Proving Triangles Are Similar

Grade 6

The AA condition

Two triangles are similar if two pairs of corresponding angles are equal — this is the AA (angle-angle) condition. Because the angles of a triangle sum to 180°, two equal angles guarantee the third is equal too, so AA is sufficient.

What to write in a proof

Name each pair of equal angles, give the reason (vertically opposite, alternate angles, common angle, angle in a semicircle, etc.), then state "therefore the triangles are similar by AA".

Look for shared angles (a common vertex is a giveaway), vertically opposite angles, and alternate angles in parallel-line diagrams — these are the three most common sources of equal angles in similarity proofs.

Worked example

EXAMPLE

In a triangle, DE is parallel to BC. Prove that triangle ADE is similar to triangle ABC.

Angle A is common to both triangles. Angle ADE = angle ABC (corresponding angles, DE parallel to BC). Two pairs of angles are equal, so triangle ADE is similar to triangle ABC (AA).

Try this one

\( ABCD \) is a parallelogram. The diagonal \( AC \) is drawn, splitting it into triangle \( ABC \) and triangle \( CDA \).

Prove that triangle \( ABC \) is congruent to triangle \( CDA \). State the congruence condition you use. [3]

Nova's hint:

Use the properties of a parallelogram, together with the diagonal, to match up parts of the two triangles.

Ask yourself: 'which pairs of sides can I show are equal, and is there a side that both triangles share?' Then decide which congruence condition those facts satisfy.

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