Angle-Angle (AA) Similarity

8.G.5Grade 8Geometry20 practice questions

You don't need to measure every side to know two triangles are similar. Two matching angles are enough, and that lets you measure things you can't reach, like a tree or a pond.

Know these first

Triangle Angle Sum 8.G.5
Similarity Through Dilations 8.G.4

This unlocks

More lessons that build on this are coming.

What you'll be able to do

Use the AA criterion to decide whether two triangles are similar.
Find equal angles from shared angles, vertical angles, right angles and parallel lines.
Match the sides that sit opposite equal angles.
Find a missing length with a proportion, including shadows and mirrors.
Solve for x when matching angles are given as expressions.

The big idea

Two triangles are similar when all three pairs of angles match. You only need to check two: if two angles of one triangle equal two angles of another, the triangles are similar (AA). The third angles must match too, because each triangle's angles add to 180°.

Equal angles come from a few places: an angle the two triangles share, vertical angles where two lines cross, right angles, and the equal angles that parallel lines make.

Once you know the triangles are similar, match the sides that sit across from equal angles and find the scale factor. That's how a shadow tells you a tree's height: the sun hits the tree and a fence post at the same angle, and both stand at right angles to the ground.

Words you need

AA criterion: Two pairs of equal angles are enough to prove two triangles similar.
Vertical angles: The opposite angles where two straight lines cross. They are always equal.
Indirect measurement: Finding a length you can't measure by using a similar triangle you can.
Corresponding sides: Sides across from equal angles. Their ratio is the scale factor.

Worked examples

Example 1

Triangle ABC has angles of 50° and 70°. Triangle DEF has angles of 70° and 60°. Are they similar?

Find each missing angle: 180 − 50 − 70 = 60 and 180 − 70 − 60 = 50.
Both triangles have angles of 50°, 60° and 70°. Two pairs match, so they are similar by AA.

Answer: Yes, similar

Example 2

A 1.8 m tall person casts a 3 m shadow. At the same moment, a flagpole casts a 15 m shadow. How tall is the flagpole?

The sun's rays hit both at the same angle, and both stand at right angles to the ground. Two equal angles, so the triangles are similar by AA.
Scale factor = 15 ÷ 3 = 5.
Flagpole height = 1.8 × 5 = 9 m.

Answer: 9

Example 3

In triangle ABC, D is on AB and E is on AC, with DE parallel to BC. AD = 4, DB = 6 and AE = 6. Find EC.

Triangles ADE and ABC share angle A, and DE ∥ BC makes angle ADE equal angle ABC (corresponding angles). So they are similar by AA.
AB = 4 + 6 = 10, so the scale factor from ADE to ABC is 10 ÷ 4 = 2.5.
AC = 6 × 2.5 = 15, so EC = 15 − 6 = 9.

Answer: EC = 9

Common mistakes

Matching sides by position on the page. Match by angle: the side across from the 40° angle in one triangle goes with the side across from the 40° angle in the other.

Comparing a piece of a side with the whole side of the other triangle. In a nested triangle, compare small to big: AD with AB (the whole side, AD + DB), and DE with BC.

Calling triangles similar after finding only one pair of equal angles. One pair isn't enough. Find a second pair, or work out the third angles with 180°.

Practice: 20 questions

12 of 20 are word problems. For a word problem, sketch it first and mark what you know.

Tier 1 Foundations Use the rule directly

Triangle PQR has angles of 45° and 80°. Triangle XYZ has angles of 80° and 55°. Are they similar?
Triangle JKL has angles of 30° and 70°. Triangle MNP has angles of 30° and 75°. Are they similar?
A 2 m fence post casts a 3 m shadow. At the same moment, a tree casts a 12 m shadow. How tall is the tree? Word problem
One roof truss is a triangle with angles of 35° and 110°. Another has two angles of 35° each. Are the trusses similar? Word problem
Leo's triangular pennant has a right angle and a 40° angle. Maya's has a right angle and a 50° angle. Are the pennants similar? Word problem

Tier 2 Intermediate One more step to think through

Triangle ABC ~ triangle DEF, with angle A matching angle D. Angle A = (3x + 5)° and angle D = 65°. Find x.
In triangle ABC, DE is parallel to BC, with D on AB and E on AC. AD = 3, DB = 6 and DE = 4. Find BC.
Jun puts a mirror on the ground 6 m from a building. He stands 2 m from the mirror, with his eyes 1.5 m above the ground, and sees the top of the building in it. How tall is the building? Word problem
Two straight paths cross, forming two triangles whose far sides are parallel. In the small triangle, the side along one path is 9 m and the parallel side is 12 m. In the big triangle, the side along the same path is 60 m. How long is its parallel side? Word problem
Ana's triangle has angles of 50° and 60°. Ben's has angles of 60° and 80°. Are they similar? Word problem
Two triangles have two pairs of equal angles. The first has sides 5, 7 and 9; the matching sides of the second are 7.5, an unknown side and 13.5. Find the unknown side.
Two triangular picture frames are similar. One has angles of 40° and 75°. The matching angles of the other are (2x)° and (x + 55)°. Find x. Word problem

Tier 3 Advanced Several steps - find the hidden structure

A triangular frame ABC has a crossbar DE parallel to the ground side BC, with D on AB and E on AC. AD = 2 ft, DB = 4 ft and DE = 3 ft. How long is BC? Word problem
A 1.6 m tall girl stands 4 m from a lamp post. Her shadow is 2 m long, and its tip lines up with the top of the lamp. How tall is the lamp post? Word problem
In triangle ABC, DE is parallel to BC, with D on AB and E on AC. AB = 15, AD = 6 and AC = 20. Find AE.
Triangle ABC ~ triangle DEF, with angle A matching D and angle B matching E. Angle A = (x + 10)°, angle B = (2x)°, angle D = 50° and angle E = 80°. Find x.
A shelf bracket is a triangle ABC. A support DE runs parallel to BC, with D on AB and E on AC. AD = 10 cm, AB = 25 cm and BC = 30 cm. How long is the support DE? Word problem

Tier 4 Challenge Stretch problems, some above grade level

To find the width AB of a pond, Ravi lays out two triangles that meet at point C, where two straight lines cross. AB is parallel to DE. AC = 30 m, CD = 12 m and DE = 10 m. How wide is the pond? Word problem
Triangle PQR has angles of 40° and 65°. In a similar triangle, the matching angles are (x + 15)° and (2x + 15)°. Find x.
Two triangular sails are similar by AA. Matching sides are 3 m on the small sail and 7.5 m on the large one. The small sail has an area of 4 square meters. What is the area of the large sail? Word problem

Answers

Tier 1 Foundations

Yes, similar. All three angles: 45°, 55°, 80° and 45°, 55°, 80°. Two pairs match, so the triangles are similar by AA.
No, not similar. All three angles: 30°, 70°, 80° and 30°, 75°, 75°. The angles don't match, so they aren't similar.
8. Scale factor = 12 ÷ 3 = 4, so the missing height is 2 × 4 = 8.
Yes, similar. All three angles: 35°, 35°, 110° and 35°, 35°, 110°. Two pairs match, so the triangles are similar by AA.
Yes, similar. All three angles: 40°, 50°, 90° and 40°, 50°, 90°. Two pairs match, so the triangles are similar by AA.

Tier 2 Intermediate

x = 20. Matching angles are equal: 3x + 5 = 65, so x = 20.
BC = 12. DE ∥ BC, so triangle ADE ~ triangle ABC by AA. The scale factor is AB ÷ AD = 9 ÷ 3 = 3; use it on the matching sides.
4.5. Scale factor = 6 ÷ 2 = 3, so the missing height is 1.5 × 3 = 4.5.
80. Scale factor = 60 ÷ 9 = 20/3, so the missing side is 12 × 20/3 = 80.
No, not similar. All three angles: 50°, 60°, 70° and 40°, 60°, 80°. The angles don't match, so they aren't similar.
10.5. Scale factor = 7.5 ÷ 5 = 1.5, so the missing side is 7 × 1.5 = 10.5.
x = 20. Matching angles are equal: 2x = 40, so x = 20; check: x + 55 = 75.

Tier 3 Advanced

BC = 9. DE ∥ BC, so triangle ADE ~ triangle ABC by AA. The scale factor is AB ÷ AD = 6 ÷ 2 = 3; use it on the matching sides.
4.8 m. The lamp's triangle has a base of 4 + 2 = 6 m and the girl's has a base of 2 m, so the scale factor is 3 and the lamp is 1.6 × 3 = 4.8 m tall.
AE = 8. DE ∥ BC, so triangle ADE ~ triangle ABC by AA. The scale factor is AB ÷ AD = 15 ÷ 6 = 2.5; use it on the matching sides.
x = 40. Matching angles are equal: x + 10 = 50, so x = 40; check: 2x = 80.
DE = 12. DE ∥ BC, so triangle ADE ~ triangle ABC by AA. The scale factor is AB ÷ AD = 25 ÷ 10 = 2.5; use it on the matching sides.

Tier 4 Challenge

25. Scale factor = 30 ÷ 12 = 2.5, so the missing width is 10 × 2.5 = 25.
x = 25. Matching angles are equal: x + 15 = 40, so x = 25; check: 2x + 15 = 65.
25. The scale factor is 7.5 ÷ 3 = 2.5, and areas grow by k², so the large sail is 4 × 2.5² = 25 square meters.

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