# Quadrilaterals

## Exercise 3.3

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

**Answer:** (a) AD = Opposite Sides are equal

(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°`

Question 10: Explain how this figure is a trapezium. Which of its two sides are parallel?

**Answer:** Sum of internal angles on side of transversal ML is 180°. So, SM||KL. Since two sides are parallel, so this is a trapezium.

Question 11: In the given figure, find ∠C if AB||DC

**Answer:** AB||DC, so sum of internal angles on one side of transversal BC = 180°

Or, ∠C + 120° = 180°

Or, ∠C = 180° - 120° = 60°

Question 12: In the given figure, find the measure of ∠P and ∠S if SP||RQ.

**Answer:** SP||RQ, so, ∠S = ∠R = 90°

Sum of internal angles of quadrilateral PQRS = 360°

Or, ∠P + 90° + 90° + 130° = 360°

Or, ∠P + 310° = 360°

Or, ∠P = 360° - 30° = 50°