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 = 1800
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 = 1800
1000 + x = 1800
x = 180 – 100
x = 800
the opposite angles of a parallelogram are congruent
Ð ADC = Ð ABC
but Ð ABC = 1000
Ð ADC = y = 1000
Similarly ÐBCD = ÐBAD
but Ð BCD = 800
Ð BAD = z = 800
(ii) 
The adjacent angles of a parallelogram are supplementary.
500 + x = 1800
x = 1800 – 500
x = 1300
the opposite angles of a parallelogram are congruent
x = y
but x = 1300
y = 1300
z and x are corresponding angles
z = x
but x = 1300
z = 1300
(iii) 
diagonals are intersecting at right angles
The given quadrilateral is a rhombus
x = 900 vertically opposite angles
x + y + 300 = 1800 angle sum property of triangle
900 + y + 300 = 1800
y + 1200 = 1800
y = 1800 – 1200
y = 600
z = y alternate angles
but y = 600
z = 600
(iv) 
The adjacent angles of a parallelogram are supplementary.
800 + x = 1800
x = 1800 – 800
x = 1000
the opposite angles of a parallelogram are congruent
y = 800
z = y alternate angles
but y = 800
z = 800
(v) 
the opposite angles of a parallelogram are congruent
y = 1120
x + y + 400 = 1800 angle sum property of triangle
x + 1120 + 400 = 1800
x + 1520 = 1800
x = 1800 – 1520
x = 280
z = x alternate angles
but x = 280
z = 280
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 3x and 2x
The adjacent angles of a parallelogram are supplementary
3x + 2x = 1800
5x = 1800
x = 180/5
x = 360
3x = 3 X 36 = 1080
2x = 2 X 36 = 720
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 = 1800
2x = 1800
x = 180/2
x = 900
opposite angles of parallelogram are equal
therefore, each of the angle of parallelogram is 900
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 = 400 alternate angles
y + z = 700 Exterior angle property
400 + z = 700
z = 700 – 400
z = 300
ÐPOH + 700 = 1800 linear pair
ÐPOH = 1800 – 700
ÐPOH = 1100
ÐPOH = x opposite angles
x = 1100
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
3x = 18
x = 18/3
x = 6 cm
GU = SN
3y – 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 = 1800
1200 + ÐKSI = 1800
ÐKSI = 1800 – 1200
ÐKSI = 600
Similarly, ÐCLU = ÐUEC opposite angles of parallel
ÐUEC = 700
700 + 600 + x = 1800
1300 + x = 1800
x = 1800 – 1300
x = 500
10). Explain how this figure is a trapezium. Which of its two sides are parallel? (Fig 3.26)
ÐM + ÐN = 1000 + 800 = 1800
One pair of adjacent angles are supplementary
□KLMN is trapezium
KL II MN
11). Find mÐC in Fig 3.27 if AB II DC.
AB II DC and BC is transversal
ÐB + ÐC = 1800 interior angles
1200 + ÐC = 1800
ÐC = 1800 – 1200
ÐC = 600
12). Find the measure of ÐP and ÐS if SP II RQ in Fig 3.28. (If you find mÐR, is there more than one method to find mÐP?)
SP II RQ and SR is transversal
ÐR + ÐS = 1800 interior angles
900 + ÐS = 1800
ÐS = 1800 – 900
ÐS = 900
ÐP + ÐQ + ÐR + ÐS = 3600
the sum of the angles of a quadrilateral is 3600
ÐP + 1300+ 900 + 900 = 3600
ÐP + 3100 = 3600
ÐP = 3600 – 3100
ÐS = 500
Click here for the solutions of Std 8 Maths
1). Rational Numbers
2). Linear Equations in One Variable
3). Understanding Quadrilaterals
4). Data Handling
