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Understanding Quadrilaterals

    Execise 3.3

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8th class > Understanding Quadrilaterals > Execise 3.3

Execise 3.3

  1. Given a parallelogram ABCD. Complete each statement along with the definition or property used.
ElementEqual toProperty Used
AD
∠DCB∠
OC
m ∠DAB + m ∠CDA°

Explanation:

(i) AD = BC as opposite sides of a parallelogram are equal

(ii) ∠DCB = ∠DAB as opposite angles of a parallelogram are equal

(iii) OC = OA as diagonals of a parallelogram bisect each other

(iv) m ∠DAB + m ∠CDA = 180° as adjacent angles of a parallelogram are supplementary.

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

Instructions

Where one angle = 100°.
Adjacent angles of a parallelogram are supplementary: z + 100 = 180° ⇒ z = °
In parallelogram, opposite angles are equal ⇒ y = °
Also, x = z = °

Instructions

Opposite angles are equal and adjacent angles are supplementary : z + 50° = 180° ⇒ z = °
x and z are angles ⇒ x = °
Opposite angles of a parallelogram are equal : x = y = °
So, x = y = z = 130°

Instructions

From the figure, we can see: x = °
Sum of all angles in a triangle = 180° ⇒ y + x + 30° = 180° ⇒ y + 90° + 30° = 180° ⇒ y = °
z = y (as they are angles ) ⇒ z = °

Instructions

The 80° angle is to z. It is also to y.
Thus, y = z = °
Sum of adjacent angles of a parallelogram is 180° ⇒ x + 80° = 180° ⇒ x = °

Can a quadrilateral ABCD be a parallelogram if :

Instructions

∠D + ∠B = 180°
AB = DC = 8 cm, AD = 4 cm and BC = 4.4 cm
∠A = 70° and ∠C = 65°
Yes
No

Explanation

(i) Yes, since the opposite angles i.e. ∠D and ∠B are equal.

(ii) No, since the opposite sides i.e. AD and BC must be equal.

(iii) No, since the opposite angles must be of the same measure, which isn't the case with ∠A and ∠C.

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

Solution:

ABCD is quadrilateral whose opposite angles are equal as shown above.

ABCD is a kite where ∠B = ∠

In a kite, the angle between unequal sides is equal.

Let's prove the same.

Drawing a line from A to C and we get two triangles with common base AC.

In ∆ABC and ∆ADC:

AB =

BC =

AC is to both

Thus, ∆ABC ≅ ∆ADC by congruence.

Hence, corresponding parts of congruent triangles are equal.

Therefore, ∠B = ∠

However, the quadrilateral ABCD is not a parallelogram as the measures of the remaining pair of opposite angles, ∠A and ∠C are not equal since they form angles between equal sides.

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.

Instructions

The angles are 3x and 2x respectively.
∠A + ∠B = 180° ⇒ 3x + 2x = 180° ⇒ 5x = 180° ⇒ x = °
Thus, one of the angles = 3 × (36) = ° while the 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.

Instructions

In a parallelogram, adjacent angles are supplementary: their sum is 180°.
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° ⇒ 2x = 180° ⇒ x = °
Since a parallelogram has opposite angles that are equal and the sum of all internal angles of a parallelogram is 360°, the other two angles must also be 90°.
Therefore, the measure of each angle in the parallelogram is °.

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

Instructions

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

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

Instructions

In a parallelogram, the opposite sides have equal length. In GUNS, SG = NU
3x = 18 ⇒ x = 183 ⇒ x = cm
Also SN = GU ⇒ 26 = 3y - 1 ⇒ 3y = 26 + 1 ⇒ y = 273 ⇒ y = cm
Thus, x = 6 cm while y = 9 cm.

Instructions

The diagonals of a parallelogram each other.
Thus in parallelogram RUNS, considering diagonal SU: y + 7 = 20 ⇒ y = 20 - 7 ⇒ y = cm.
Considering diagonal RN: x + y = 16 ⇒ x + 13 = 16 ⇒ x = cm.

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

Instructions

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

Instructions

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

Instructions

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?)

Instructions

SPQR is a trapezium. So, ∠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 = ° - ° ⇒ ∠P = °