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.

AA Criterion of Similarly on Quadrilateral

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 \(\frac{PQ}{SU}\) = \(\frac{PT}{TS}\).

Similarly on Quadrilateral

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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