Similar Triangles Review- Concepts and Practice

What Similar Triangles Actually Are

Similar triangles are triangles that have exactly the same shape but not necessarily the same size. Two triangles are similar if their corresponding angles are equal and their corresponding sides are in proportion.

This isn't the same as congruent triangles. Congruent means identical in both shape and size. Similar means the shape matches, but one can be a scaled-up (or scaled-down) version of the other.

Think of it like a photograph and a poster of that same photograph. Same image, different dimensions. That's similar.

The Core Properties You Need to Know

Three things define similar triangles:

• Corresponding angles are equal — if angle A in triangle 1 equals angle D in triangle 2, then angle B equals angle E, and angle C equals angle F
• Corresponding sides are proportional — the ratio of any two sides in one triangle equals the ratio of the matching sides in the other
• The similarity ratio works both ways — if triangle ABC is similar to triangle DEF with ratio k, then triangle DEF is similar to triangle ABC with ratio 1/k

That's it. No tricks. Memorize those three points.

How to Prove Triangles Are Similar

You don't have to check every angle and every side ratio. Three shortcut methods exist:

AA (Angle-Angle) Similarity

If two angles of one triangle are equal to two angles of another triangle, the triangles are similar. This works because if two angles match, the third must match too (angles in a triangle sum to 180°).

This is the fastest method. Find two matching angles, and you're done.

SSS (Side-Side-Side) Similarity

If all three pairs of corresponding sides are in the same proportion, the triangles are similar.

You'll need to calculate three ratios and confirm they're all equal.

SAS (Side-Angle-Side) Similarity

If two sides are proportional and the included angle between them is equal, the triangles are similar.

Make sure it's the included angle — the angle sandwiched between the two sides you're comparing.

Similarity Criteria Comparison

Use AA whenever possible. It's the path of least resistance.

Using Similar Triangles to Find Missing Sides

This is where it gets practical. Given two similar triangles, you can find any missing side using proportions.

Setting Up the Ratio

The key is writing the proportion correctly. Match corresponding sides.

If triangle ABC is similar to triangle DEF:

• AB corresponds to DE
• BC corresponds to EF
• AC corresponds to DF

The similarity statement tells you the order. When you write △ABC ~ △DEF, the first letter corresponds to the first, second to second, third to third.

The Proportion Equation

Once you identify corresponding sides, set up the proportion:

AB/DE = BC/EF = AC/DF

Solve for your missing side using cross-multiplication. That's it.

How to Solve Similar Triangle Problems

Follow this step-by-step process:

1. Identify the similar triangles — look for the similarity statement or determine it using AA/SAS/SSS
2. Write the proportionality statement — list the three pairs of corresponding sides
3. Set up the proportion — include your known values and the unknown
4. Solve using cross-multiplication — multiply diagonally and isolate the variable
5. Check your answer — verify the ratio is consistent across all three pairs

Practice Problems

Problem 1

△ABC is similar to △DEF. AB = 6, BC = 8, AC = 10. DE = 3. Find EF and DF.

Solution:

The scale factor from △DEF to △ABC is 6/3 = 2. Multiply the other sides by 2:

• EF = 8 × 2 = 16
• DF = 10 × 2 = 20

Problem 2

In the diagram, a 6-foot person casts a 9-foot shadow. A nearby tree casts a 27-foot shadow at the same time. How tall is the tree?

Solution:

The person and their shadow form one right triangle. The tree and its shadow form a similar right triangle. Set up the proportion:

6/9 = tree height/27

Cross-multiply: 6 × 27 = 9 × height

162 = 9 × height

Height = 18 feet

Problem 3

Given △PQR ~ △STU, angle P = 45° and angle Q = 65°. Find angle T.

Solution:

Angle R = 180° - 45° - 65° = 70°. In similar triangles, corresponding angles are equal. Angle P corresponds to angle S, angle Q to angle T, and angle R to angle U. So angle T = angle Q = 65°.

Common Mistakes to Avoid

• Writing the wrong correspondence — if △ABC ~ △DEF, don't mix up which letters match
• Forgetting to check all three ratios — if two ratios match but the third doesn't, the triangles aren't similar
• Using non-corresponding sides — always match sides that are across from matching angles
• Mixing up similarity and congruence — similar doesn't mean equal, it means proportional

Where Similar Triangles Show Up

You'll encounter similar triangles in:

• Shadow problems — using light angles to find heights
• Indirect measurement — measuring things you can't reach directly
• Trigonometry prep — similar right triangles are the foundation for sine, cosine, and tangent
• Geometry proofs — often a step in larger proofs
• Architecture and engineering — scale models and proportional design

Once you understand similar triangles, a lot of other geometry concepts click faster.

Quick Reference

When working with similar triangles, remember:

• AA is your best friend — check for two equal angles first
• Corresponding sides are proportional, not equal
• The scale factor multiplies all sides consistently
• Corresponding angles are equal