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Example 27: Find ∠DCA in the given square ABCD. D C
1
Solution: As ABCD is a square, AD = DC and ∠ADC = 90°.
Now in ∆ADC, AD = DC

⇒ ∠1 = ∠2 (Angles opp. to equal sides are equal)
2
Also, ∠1 + ∠2 + ∠ADC = 180° (Angle sum property)
A B
Or ∠1 + ∠1 + 90° = 180°

2 × ∠1 = 180° – 90° = 90° ⇒ ∠1 = 45° \ ∠DCA = 45°

Quick Check

Which of the following quadrilaterals is a parallelogram? Give reasons.
1. 2. 3. 4.

Practice Time 3D

1. In the given figure, ∠EBC = ∠ECB = 60° and EC = 8 cm, then find the length of the side of the square
ABCD.
E

B 60° 60° C

A D
2. In a quadrilateral PQRS, the bisectors of ∠Q and ∠R meet at point O. If ∠P = 60°, ∠S = 80°, find
∠QOR.

3. In the figure given alongside, ABCD is a rhombus and diagonals intersect at O. If D C
∠OAB : ∠OBA = 3 : 2, find the angles of the DAOD.
O
4. In a rectangle ABCD, AC and BD are diagonals that intersect at O. If AO = (2y + 3) units
and DO = (3y + 1) units, find the value of y.
5. One of the diagonals of a rhombus is equal to one of its sides. Find the angles of A B
the rhombus.
6. ABCD is a rhombus. If ∠ACD = 40°, find ∠ADB.

D C
40°

O

A B

75 Quadrilaterals