DWG NO. 07.3 — Lesson 3 of 5

Proving Triangles Similar (AA, SSS∼, SAS∼)

Unit 7: Similarity · ~15–30 min

Objective

Prove that two triangles are similar using the AA, SSS∼, or SAS∼ similarity criteria.

Three shortcuts for triangle similarity

Just like congruence had shortcuts (SSS, SAS, ASA), similarity has its own set — and they need less information, because similarity only requires proportional sides, not equal ones.

AA (Angle-Angle) Similarity: If two angles of one triangle are congruent to two angles of another triangle, the triangles are similar. This is the one you'll use most — and it's enough on its own, because if two angles match, the third must match too (angle sum is always 180°).
SSS∼ (Side-Side-Side) Similarity: If all three pairs of corresponding sides are proportional (same ratio), the triangles are similar.
SAS∼ (Side-Angle-Side) Similarity: If two pairs of corresponding sides are proportional and the included angles are congruent, the triangles are similar.

Notice the difference from congruence: instead of requiring sides to be equal, similarity only requires them to be proportional by the same constant scale factor.

Two congruent angle pairs (marked in gold) are enough to guarantee AA similarity.

Worked Example 1 · Proving similarity with AA

ProblemIn ▵ABC and ▵DEF, m∠A = 50°, m∠B = 70°, m∠D = 50°, and m∠E = 70°. Are the triangles similar?

1.∠A ≅ ∠D (both 50°) and ∠B ≅ ∠E (both 70°).

2.Two pairs of congruent angles satisfy AA Similarity.

▵ABC ∼ ▵DEF by AA Similarity

Worked Example 2 · Checking SSS∼

Problem▵PQR has sides 4, 6, 8. ▵XYZ has sides 6, 9, 12. Are the triangles similar?

1.Order each triangle's sides smallest to largest and compare ratios: 4/6 = 2/3, 6/9 = 2/3, 8/12 = 2/3.

2.All three ratios equal 2/3, so the sides are proportional.

▵PQR ∼ ▵XYZ by SSS∼, scale factor 2:3

Guided practice

1.▵JKL has m∠J = 40° and m∠K = 85°. ▵MNO has m∠M = 40° and m∠O = 55°. Are the triangles similar? Justify your answer.
Show answer
In ▵MNO, m∠N = 180° − 40° − 55° = 85°. So ∠J ≅ ∠M and ∠K ≅ ∠N, giving ▵JKL ∼ ▵MNO by AA Similarity.
2.▵ABC has sides 3, 5, 7. ▵DEF has sides 9, 14, 21. Are they similar by SSS∼? Show the ratios.
Show answer
3/9 = 1/3, 5/14 ≠ 1/3 (it's 5/14). Since the ratios aren't all equal, the triangles are not similar.
3.In ▵RST and ▵UVW, RS/UV = 2/5 and ST/VW = 2/5, with ∠S ≅ ∠V (the included angle). Which similarity criterion applies?
Show answer
Two proportional sides with the included congruent angle is SAS∼, so ▵RST ∼ ▵UVW.
4.A flagpole casts a 12 ft shadow at the same time a 5 ft student casts a 3 ft shadow. Explain why the triangles formed are similar, then find the flagpole's height.
Show answer
Both triangles have a right angle (pole/student meeting the ground) and share the same angle of elevation to the sun, so they're similar by AA. Setting up 5/3 = h/12 gives 3h = 60, so h = 20 feet.
5.Explain why AA Similarity only needs two angle pairs instead of three, the way you might expect.
Show answer
The angle measures in any triangle always add to 180°. If two angles of one triangle match two angles of another, the third angle in each triangle is forced to be 180° minus the sum of the other two — which will automatically match as well.

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