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AAA similarity criterion

If three angles of one triangle are respectively equal to three angles of another triangle, the triangles are similar.

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Student-friendly explanation

The third angle automatically matches because the angle sum of a triangle is 180°. So once two corresponding angles are equal, the full shape pattern is fixed.

How to write this in exams

  1. 1

    Start with the exact idea

    If three angles of one triangle are respectively equal to three angles of another triangle, the triangles are similar.

  2. 2

    Then show how to use it

    1. Check the first two corresponding angles. 2. Confirm the third angle using angle sum if needed. 3. State AAA similarity clearly.

  3. 3

    Add one concrete example

    If one triangle has angles 35°, 75°, 70° and another triangle has the same angle pattern, the triangles are similar by AAA.

  4. 4

    Avoid this incomplete answer

    Thinking side lengths must also be given. For AAA, matching angles are already sufficient.

Definition

If three angles of one triangle are respectively equal to three angles of another triangle, the triangles are similar.

Example

If one triangle has angles 35°, 75°, 70° and another triangle has the same angle pattern, the triangles are similar by AAA.

Rule to remember

AAA similarity: three corresponding angles equal implies similarity.

Memory hook

Three equal angles, one same shape.

Examples and method

Worked example

In triangles ABC and DEF, if ∠A = ∠D = 40° and ∠B = ∠E = 65°, then ∠C = ∠F = 75°. The triangles are similar by AAA.

Method to apply

1. Check the first two corresponding angles. 2. Confirm the third angle using angle sum if needed. 3. State AAA similarity clearly.

Diagram support

Mark the equal angles on the corresponding corners and keep the vertex order the same.

How CBSE asks it

A proof or short-answer question may give angles and ask for the correct similarity reason.

Avoid common mistakes

Common confusion

Students sometimes wait for side lengths even after all the corresponding angles have already matched.

Common wrong answer

Thinking side lengths must also be given. For AAA, matching angles are already sufficient.

Exam tip

If two corresponding angles match, the third one will also match because the total is always 180°.

Quick check

What does AAA similarity tell you when all corresponding angles of two triangles match?

AAA similarity tells us that the triangles are similar because equal angles fix the shape completely. Their sides may be different in size, but the triangle pattern remains the same.

Answer writing and exam use

1-mark answer

If three angles of one triangle are respectively equal to three angles of another triangle, the triangles are similar.

2-mark answer

If three angles of one triangle are respectively equal to three angles of another triangle, the triangles are similar. AAA similarity: three corresponding angles equal implies similarity. If one triangle has angles 35°, 75°, 70° and another triangle has the same angle pattern, the triangles are similar by AAA.

3-mark answer

The third angle automatically matches because the angle sum of a triangle is 180°. So once two corresponding angles are equal, the full shape pattern is fixed. AAA similarity: three corresponding angles equal implies similarity. In triangles ABC and DEF, if ∠A = ∠D = 40° and ∠B = ∠E = 65°, then ∠C = ∠F = 75°. The triangles are similar by AAA. A proof or short-answer question may give angles and ask for the correct similarity reason. Thinking side lengths must also be given. For AAA, matching angles are already sufficient.
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