Triangle Similarity — I can determine whether triangles are...

SECTION 1

Historical Context & Motivation

Have you ever looked at a map and realized it's just a tiny version of the real world? Or noticed that a small photo and a large poster of the same picture look identical, just different sizes? People have been thinking about this idea — same shape, different size — for thousands of years. The concept is called similarity, and it has been one of the most useful tools in math since ancient times.

~600 BCE

Thales of Miletus

The Greek mathematician Thales used similar triangles to measure the height of the Great Pyramid of Egypt. He compared the pyramid's shadow to his own shadow and set up a proportion to find the answer.

~300 BCE

Euclid's Elements

Euclid wrote down rules about similar triangles in his famous math book. He proved that triangles with the same angle measurements always have sides in the same ratio.

~200 BCE

Eratosthenes Measures Earth

The Greek scholar Eratosthenes used similar triangles and shadow measurements to estimate the circumference of the entire Earth — and he was remarkably close!

Today

Modern Applications

Engineers, architects, and even video game designers use triangle similarity every day. It helps them scale blueprints, create 3D graphics, and measure distances that are hard to reach.

So here's the big question: how do we know when two triangles are similar? And once we know they are, how can we use that to find a side length we don't know? That's exactly what this lesson is about.

SECTION 2

Core Principles & Definitions

Before we jump into problems, let's get clear on some important ideas. Two triangles are similar if they have the exact same shape but not necessarily the same size. Think of it like zooming in or out on a photo — everything stays in the same proportion.

Same Shape, Different Size

Similar triangles look alike. All their angles match up, and their sides are in the same ratio. One triangle is basically a scaled copy of the other.

Corresponding Parts

Corresponding means "matching." Corresponding angles are the angles that sit in the same position in each triangle. Corresponding sides are the sides opposite those matching angles.

Scale Factor

The scale factor is the number you multiply one triangle's sides by to get the other triangle's sides. For example, a scale factor of 2 means the bigger triangle's sides are twice as long.

Proportions

A proportion is an equation that says two ratios (fractions) are equal. We write proportions to find missing side lengths in similar triangles.

✦ KEY TAKEAWAY

Think of similar triangles like two different-sized T-shirts with the same design. A small and a large T-shirt look the same — same logo, same shape — but one is just bigger. The design didn't get stretched or squished; it was scaled up evenly. That's exactly what happens with similar triangles: every side gets multiplied by the same number.

SECTION 3

Visual Explanation

Let's look at two similar triangles side by side. Notice how the angles match and the sides are in the same ratio.

Triangle ABC and Triangle DEF have all the same angles (53°, 37°, and 90°). Every side of DEF is exactly half the matching side of ABC. The scale factor from ABC to DEF is ½.

In the diagram above, look at how the sides match up. Side AB = 12 corresponds to side DE = 6. Side AC = 10 corresponds to side DF = 5. Side BC = 8 corresponds to side EF = 4. Every pair gives the same ratio: 6 ÷ 12 = ½, 5 ÷ 10 = ½, and 4 ÷ 8 = ½. When all the ratios of corresponding sides are equal, the triangles are similar.

SECTION 4

Mathematical Framework

There are three main ways to prove that two triangles are similar. You don't have to check everything — just one of these tests is enough!

AA (Angle-Angle) Test

ANGLE-ANGLE (AA) SIMILARITY

If two angles of one triangle equal two angles of another triangle, the triangles are similar.

Why only two? Because the three angles of any triangle always add up to 180°. If two angles match, the third one must match too!

SSS (Side-Side-Side) Test

SIDE-SIDE-SIDE (SSS) SIMILARITY

a₁ / a₂ = b₁ / b₂ = c₁ / c₂

Here, a₁, b₁, c₁ are the sides of the first triangle and a₂, b₂, c₂ are the matching sides of the second. If all three ratios are equal, the triangles are similar.

SAS (Side-Angle-Side) Test

SIDE-ANGLE-SIDE (SAS) SIMILARITY

a₁ / a₂ = b₁ / b₂ AND the included angle is equal

If two pairs of corresponding sides have the same ratio AND the angle between those two sides is the same in both triangles, the triangles are similar.

Setting Up a Proportion

PROPORTION FOR MISSING SIDES

side in Triangle 1 / matching side in Triangle 2 = another side in Triangle 1 / its matching side in Triangle 2

Once you know two triangles are similar, set up a proportion using two pairs of corresponding sides. Put the unknown in one spot, then cross-multiply to solve.

SECTION 5

The Three Similarity Tests — A Closer Look

Let's see all three similarity tests in one visual so you can compare them. Each test gives you a different shortcut for proving triangles are similar.

The three similarity tests shown side by side. AA checks angles. SSS checks all side ratios. SAS checks two side ratios and the angle between them.

💡 Quick Tip

The AA test is the one you'll use most often. It's the easiest because you only need to find two matching angles. If two angles match, the third always will too!

SECTION 6

Worked Example

Let's work through a complete problem together. Read each step carefully — this is exactly the process you'll use on your own.

📐 Problem

Triangle PQR has sides PQ = 6, QR = 9, and PR = 12. Triangle STU has sides ST = 4 and TU = 6. The angles at Q and T are both 55°, and the angles at R and U are both 80°. Are the triangles similar? If so, find the length of side SU.

Finding a Missing Side Using Similar Triangles

Step 1 — Check for similarityWe know angle Q = angle T = 55° and angle R = angle U = 80°. That's two pairs of matching angles. By the AA test, the triangles are similar!

△PQR ~ △STU (by AA)

Step 2 — Match up corresponding sidesSince angle Q matches angle T, the sides opposite them match: PR ↔ SU. Since angle R matches angle U, the sides opposite them match: PQ ↔ ST. The remaining pair is QR ↔ TU.

PQ ↔ ST (6 ↔ 4), QR ↔ TU (9 ↔ 6), PR ↔ SU (12 ↔ ?)

Step 3 — Find the scale factorPick any pair of known sides. Let's use PQ and ST. Divide: ST ÷ PQ = 4 ÷ 6 = 2/3. Let's check with the other pair: TU ÷ QR = 6 ÷ 9 = 2/3. Great — the scale factor is 2/3.

Scale factor = 2/3

Step 4 — Set up a proportion and solveWe need SU. We know PR = 12 and the scale factor is 2/3. So: SU = PR × (2/3) = 12 × (2/3). Multiply: 12 × 2 = 24, then 24 ÷ 3 = 8.

SU = 8

Step 5 — Check your answerVerify all ratios: 4/6 = 2/3 ✓, 6/9 = 2/3 ✓, 8/12 = 2/3 ✓. All three ratios are equal, so our answer makes sense.

All ratios = 2/3 ✓ — Answer confirmed!

SECTION 7

Similar vs. Congruent — What's the Difference?

You might have heard the word congruent before. Congruent triangles are like identical twins — same shape AND same size. Similar triangles are like siblings — they look alike but can be different sizes. Let's compare these two ideas.

Comparing similar and congruent triangles

Feature	Similar Triangles	Congruent Triangles
Same shape?	Yes	Yes
Same size?	Not necessarily	Yes, always
Angles	All corresponding angles are equal	All corresponding angles are equal
Sides	Sides are proportional (same ratio)	Sides are exactly equal
Symbol	~ (tilde)	≅ (equals with tilde)
Scale factor	Can be any positive number	Always exactly 1

✦ KEY TAKEAWAY

Every congruent pair of triangles is also similar (with a scale factor of 1). But similar triangles are not always congruent. Think of it this way: all squares are rectangles, but not all rectangles are squares. Congruent is a special case of similar.

SECTION 8

Connection to Advanced Topics

Triangle similarity is a stepping stone to bigger ideas in math. Once you master it, you're ready for some really cool concepts. Here's a sneak peek at what's ahead.

How triangle similarity connects to future math topics

What You Learn Now	What Comes Next
Setting up proportions with similar triangles	Trigonometry — using ratios of sides (sine, cosine, tangent) to solve problems with angles
Scale factors between similar figures	Dilations in coordinate geometry — enlarging or shrinking shapes on the x-y plane
Matching corresponding parts	Formal geometric proofs — writing logical arguments to prove shapes are similar or congruent
Using shadows and indirect measurement	Surveying and engineering — measuring real-world distances using triangle similarity

In high school geometry, you'll write formal proofs using the AA, SSS, and SAS similarity tests. In trigonometry, the ratios of sides in similar right triangles become the foundation for sine, cosine, and tangent. The proportional thinking you practice now will help you with all of these topics!

SECTION 9

Practice Problems

Try these five problems on your own. They go from easier to harder. Write out your work before checking the answers!

PROBLEM 1 — CONCEPTUAL

Triangle ABC has angles of 40°, 60°, and 80°. Triangle DEF has angles of 40°, 80°, and 60°. Are these triangles similar? Explain why or why not.

PROBLEM 2 — BASIC CALCULATION

Triangle GHI is similar to Triangle JKL. The sides of GHI are 5, 10, and 15. Side JK (corresponding to GH = 5) is 8. What are the other two sides of Triangle JKL?

PROBLEM 3 — INTERMEDIATE

Two triangles have the following side lengths. Triangle 1: 6, 8, 10. Triangle 2: 9, 12, 18. Are these triangles similar? Show your work using the SSS test.

PROBLEM 4 — APPLIED

A flagpole casts a shadow that is 24 feet long. At the same time, a 5-foot-tall student standing nearby casts a shadow that is 8 feet long. The sun hits both the student and the flagpole at the same angle, creating two similar right triangles. How tall is the flagpole?

PROBLEM 5 — CRITICAL THINKING

Marcus says: "If I double every side of a triangle, the new triangle is similar to the original with a scale factor of 2." Priya says: "If I add 3 to every side of a triangle, the new triangle is also similar to the original." Who is correct? Explain with a specific example.

SUMMARY

Lesson Summary

Similar triangles have the same shape but can be different sizes. Their corresponding angles are equal, and their corresponding sides are in the same ratio. You can prove triangles are similar using three tests: AA (Angle-Angle) — match two pairs of angles; SSS (Side-Side-Side) — show all three side ratios are equal; or SAS (Side-Angle-Side) — show two side ratios are equal and the angle between them matches.

Once you know triangles are similar, find the scale factor by dividing a pair of corresponding sides. Then set up a proportion and cross-multiply to solve for any missing side. This skill connects directly to trigonometry, coordinate geometry, and real-world measurement problems you'll encounter later on.