7.3 Similarity proofs and applications

Similarity Theorems and Proofs

Two triangles are similar when they have the same shape but not necessarily the same size. Their corresponding angles are congruent, and their corresponding sides are in proportion. Proving similarity is one of the most tested skills in this unit because it connects proportional reasoning to geometric structure. You'll use these proofs to find missing side lengths, solve indirect measurement problems, and work with scale factors across dimensions.

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Triangle Similarity Theorems

There are three main ways to prove two triangles are similar. Each one requires you to show a specific combination of angle and/or side relationships.

AA (Angle-Angle) Similarity Theorem: If two angles of one triangle are congruent to two angles of another triangle, the triangles are similar. Since the angles of any triangle sum to 180°, matching two angles automatically guarantees the third angle matches too.

AA is the most commonly used theorem in proofs because you only need two pieces of information. Look for parallel lines (which create congruent alternate interior angles) or shared angles as your go-to strategies.

SAS (Side-Angle-Side) Similarity Theorem: If two sides of one triangle are proportional to two corresponding sides of another triangle and the included angles are congruent, the triangles are similar. The angle must be between the two sides you're comparing.

SSS (Side-Side-Side) Similarity Theorem: If all three pairs of corresponding sides are proportional, the triangles are similar. You don't need any angle information here, but you do need all three ratios to be equal.

A common mistake with SSS and SAS similarity is confusing them with the congruence postulates. For congruence, sides must be equal. For similarity, sides must be proportional. Always set up ratios, not equations, when testing for similarity.

Applications of Similarity Proofs

Once you've proven two triangles are similar, the proportional relationship extends beyond just the sides.

Corresponding sides are proportional by definition. If $\triangle ABC \sim \triangle DEF$, then:

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

This ratio is the scale factor between the two triangles.

Altitudes, medians, angle bisectors, and perimeters of similar triangles are also proportional, and they share the same scale factor as the corresponding sides. So if the scale factor between two similar triangles is $\frac{3}{5}$, the ratio of their altitudes is also $\frac{3}{5}$, and so is the ratio of their perimeters.

These properties extend to similar polygons in general. Any two similar polygons have congruent corresponding angles and proportional corresponding sides. This means you can apply the same proportional reasoning to rectangles, pentagons, or any other shape once similarity is established.

Similar Figures in Real-World Problems

Indirect measurement is the classic application. You can find the height of a building or the distance across a river without measuring directly by setting up similar triangles and solving a proportion.

For example, in a shadow problem:

1. A person who is 6 ft tall casts a 4 ft shadow.
2. At the same time, a flagpole casts a 20 ft shadow.
3. The sun hits both at the same angle, so the two right triangles formed (object + shadow + sun ray) are similar by AA.
4. Set up the proportion: $\frac{6}{4} = \frac{h}{20}$
5. Cross-multiply and solve: $h = 30$ ft.

Scale models and maps work the same way. A scale factor like 1:500 on an architectural model means every 1 cm on the model represents 500 cm (5 m) in real life. You use proportions with the scale factor to convert between model and actual measurements.

Similarity and Proportionality Across Dimensions

The scale factor between similar figures affects lengths, areas, and volumes differently. This is one of the trickiest parts of the unit, so pay close attention to the exponents.

If the scale factor between two similar figures is $k$:

• Lengths (sides, perimeters, altitudes) scale by $k$
• Areas scale by $k^2$
• Volumes scale by $k^3$

For example, if two similar rectangular prisms have a scale factor of $\frac{2}{3}$:

• The ratio of their corresponding side lengths is $\frac{2}{3}$
• The ratio of their surface areas is $\left(\frac{2}{3}\right)^2 = \frac{4}{9}$
• The ratio of their volumes is $\left(\frac{2}{3}\right)^3 = \frac{8}{27}$