Find the Remaining Angles in the Following Quadrilaterals
(i)
In quadrilateral P R A E, opposite sides are parallel. Given ∠P = 40°. Find all remaining angles.
It is a parallelogram.
Adjacent angles are supplementary:
∠R = 180° − 40° = 140°
Opposite angles are equal:
∠A = 40°, ∠E = 140°
(ii)
In quadrilateral S R Q P, opposite sides are parallel. Given ∠P = 110°. Find all remaining angles.
It is a parallelogram.
∠Q = 180° − 110° = 70°
Opposite angles are equal:
∠R = 110°, ∠S = 70°
(iii)
In quadrilateral X W V U, all sides are equal and a diagonal is drawn. The angle between the diagonal and side at V is 30°. Find all angles.
It is a rhombus.
Diagonal bisects the angle:
∠V = 2 × 30° = 60°
Opposite angle: ∠X = 60°
Adjacent angles = 180° − 60° = 120°
∠U = 120°, ∠W = 120°
(iv)
In quadrilateral O I E A, all sides are equal and a diagonal is drawn. The angle between the diagonal and side at E is 20°. Find all angles.
It is a rhombus.
Diagonal bisects the angle:
∠E = 2 × 20° = 40°
Opposite angle: ∠O = 40°
Adjacent angles = 180° − 40° = 140°
∠A = 140°, ∠I = 140°
Final Answers Summary:
- (i) 40°, 140°, 40°, 140°
- (ii) 110°, 70°, 110°, 70°
- (iii) 60°, 120°, 60°, 120°
- (iv) 40°, 140°, 40°, 140°
Construction Using Diagonal Properties (With Diagrams)
Question 2: Parallelogram
Construct a parallelogram with diagonals 7 cm and 5 cm intersecting at 140°.
- Draw AC = 7 cm.
- Find midpoint O.
- Draw angle 140° at O.
- Mark OB = OD = 2.5 cm.
- Join A-B-C-D-A.
Question 3: Rhombus
Construct a rhombus with diagonals 5 cm and 4 cm.
- Draw AC = 5 cm.
- Find midpoint O.
- Draw perpendicular at O.
- Mark OB = OD = 2 cm.
- Join A-B-C-D-A.