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°