The diagonals PR and QS of a cyclic quadrilateral PQRS intersect at X. The tangent at P is parallel to QS. Prove that $P Q = P S$ .
If m $∠ P Q S = 50^{o}$ , then m( $∠ P R S$ ) is

Open in App

Solution

Chords $P S$ subtends.
$∠ P Q S$ and $∠ P R S$ in the same segment.
$∠ P Q S = ∠ P R S$
$∠ P Q S = 80 °$ [L subtended by same chord / arc in same segment in equal]
$∠ P Q R = 80 + 50 = 130 ∠ P Q R + ∠ P S R = 180$ (sum of opposite angle of cyclic quadrilaterals is 180°)
$130 + ∠ P S R = 180 ∠ P S R = 50 °$
And as we know $∠ P Q S = 50 ∴ ∠ P R S = 360 - (∠ P Q S + ∠ P S R) = 360 - 100 = 260$

Suggest Corrections

Similar questions

Q. The diagonals PR and QS of a cyclic quadrilateral PQRS intersect at X. The tangent at P is parallel to QS. Prove that PR bisects

∠

QRS.

Q. PQRS is a quadrilateral in which diagonals PR and QS intersect at O. Prove that PQ + QR + RS + SP < 2 (PR + QS).

Q. Construct a cyclic quadrilateral

P Q R S

such that

P Q = Q R = 4.3 c m

and diagonals

P R = 6.2 c m, Q S = 5.8 c m

. Then find

m ∠ P Q R

In Fig. PQRS is a quadrilateral in which diagonals PR and QS intersect in O. [4 MARKS]

Show that:
(i) PQ + QR + RS + SP $>$ PR + QS
(ii) PQ + QR + RS + SP $<$ 2 (PR + QS)

P Q R S

is a quadrilateral in which the diagonals intersect each other at O. Prove that

P Q + Q R + R S + S P < 2 (P R + Q S)

Join BYJU'S Learning Program

The diagonals PR and QS of a cyclic quadrilateral PQRS intersect at X. The tangent at P is parallel to QS. Prove that PQ=PS.If m∠PQS=50o, then m(∠PRS) is

The diagonals PR and QS of a cyclic quadrilateral PQRS intersect at X. The tangent at P is parallel to QS. Prove that $P Q = P S$ .
If m $∠ P Q S = 50^{o}$ , then m( $∠ P R S$ ) is