Similarity Criteria Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Similarity Criteria.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

Three sets of conditions that guarantee two triangles are similar: AA (two pairs of equal angles), SAS~ (two pairs of proportional sides with equal included angle), and SSS~ (all three pairs of sides in the same ratio).

Think of a photo and its enlargement. They look the same but are different sizes. For triangles, you only need to check that two angles match (AA)—if the angles are the same, the shape is the same, even if the size differs. It's like verifying two buildings have the same blueprint, even if one is a scale model.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Similarity criteria (AA, SAS~, SSS~) prove two triangles have the same shape using equal angles or proportional sides.

Common stuck point: The procedure for similarity criteria is the easy part; the trap is demanding all three angle pairs. Asking "Do the triangles match by equal angles or proportional sides (not equal lengths)?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Do the triangles match by equal angles or proportional sides (not equal lengths)?

Worked Examples

Example 1

easy
Triangle ABCABC has angles 40°40°, 60°60°, 80°80°. Triangle DEFDEF has angles 40°40°, 60°60°, 80°80°. Are the triangles similar? Which criterion applies?

Answer

△ABC∼△DEF\triangle ABC \sim \triangle DEF by AA.

First step

1
Step 1: List the angle pairs: ∠A=∠D=40°\angle A = \angle D = 40°, ∠B=∠E=60°\angle B = \angle E = 60°, ∠C=∠F=80°\angle C = \angle F = 80°.

Full solution

  1. 2
    Step 2: All three angles match. However, for similarity, we only need two angles — the third is determined since angles sum to 180°.
  2. 3
    Step 3: The AA (Angle-Angle) criterion states: if two angles of one triangle equal two angles of another, the triangles are similar.
  3. 4
    Step 4: Conclude: △ABC∼△DEF\triangle ABC \sim \triangle DEF by AA.
AA is the most commonly used similarity criterion. Because all angles in a triangle sum to 180°, knowing two angles determines the third. Two triangles with the same angle measures have the same shape (though possibly different sizes), making them similar. Their corresponding sides are proportional.

Example 2

medium
In △ABC\triangle ABC: AB=6AB = 6, BC=9BC = 9, AC=12AC = 12. In △DEF\triangle DEF: DE=4DE = 4, EF=6EF = 6, DF=8DF = 8. Are the triangles similar? State the criterion.

Example 3

medium
Triangle ABC has sides 55, 1212, 1313. Triangle DEF has sides 1010, 2424, 2626. Prove the triangles are similar and state the criterion used.

Example 4

medium
In △ABC\triangle ABC, DD is on ABAB and EE is on ACAC so that DE∥BCDE \parallel BC. Prove △ADE∼△ABC\triangle ADE \sim \triangle ABC.

Example 5

medium
△ABC∼△XYZ\triangle ABC \sim \triangle XYZ with AB=9,XY=6AB = 9, XY = 6. If BC=12BC = 12, find YZYZ.

Example 6

medium
△PQR\triangle PQR has ∠P=40°,∠Q=60°\angle P = 40°, \angle Q = 60°. △STU\triangle STU has ∠S=40°,∠U=80°\angle S = 40°, \angle U = 80°. Are the triangles similar? Which correspondence?

Example 7

medium
△ABC\triangle ABC has AB=15,AC=20,∠A=50°AB = 15, AC = 20, \angle A = 50°. △DEF\triangle DEF has DE=9,DF=12,∠D=50°DE = 9, DF = 12, \angle D = 50°. Are they similar?

Example 8

medium
A flag pole's shadow at 33 PM is 2020 ft. A nearby 55 ft mailbox casts a 44 ft shadow. How tall is the flag pole? Justify with a similarity criterion.

Example 9

hard
In right triangle ABCABC with right angle at CC, the altitude from CC meets ABAB at DD. Prove that △ACD∼△ABC\triangle ACD \sim \triangle ABC.

Example 10

hard
△ABC∼△DEF\triangle ABC \sim \triangle DEF with sides in ratio 2:32:3. If the perimeter of △ABC\triangle ABC is 2424, what is the perimeter of △DEF\triangle DEF?

Example 11

hard
△ABC\triangle ABC has AB=8,BC=6,CA=10AB = 8, BC = 6, CA = 10. Point PP is on ABAB with AP=6AP = 6 and QQ is on ACAC with AQ=7.5AQ = 7.5. Show △APQ∼△ABC\triangle APQ \sim \triangle ABC and find the ratio.

Example 12

hard
In a triangle, the angle bisector from AA divides side BCBC at DD. Given △ABD\triangle ABD and △ACD\triangle ACD share an angle at DD — explain why they are NOT in general similar.

Example 13

challenge
In a right triangle with legs a,ba, b and hypotenuse cc, the altitude from the right angle to the hypotenuse has length hh. Prove h=ab/ch = ab/c using similar triangles.

Example 14

challenge
Two similar triangles have perimeters 3030 and 5050. The smaller's area is 3636. Find the larger's area.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
A tree casts a 15 m shadow, and at the same time a 2 m stick casts a 3 m shadow. How tall is the tree? Which similarity criterion justifies this method?

Example 2

hard
In △PQR\triangle PQR and △STU\triangle STU: PQ=8PQ = 8, QR=12QR = 12, ∠Q=50°\angle Q = 50°; ST=6ST = 6, TU=9TU = 9, ∠T=50°\angle T = 50°. Show the triangles are similar and find the scale factor.

Example 3

easy
Triangle ABCABC has angles 30°30°, 90°90°, 60°60°. Triangle XYZXYZ has angles 30°30°, 90°90°, 60°60°. Are they similar?

Example 4

easy
△ABC\triangle ABC has sides 6,8,106, 8, 10 and △DEF\triangle DEF has sides 3,4,53, 4, 5. Which criterion shows they're similar?

Example 5

easy
Are all isoceles right triangles similar to each other?

Example 6

easy
△ABC\triangle ABC and △DEF\triangle DEF share angle A=D=70°A = D = 70°, and AB/DE=AC/DF=1.5AB/DE = AC/DF = 1.5. Which criterion proves similarity?

Example 7

medium
△ABC\triangle ABC has AB=8,BC=10,AC=12AB = 8, BC = 10, AC = 12. △DEF\triangle DEF has DE=12,EF=15,DF=18DE = 12, EF = 15, DF = 18. Are they similar? Which criterion?

Example 8

medium
A right triangle has legs 33 and 44. A similar right triangle has hypotenuse 2525. Find its legs.

Example 9

medium
Two triangles have side ratios 5:7:95:7:9. Are they similar to triangles with side ratios 10:14:1810:14:18?

Example 10

medium
Two triangles have ∠A=∠D=50°\angle A = \angle D = 50° and AB/DE=4AB/DE = 4. If AC=12AC = 12, find DFDF assuming similarity by SAS∼\sim.

Example 11

hard
In △ABC\triangle ABC, the altitude from CC to ABAB has length hh. If AC=6,BC=8AC = 6, BC = 8, and ∠C=90°\angle C = 90°, find hh using similarity.

Example 12

hard
△ABC∼△DEF\triangle ABC \sim \triangle DEF with area ratio 25:8125:81. The longest side of △ABC\triangle ABC is 1010. Find the longest side of △DEF\triangle DEF.

Example 13

hard
Two similar triangles have areas 5050 and 9898 cm2^2. Find the ratio of their corresponding sides in simplest form.
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Background Knowledge

These ideas may be useful before you work through the harder examples.

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