’t measure directly
• Blueprint reading is a direct application of similar figures and proportional reasoning

Return to Geometry for more topics in this section.