A quadrilateral is a closed polygon that has four sides, four vertices. We can use theorems related to the AA criterion of similarly to explore the concept quadrilateral. Have a look at the following sections to know about the theorems on the angle-angle similarly criterion and the solved questions.

## Theorems on AA Criterion of Similarly on Quadrilateral

Here we will provide the prove related to the theorems on the AA criterion of similarly.

Theorem 1: In quadrilateral ABCD, AB ∥ CD and O is the point of intersection of two diagonals, such that OA × OD = OB × OC.

Given: ABCD is a quadrilateral, the side AB is parallel to CD.
Point O is the intersection of two diagonals.

To Prove: OA × OD = OB × OC

Proof:

Statement Reason
In ∆ABC, ∆BPC
∠ACB = ∠PCB
∠ABC = ∠BPC = 90°
Vertically opposite angles are equal
Alternate angles
∆OAB ∼ ∆OCD By AA criterion of similarity
Therefore, $$\frac { OA }{ OC }$$ = $$\frac { OB }{ OD }$$ Corresponding sides of similar triangles are proportional

Hence, OA × OD = OB × OC.

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Theorem 2: In quadrilateral PQRS, PQ ∥ RS. T is a point on PS. QT is joined and produced to meet RS produced at U, then $$\frac { PQ }{ SU }$$ = $$\frac { PT }{ TS }$$.

Given: PQRS is a quadrilateral, sides PQ and RS are parallel to each other. T is a point on the side PS.

Construction:
Let us take a quadrilateral PQRS, where PQ is parallel to RS.
Locate any point T on PS and join QT and extend it till U and U is a straight line extending from RS.

To Prove: $$\frac { PQ }{ SU }$$ = $$\frac { PT }{ TS }$$

Proof:

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

Hence, proved.

### Questions on Quadrilateral AA Criterion of Similarly

Question 1:
In a quadrilateral PQRS, PQ is parallel to RS and the meeting point of diagonals. If OP = 20 cm, OR = 5 cm, OS = 6 cm, find OQ.

Solution:
Given that,
PQRS is a quadrilateral where PQ is parallel to RS
OP = 20 cm, OR = 5 cm, OS = 6 cm
According to the theorem on AA criterion on similarity on quadrilateral OP × OS = OQ × OR
20 x 6 = OQ x 5
120 = OQ x 5
OQ = 24
Therefore, OQ = 24 cm.

Question 2:
In quadrilateral ABCD, AB ∥ CD. P is a point on AD. BP is joined and produced to meet CD produced at U. If AB = 10 cm, DU = 5 cm, PD = 12 cm, find PA.

Solution:
Given that,
AB = 10 cm, DU = 5 cm, PD = 12 cm
ABCD is a quadrilateral, AB is parallel to CD
P is a point on AD. BP is joined and produced to meet CD produced at U
$$\frac { AB }{ DU }$$ = $$\frac { AP }{ PD }$$
$$\frac { 10 }{ 5 }$$ = $$\frac { AP }{ 12 }$$
2 = $$\frac { AP }{ 12 }$$
AP = 2 x 12 = 24