Angle-Angle Criterion for Triangle Similarity

Students apply the AA (Angle-Angle) similarity criterion: if two angles of one triangle are congruent to two angles of another, the triangles are similar. They use AA to identify similar triangles and set up proportions to find unknown side lengths.

Try three of them

Real questions from this skill, graded here. No account, and nothing is saved.

Grade 8 · Geometry: Transformations · question 1 of 3

0 of 20

20 rungs. A star at the top.

: 20 = 2 : 8

How it is taught

0:00 of 4:30

AA similarity says two pairs of congruent angles make similar triangles.

Same angles mean the third angles match too.

Read it: Similar triangles by their angles

Triangles ABC and DEF have two pairs of equal angles, so they are similar. AB = 3 matches DE = 9, 3 times as long, so every side of DEF is 3 times its match: AC = 7 gives DF = 21.

Mark the matching angles: two pairs equal is enough.
Find the scale factor from one pair of matching sides.
Use the same factor on the side you want.

If two angles match, the third must too, since each triangle's angles make 180°.

Triangles with the same angles have the same shape, so their sides grow by one factor.

That is the ratio 3 : 9 = 7 : 21.

Two pairs of equal angles make two triangles similar, and similar sides share one scale factor.

Adding the difference: 9 is 6 more than 3, so DF = 7 + 6 = 13.

Similar sides multiply, not add. 9 is 3 times 3, so DF is 3 times 7: 21.

Triangles ABC and DEF are similar. AB = 5 matches DE = 25, and AC = 8 matches DF. How long is DF? — 40. DE is 5 times AB, so DF is 5 times AC: 8 × 5 = 40.

Find EF when △ABC ~ △DEF

Look at these two triangles. Same shape, different size — but are they actually similar?

Here's our first triangle, A B C.

And here's a bigger triangle, D E F.

Angle A and angle D both measure fifty degrees. They match!

Look — angle B and angle E both measure seventy degrees. Another match!

Grade 8 · Two matching angles → similar triangles

Two pairs of angles match — so by the A-A criterion, the triangles are similar.

Side A B is six. The matching side D E is nine. And side B C is eight.

But how long is E F? That's our missing side.

Similar triangles have proportional sides. So six over nine equals eight over E F.

Cross multiply. Six times E F equals nine times eight, which is seventy-two.

Divide both sides by six. E F equals twelve.

Just two matching angles unlocked the missing side. Big idea: angle, angle — similar!

What you see

While they climb, your page writes one true line to ask about tonight, and a short note each Sunday.

MayaGrade 3 · Mathematics

Did you know?

A chessboard has 64 squares — share them out between the 8 rows and each row holds exactly 8.

Today’s goal: 1 star — one skill mastered, every rung of its ladder climbed
Climbing now: Divide within 100

An example student.

MayaGrade 3 · Mathematics

Maya earned 2 stars this week — Interpret multiplication as equal groups, which took 6 answers over 3 days before its star landed 3 days ago and Multiply by 6, 7, 8 and 9. Next up: Divide within 100.

An example week.

More of the library

Every skill has a page

Standard

CCSS.8.G.A.5

Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles

Grade 8

Quoted from the Common Core State Standards.

Where this sits

Comes after

Leads toNothing yet

What a star here means

A star here is the top of a twenty-rung ladder, in four blocks of equal length — three of facts and one of the skill used in a real question — where a right answer climbs one rung and a wrong one drops back to the start of the block, never below it. The star is kept for good. How we grade

All of Grade 8

This is one rung of 48 in Grade 8 mathematics, in the order they build on each other.

Practise this skill free

14 days of every skill, practice packets, Fluency sheets and up to four students. No card.

Start free