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. |
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