## Exercise 3.3

Question 1: Given a parallelogram ABCD. Complete each statement along with the definition or property used.

(b) Angle DCB = Opposite angles are equal
(c) OC = Diagonals bisect each other
(d) Angle DAB + angle CDA = 180°

Question 2: Consider the following parallelograms. Find the values of the unknowns x, y, z.

(i)

Solution: Here, x = 180° - 100° = 80°
As opposite angles are equal in a parallelogram
So, y = 100° and z = 80°

(ii)

Solution: x, y and z will be complementary to 50°.
So, Required angle = 180° - 50° = 130°

(iii)

Solution: z being opposite angle= 80°
x and y are complementary, x and y
= 180° - 80° = 100°

(iv)

Solution: As angles on one side of a line are always complementary
So, x = 90°
So, y = 180° - (90° + 30°) = 60°
The top vertex angle of the above figure = 60° xx 2=120°
Hence, bottom vertex Angle = 120° and
z = 60°

(v)

Solution: y= 112°, as opposite angles are equal in a parallelogram
As adjacent angles are complementary so angle of the bottom left vertex
=180°-112°=68°
So, z=68°-40°=28°

Another way of solving this is as follows:
As angles x and z are alternate angles of a transversal so they are equal in measurement.

3. Can a quadrilateral ABCD be a parallelogram if

(i) Angle D + angle B = 180°?
(ii) AB = DC = 8 cm, AD = 4 cm and BC 4.4 cm?
(iii) Angle A = 70° and angle C = 65° ?

Solution: (i)It can be , but not always as you need to look for other criteria as well.
(ii) In a parallelogram opposite sides are always equal, here AD BC, so its not a parallelogram.
(iii) Here opposite angles are not equal, so it is not a parallelogram.

Question 5: The measures of two adjacent angles of a parallelogram are in the ratio 3 : 2. Find the measure of each of the angles of the parallelogram.

Solution: Opposite angles of a parallelogram are always add upto 180°.
So, 180°= 3x + 2x
Or, 5x = 180°
Or, x = 36°
So, angles are; 36° xx 3 = 108°
And 36° xx 2 = 72°

Question 6: Two adjacent angles of a parallelogram have equal measure. Find the measure of each of the angles of the parallelogram.

Solution: 90°, as they add up to 180°

Question 7: The adjacent figure HOPE is a parallelogram. Find the angle measures x, y and z. State the properties you use to find them.

Solution: Angle opposite to y = 180° - 70°=110°
Hence, y = 110°
x = 180° - (110° + 40°) = 30°, (triangle’s angle sum)
z = 30° (Alternate angle of a transversal)

Question 8: The following figures GUNS and RUNS are parallelograms. Find x and y. (Lengths are in cm)

Solution As opposite sides are equal in a parallelogram
So, 3y – 1 = 26
Or, 3y = 27
Or, y = 9
Similarly, 3x = 18
Or, x = 6

Solution: As you know diagonals bisect each other in a parallelogram.
So, y + 7 = 20
Or, y = 20 – 7 = 13
Now, x + y = 16
Or, x + 13 = 16
Or, x = 16 – 13 = 3

Question 9: In the given figure both RISK and CLUE are parallelograms. Find the value of x.

Solution: In parallelogram RISK
∠ISK = 180° - 120° = 60°
Similarly, in parallelogram CLUE
∠CEU = 180° - 70° = 110°
Now, in the triangle
x = 180° - (110° - 60°) = 10°