AA Criterion of Similarity

Here we will prove that in a right-angled triangle, if a perpendicular is drawn from the right-angled vertex to the hypotenuse, the triangles on each side of it are similar to the whole triangle and to one another.

Given: Let XYZ be a right angle in which ∠YXZ = 90° and XM ⊥ YZ.

AA Criterion of Similarity

Therefore, ∠XMY = ∠XMZ = 90°.

To prove: ∆XYM ∼ ∆ZXM ∼ ∆ ZYX.



1. In ∆XYM and ∆XYZ,

(i) ∠XMY = ∠YXZ = 90°.

(ii) ∠XYM = ∠XMZ

2. Therefore, ∆XYM ∼ ∆ZYX.

3. In ∆XYZ and ∆XMZ,

(i) ∠YXZ = ∠XMZ = 90°.

(ii) ) ∠XZY= ∠XZM.

4. Therefore, ∆ZYX ∼ ∆ ZXM.

5. Therefore, ∆XYM ∼ ∆ZXM ∼ ∆ ZYX. (Proved)



(i) Given.

(ii) Common angle.

2. By AA criterion of similarity.


(i) Given.

(ii) Common angle.

4. By AA criterion of similarity.

5. From statement 2 and 4.

9th Grade Math

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