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Example 19: In the given parallelogram, find the values of x and y. E D

Solution: In the parallelogram DEFY, (5y)°
∠Y = 45° + 70°
= 115° (7x – 5)° 45° 70°
Now, ∠F + ∠Y = 180° (Adjacent angles of a ||gm F Y
are supplementary)

\ (7x – 5)° + 115° = 180° [∠F = (7x – 5)° and ∠Y = 115°)
7x° + 110° = 180° ⇒ 7x° = 70°

x = 10
Thus, ∠F = (7x – 5)° = (7 × 10 – 5)° = (70 – 5)° = 65°

Since, in a parallelogram, opposite angles are equal.
\ ∠D = ∠F

(5y)° = 65° ⇒ 5y = 65
y = 13

Thus, x = 10 and y = 13.
Example 20: Two adjacent angles of a parallelogram are in the ratio D C
2 : 3. Find the measure of each of its angles.

Solution: Let ABCD be the given parallelogram.
Then, ∠A and ∠B are its adjacent angles. 2x° 3x°
A B
Let ∠A = (2x)° and ∠B = (3x)°.

Then, ∠A + ∠B = 180° (Since, sum of adjacent angles of a ||gm is 180°)
⇒ 2x + 3x = 180

⇒ 5x = 180
⇒ x = 36

Therefore, ∠A = (2 × 36)° = 72° and ∠B = (3 × 36)° = 108°.
Since, opposite angles of a parallelogram are equal, we have

∠A = ∠C and ∠B = ∠D
⇒ ∠C = 72° and ∠D = 108°

Therefore, ∠A = 72°, ∠B = 108°, ∠C = 72°, and ∠D = 108°.
Example 21: If the opposite angles of a parallelogram are (3x + 5)° and (61 – x)°, then calculate all
the four angles of the parallelogram.

Solution: We know that the opposite angles are equal in a parallelogram. D C
(61 – x)°
Therefore, (3x + 5)° = (61 – x)°
⇒ 3x + x = 61 – 5
(3x + 5)°
4x = 56 A B

x = 14

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