# Understanding Quadrilaterals

## Exercise 3.3

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

(i) AD = …… (ii) Ð DCB = ……

(iii) OC = …… (iv) *m Ð *DAB + *m Ð *CDA = ……

Answer:

(i) AD = BC

The opposite sides of a parallelogram are congruent.

(ii) Ð DCB = Ð DAB

The opposite angles of a parallelogram are congruent.

(iii) OC = OA

The diagonals of a parallelogram bisect each other.

(iv) *m Ð *DAB + *m Ð *CDA = 180^{0}

The adjacent angles of a parallelogram are supplementary.

2). Consider the following parallelograms. Find the values of the unknowns *x*, *y*, *z*.

(i) The adjacent angles of a parallelogram are supplementary.

Ð ABC + Ð BCD = 180^{0}

100^{0} + *x* = 180^{0}

*x* = 180 – 100

*x* = 80^{0}

the opposite angles of a parallelogram are congruent

Ð ADC = Ð ABC

but Ð ABC = 100^{0}

Ð ADC = *y *= 100^{0}

Similarly ÐBCD = ÐBAD

but Ð BCD = 80^{0}

Ð BAD = *z* = 80^{0}

(ii)

The adjacent angles of a parallelogram are supplementary.

50^{0} + *x* = 180^{0}

*x* = 180^{0 }– 50^{0}

*x* = 130^{0}

the opposite angles of a parallelogram are congruent

*x* = *y*

but *x* = 130^{0}

*y* = 130^{0}

*z* and *x* are corresponding angles

*z* = *x*

but* x *= 130^{0}

*z* = 130^{0}

(iii)

diagonals are intersecting at right angles

The given quadrilateral is a rhombus

*x* = 90^{0 } vertically opposite angles

*x* + *y* + 30^{0} = 180^{0} angle sum property of triangle

90^{0 }+ *y* + 30^{0} = 180^{0}

*y* + 120^{0} = 180^{0}

*y* = 180^{0} – 120^{0}

*y* = 60^{0}

*z* = *y *alternate angles

but* y *= 60^{0}

*z* = 60^{0}

(iv)

The adjacent angles of a parallelogram are supplementary.

80^{0} + *x* = 180^{0}

*x* = 180^{0 }– 80^{0}

*x* = 100^{0}

the opposite angles of a parallelogram are congruent

*y* = 80^{0}

*z* = *y *alternate angles

but* y *= 80^{0}

*z* = 80^{0}

(v)

the opposite angles of a parallelogram are congruent

*y* = 112^{0}^{ }

*x* + *y* + 40^{0} = 180^{0} angle sum property of triangle

*x* + 112^{0 }+ 40^{0} = 180^{0}

*x* + 152^{0} = 180^{0}

*x* = 180^{0} – 152^{0}

*x* = 28^{0}

*z* = *x *alternate angles

but* x *= 28^{0}

*z* = 28^{0}

3). Can a quadrilateral ABCD be a parallelogram if

(i) Ð D + Ð B = 180°?

(ii) AB = DC = 8 cm, AD = 4 cm and BC = 4.4 cm?

(iii) Ð A = 70° and Ð C = 65°?

Solution:

(i) Ð D + Ð B = 180°?

A quadrilateral can be a parallelogram if

i). The opposite angles are congruent and

ii). The adjacent angles are supplementary

as the information is incomplete

ABCD may or may not be a parallelogram

(ii) AB = DC = 8 cm, AD = 4 cm and BC = 4.4 cm?

A quadrilateral can be a parallelogram if the opposite sides are congruent

Here AB = DC but AD ≠ BC

ABCD is not a parallelogram

(iii) Ð A = 70° and Ð C = 65°?

A quadrilateral can be a parallelogram if the opposite angles are congruent

ÐA ≠ ÐC

ABCD is not a parallelogram

4). Draw a rough figure of a quadrilateral that is not a parallelogram but has exactly two opposite angles of equal measure.

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.

Let the common multiple be *x*

The measures of the two adjacent angles are 3*x* and 2*x*

The adjacent angles of a parallelogram are supplementary

3*x* + 2*x *= 180^{0}

5*x* = 180^{0}

*x* = 180/5

*x* = 36^{0}

3*x* = 3 X 36 = 108^{0}

2*x* = 2 X 36 = 72^{0}

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

Let the two equal adjacent angles be *x*

The adjacent angles of a parallelogram are supplementary

*x* + *x *= 180^{0}

2*x* = 180^{0}

*x* = 180/2

*x* = 90^{0}

opposite angles of parallelogram are equal

therefore, each of the angle of parallelogram is 90^{0}

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

*y* = 40^{0} alternate angles

*y* + *z* = 70^{0} Exterior angle property

40^{0} + *z* = 70^{0}

*z* = 70^{0 }– 40^{0}

*z* = 30^{0}

^{ Ð}POH + 70^{0} = 180^{0} linear pair

ÐPOH = 180^{0 }– 70^{0}

ÐPOH = 110^{0}

ÐPOH = *x* opposite angles

*x* = 110^{0}

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

i).

opposite sides of a parallelogram are congruent

GS = UN and GU = SN

GS = UN

3*x* = 18

*x *= 18/3

*x *= 6 cm

GU = SN

3*y* – 1 = 26

3* y* = 26 + 1

3* y* = 27

*y* = 27/3

*y* = 9 cm

ii).

The diagonals of a parallelogram bisect each other

*x* + *y* = 16 and *y* + 7 = 20

*y* + 7 = 20

*y* = 20 – 7

*y* = 13 cm

*x* + *y* = 16

*x* + 13 = 16

*x* = 16 – 13

*x* = 3 cm

9). In the above figure both RISK and CLUE are parallelograms. Find the value of *x*.

Adjacent angles of a parallelogram are supplementary

ÐRKS + ÐKSI = 180^{0}

120^{0} + ÐKSI = 180^{0}

ÐKSI = 180^{0 }– 120^{0}

ÐKSI = 60^{0}

Similarly, ÐCLU = ÐUEC opposite angles of parallel

ÐUEC = 70^{0}

70^{0} + 60^{0} + *x* = 180^{0}

130^{0} + *x* = 180^{0}

*x* = 180^{0 }– 130^{0}

*x* = 50^{0}

^{ }10). Explain how this figure is a trapezium. Which of its two sides are parallel? (Fig 3.26)

ÐM + ÐN = 100^{0} + 80^{0} = 180^{0}

One pair of adjacent angles are supplementary

□KLMN is trapezium

KL II MN

11). Find *mÐ*C in Fig 3.27 if AB II DC.

AB II DC and BC is transversal

ÐB + ÐC = 180^{0 }interior angles

120^{0} + ÐC = 180^{0}

ÐC = 180^{0 }– 120^{0}

ÐC = 60^{0}

^{ }12). Find the measure of ÐP and ÐS if SP II RQ in Fig 3.28. (If you find *mÐ*R, is there more than one method to find *mÐ*P?)

SP II RQ and SR is transversal

ÐR + ÐS = 180^{0 }interior angles

90^{0} + ÐS = 180^{0}

ÐS = 180^{0 }– 90^{0}

ÐS = 90^{0}^{ }

ÐP + ÐQ + ÐR + ÐS = 360^{0 }

the sum of the angles of a quadrilateral is 360^{0}

ÐP + 130^{0}+ 90^{0 }+ 90^{0} = 360^{0}

ÐP + 310^{0} = 360^{0}

ÐP = 360^{0 }– 310^{0}

ÐS = 50^{0}

Click here for the solutions of Std 8 Maths

1). Rational Numbers

2). Linear Equations in One Variable

3). Understanding Quadrilaterals

4). Data Handling