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Triangle
Similarity

Learn to identify similar triangles, understand the key tests for similarity, and use proportional reasoning to find missing side lengths.

Geometry

What Does "Similar" Mean?

In everyday language, "similar" means "alike." In mathematics, similar triangles have a much more precise meaning: they are triangles with the exact same shape but not necessarily the same size.

A
B
C
D
E
F
\(\sim\)

Two Conditions for Similarity

For two triangles to be similar, both of the following must be true:

1. Equal Angles
All three pairs of corresponding angles are equal: \(\angle A = \angle D\), \(\angle B = \angle E\), \(\angle C = \angle F\).
2. Proportional Sides
All three pairs of corresponding sides share the same ratio: \(\frac{AB}{DE} = \frac{BC}{EF} = \frac{AC}{DF}\).

Why Similarity Matters

Similarity is one of the most practical tools in geometry. It lets you find measurements you cannot measure directly by using proportions.

★ Architecture
Scale models of buildings use similar triangles to keep every proportion accurate.
★ Shadows
You can calculate the height of a tree by comparing its shadow to the shadow of a known object.
★ Maps
Every map is a scaled-down similar figure, with distances reduced by a constant factor.

The AA Similarity Postulate

You do not need to check all three angle pairs. Because the angles of any triangle always add up to \(180^\circ\), knowing just two pairs of equal angles is enough.

\(50^\circ\)
\(60^\circ\)
A
B
C
\(70^\circ\) (forced)
\(50^\circ\)
\(60^\circ\)
D
E
F
\(70^\circ\) (forced)
AA
\(\sim\)

SAS and SSS Similarity

Besides AA, there are two other ways to prove triangles are similar:

SAS Similarity
Two pairs of corresponding sides are proportional, and the included angle (the angle between those sides) is equal.
SSS Similarity
All three pairs of corresponding sides are proportional: they share the same scale factor \(k\).

Similar vs. Congruent

Students often confuse these two ideas. Here is the key difference:

Similar (\(\sim\)) Congruent (\(\cong\))
Shape Same Same
Size Can differ Must be equal
Angles All equal All equal
Sides Proportional (ratio = \(k\)) Exactly equal (\(k = 1\))
Symbol \(\sim\) \(\cong\)

Setting Up Proportions

The most important skill with similar triangles is writing correct proportions to find a missing side. Follow these steps:

1
Match the vertices. Write the similarity statement carefully: \(\triangle ABC \sim \triangle DEF\). The order tells you which vertices correspond.
2
Pair corresponding sides. Side \(AB\) matches \(DE\), side \(BC\) matches \(EF\), and side \(AC\) matches \(DF\).
3
Write the proportion. Place one known ratio on one side, and a ratio with the unknown on the other: \(\frac{AB}{DE} = \frac{BC}{EF}\).
4
Cross-multiply and solve. Multiply diagonally, then divide to isolate the unknown.

Example 1: Finding a Missing Side

A
B
C
9
12
15
D
E
F
6
8
\(x\)
\(\sim\)

Example 2: Using the Scale Factor

Problem: \(\triangle PQR \sim \triangle STU\). If \(PQ = 6\), \(QR = 10\), \(PR = 8\), and \(ST = 15\), find all sides of \(\triangle STU\).

1
Find the scale factor. \(PQ\) corresponds to \(ST\): \(k = \frac{ST}{PQ} = \frac{15}{6} = \frac{5}{2} = 2.5\).
2
Multiply each side by \(k\). \(TU = QR \times 2.5 = 10 \times 2.5 =\) 25.
3
Third side: \(SU = PR \times 2.5 = 8 \times 2.5 =\) 20.

Example 3: Prove and Solve

D
E
A
B
C
4
6
7.5
\(x\)
BC = 9
DE = 15
\(BC \parallel DE\)

Cross-Multiplying Refresher

Cross-multiplication is the key technique for solving proportions. Here is the pattern:

\(\frac{\color{#818cf8}a}{\color{#818cf8}b}\)
\(=\)
\(\frac{\color{#4ade80}c}{\color{#4ade80}d}\)
\(\rightarrow\)
\(\color{#818cf8}a \color{#e2e8f0}{\cdot} \color{#4ade80}d = \color{#818cf8}b \color{#e2e8f0}{\cdot} \color{#4ade80}c\)

Identifying Corresponding Parts

The trickiest part of similarity problems is matching up the right vertices. Here are strategies:

1
Look at the angles. The smallest angle in one triangle corresponds to the smallest angle in the other.
2
Look at the sides. The shortest side in one triangle corresponds to the shortest side in the other.
3
Redraw if needed. If one triangle is rotated or flipped, sketch both in the same orientation to see the match clearly.

Common Mistakes to Avoid

✕ Mixing Up Vertices
Writing \(\frac{AB}{EF}\) instead of \(\frac{AB}{DE}\). Always follow the order given in the similarity statement.
✕ Confusing Similar and Congruent
Setting sides equal instead of proportional. Similar means same shape, not same size.
✕ Inconsistent Ratios
Putting a large-triangle side over a small-triangle side in one ratio, then reversing the order in the next ratio.

Similarity Tests Summary

Test What You Need Example
AA 2 pairs of equal angles \(\angle A = \angle D\), \(\angle B = \angle E\)
SAS 2 proportional sides + included angle equal \(\frac{AB}{DE} = \frac{AC}{DF}\), \(\angle A = \angle D\)
SSS All 3 side ratios equal \(\frac{3}{6} = \frac{4}{8} = \frac{5}{10}\)

Question 1 of 5

If \(\triangle ABC \sim \triangle DEF\), \(AB = 8\), \(DE = 12\), and \(BC = 14\), what is \(EF\)?

Question 2 of 5

Two triangles have angles of \(40^\circ\) and \(75^\circ\) in one, and \(40^\circ\) and \(65^\circ\) in the other. Are these triangles similar?

Question 3 of 5

Which similarity test requires you to check that all three pairs of sides share the same ratio?

Question 4 of 5

\(\triangle JKL \sim \triangle MNO\). The sides of \(\triangle JKL\) are \(5, 12, 13\). The shortest side of \(\triangle MNO\) is \(15\). What is the longest side of \(\triangle MNO\)?

Question 5 of 5

A flagpole casts a \(20\)-foot shadow. At the same time, a \(6\)-foot person casts a \(4\)-foot shadow. How tall is the flagpole?