PQRS is a kite and Q lies on AB, such that PR is parallel to AB If angle PQA - Maths - Understanding Quadrilaterals

Question

PQRS is a kite and Q lies on AB, such that PR is parallel to AB
If angle PQA = 20 degree and angle BQR = 30 degree Find (i) angle
PSR (i) angle SPQ (iii) angle SRQ [HINT : Diagonal PR bisects angle
P as well as angle R]

Ajanta Trivedi · Accepted Answer

given: PQRS is a kite and $A B ∥ P R$ $∠ P Q A = 20 a n d ∠ B Q R = 30$ To Find that: $(i) ∠ P S R (i i) ∠ S P Q (i i i) ∠ S R Q$ solution: since PS = PQ and RS = RQ $∠ P S O = ∠ P Q O (1) ∠ R S O = ∠ R Q O (2)$ $∠ P S O + ∠ R S O = ∠ P Q O + ∠ R Q O ∠ P S R = ∠ P Q R$ since the sum of the angles made on the same side of the straight line is 180 deg $∠ P Q R = 180 - (∠ P Q A + ∠ R Q B) = 180 - (20 + 30) = 180 - 50 = 130$ therefore $∠ P S R = 130 \deg$ (3) $∠ O P Q = ∠ P Q A = 20$ [alternate interior angles] $∠ P O Q = 90$ [since the diagonals of a kite intersect at right angle] now in the triangle POQ; $∠ P Q O = 180 - (20 + 90) = 180 - 110 = 70$ therefore $∠ P S O = ∠ P Q O = 70$ in the triangle POS; $∠ S P O = 180 - (90 + 70) = 20 ∠ S P Q = ∠ S P O + ∠ O P Q = 20 + 20 = 40$ $∠ S R Q = 360 - (∠ S P Q + ∠ P Q R + ∠ P S R) = 360 - (40 + 130 + 130) = 360 - 300 = 60$ [sum of interior angles of a quadrilateral is 360 deg] hope this helps you cheers!!

PQRS is a kite and Q lies on AB, such that PR is parallel to AB. If angle PQA = 20 degree and angle BQR = 30 degree. Find (i) angle PSR (i) angle SPQ (iii) angle SRQ. [HINT : Diagonal PR bisects angle P as well as angle R]