# AA Criterion of Similarly on Quadrilateral

Here we will prove the theorems related to AA Criterion of Similarity.

1. In the quadrilateral ABCD, AB CD. Prove that OA × OD = OB × OC.

Solution:

Proof:

 Statement Reason 1. In ∆ OAB and ∆OCD,(i) ∠AOB = ∠COD(ii) ∠OBA = ∠ODC. 1.(i) Vertically opposite angles.(ii) Alternate angles. 2. ∆ OAB ∼ ∆OCD. 2. By AA criterion of similarly. 3. Therefore, $$\frac{OA}{OC}$$ = $$\frac{OB}{OD}$$⟹ OA × OD = OB × OC. (Proved) 3. Corrosponding sides of similar triangles are proportional.

2. In the quadrilateral PQRS, PQ ∥ RS. T is any point on PS. QT is joined and produced to meet RS produced at U. Prove that $$\frac{PQ}{SU}$$ = $$\frac{PT}{TS}$$.

Solution:

Proof:

 Statement Reason 1. In ∆PQT and ∆SUT,(i) ∠PTQ = ∠STU(ii) ∠QPT = ∠TSU 1.(i) Vertically opposite angles are equal(ii) Alternate angles are equal 2. ∆PQT ∼ ∆SUT 2. By AA criterion of similarity 3. $$\frac{PQ}{SU}$$ = $$\frac{PT}{TS}$$. (Proved) 3. Corresponding sides of similar triangles are proportional.

9th Grade Math

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