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    Exercise 8.2.3

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6th class > > Exercise 8.2.3

Exercise 8.2.3

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

(i) AD =

(ii) ∠ DCB = ∠

(iii) OC =

(iv) m ∠DAB + m ∠CDA = °.

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

Where ∠B = 100°.
Adjacent angles of a parallelogram are supplementary ∠A + ∠B = 180°, z + 100° = 180°.
= z + 100 = 180 , z = .
= In parallelogram opposite angles are equal = ∠C = ∠A : ∠C = °,∠D = ∠B : ∠D = °.
Adjacent angles are supplementary : ∠A+∠D = 180° : x+50° = 180° : x = °.
= ∠C+∠D = 180° : y = 50° + 180° : y = .
Opposite angles of a parallelogram are equal : ∠ABC = ∠D : ∠ABC = 50.
= ∠ABC + z = 180° : 50° + z = 180° : z = °.
Where BCO = 30° and AOD = 90°.x=°.
Angle BOC, Sum of all angles = 180 : y+x+30° = 180°: y+90°+30° = 180° : y = °.
ODA = OBC : z = y : z=°.
Where ∠B = 80°.
Sum of adjacent angles of a parallelogram is 180° : ∠A+∠B = 180° : x+80° = 180° : x=.
Opposite angles of parallelogram are equal ∠B = ∠D : y=°.
BCO is a line ∠BCD + ∠OCD = 180° : 100°+z=180° : z=180°-100° : z=°.

3.Can a quadrilateral ABCD be a parallelogram if.

(i) ∠D + ∠B = 180°?

Using the angle sum property of a quadrilateral,∠A+∠B+∠D+∠C=360.
∠A+∠C+180=360 : ∠A+∠C=360-180. : ∠A+∠C=180.
= opposite angles also be same then, ∠D+∠B=,∠B=,∠D=.

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

The opposite sides of a parallelogram are of equal length.
AD = 4cm and BC = 4.4 cm,opposite sides AD and BC are of different Lengths,So ABCD is a parallelogram.

(iii) ∠A = 70° and ∠C = 65°?

In parallelogram opposite angles are equal A and C has different angles, so ABCD is not a parallelogram.

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.

The angles are 3x and 2x respectively.
∠A + ∠B = 180 : 3x + 2x = 180 : 5x = 180 : x = .
Thus,one of the angles = 3 x 3(36) = ,other angle = 2(36) = .

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

In a parallelogram, adjacent angles are supplementary, meaning their sum is 180 degrees.
If two adjacent angles have equal measure, let's denote each of these angles by 𝑥.
Since these two angles are supplementary: 𝑥+𝑥=180 degrees : 2𝑥=180 𝑥=degrees.
Since a parallelogram has opposite angles that are equal and the sum of all internal angles of a parallelogram is 360 degrees, the other two angles must also be 90 degrees.
Therefore, the measure of each angle in the parallelogram is degrees.

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

∠HOP + 70° = 180° since they form a linear pair : ∠HOP = 180° - 70° : ∠HOP = °
∠O = ∠E since opposite angles in a paralellogram are equal ∠O = 110 the ∠E = x = .
∠EHP = ∠HPO since they are alternate interior angles Thus, y = °.
z + 40° = 70° since they form corresponding angles z = 70° - 40°,z = °.

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

In a parallelogram, the opposite sides have equal length.In GUNS, SG = NU
3x = 18 : x = 18/3 : x = .
Also SN = GU : 26 = 3y -1 : 3y = 26 +1 : y = 27/3 : y = .
The diagonals of a parallelogram bisect each other.
Thus in parallelogram RUNS, Considering diagonal SU, y + 7 = 20 : y = 20 - 7 : y = .
Considering diagonl RN, x + y = 16 : x + 13 = 16 : x = .

9.In the above figure both RISK and CLUE are parallelograms. Find the value of x. (Lengths are in cm)

In parallelogram RISK, RKS + ∠ISK = 180° : 120° + ∠ISK = 180° : ∠ISK = 180° - 120° : ∠ISK = °.
∠I = ∠K Opposite angels of parallelogram are equal = 120°
In parallelogram CLUE, ∠L = ∠E Opposite angels of parallelogram are equal= 70°
The sum of the measures of all the interior angles of a triangle is 180º.
x + 60°+ 70° = 180° : x +130° = 180° : x = °.

10. Explain how this figure is a trapezium. Which of its two sides are parallel?(Lengths are in cm)

Since two pair of adjacent angles which form pairs of consecutive interior angles are supplementary.
∠L + ∠M = 180° : Thus, 80° + 100° = °.
NM is parallel to KL
Hence, KLMN is a trapezium as it has a pair of parallel sides KL and NM.

11.Find m∠C in if AB || DC

ABCD is a Trapezium, in which AB is parallel to DC .
∠B + ∠C = 180°: 120° + ∠C = 180°
So,∠C = 180° - 120° : ∠C = °.
Therefore, m∠C = 60°

12. Find the measure of ∠P and ∠S if SP RQ.(If you find m∠R, is there more than one method to find m∠P?)

SPQR is a trapezium.∠S + ∠R = 180°:∠S + 90° = 180°:∠S = °.
Using the angle sum property of a quadrilateral, ∠S + ∠P + ∠Q + ∠R = 360° : 90° + ∠P + 130° + 90° = 360°
∠P + 310° = 360° : ∠P = 360° - 310° : ∠P = °