[3 marks sum]

Question 13 Marks

If three angles of a quadrilateral are 90° each, show that the given quadrilateral is a rectangle.

Answer

The given quadrilateral ABCD will be a rectangle, if its each angle is 90°
Since, the sum of interior angles of a quadrilateral is 360°.
∴∠A +∠B + ∠C + ∠D = 360°
⇒ 90° + 90° + 90° + ∠D = 360°
⇒ 270° + ∠D = 360°
⇒ ∠D = 360° – 270°
⇒ ∠D = 90°
Since, each angle of the quadrilateral is 90°. ∴The given quadrilateral is a rectangle.

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Question 23 Marks

In parallelogram ABCD, ∠A = 90°
(i) What is the measure of angle B.
(ii) Write the special name of the parallelogram.

Answer

In parallelogram ABCD, ∠A = 90°

(i) ∵ In a parallelogram, adjacent angles are supplementary
∴ ∠A + ∠B = 180°
⇒ 90° + ∠B = 180°
⇒ ∠B = 180° - 90°
⇒ ∠B = 90°
(ii) The name of the given parallelogram is a rectangle.

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Question 33 Marks

In parallelogram $\text{ABCD,}$ its diagonals intersect at point $O.$ If $OA = 6 \ cm$ and $OB = 7.5 \ cm,$ find the length of $AC$ and $BD.$

Answer

$\because$ When diagonals $AC$ and $BD$ intersect each other at point $O,$
then $OA=OC=\frac{1}{2} AC$
$OB=OD=\frac{1}{2} BD$
$\therefore OA=\frac{1}{2} \times AC$
$\Rightarrow AC=2 \times OA$
$\Rightarrow A C=2 \times 6 \ cm =12 \ cm ,$
$OB=\frac{1}{2} \times BD$
$\Rightarrow BD=2 \times OB$
$\Rightarrow BD=2 \times 7.5 \ cm$
$\Rightarrow B D=15 \ cm $

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Question 43 Marks

In a parallelogram ABCD, its diagonals AC and BD intersect each other at point O.

If AC = 12 cm and BD = 9 cm ; find; lengths of OA and OD.

Answer

∵ When diagonals AC and BD intersect each other at point O,
then $O A=O C=\frac{1}{2} A C$
and $O B=O D=\frac{1}{2} B D$
$\therefore OA =\frac{1}{2} \times AC =\frac{1}{2} \times 12=6 cm$
and $OB =\frac{1}{2} \times BD =\frac{1}{2} \times 9=4.5 cm$

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Question 53 Marks

Three angles of a quadrilateral are equal. If the fourth angle is 69°; find the measure of equal angles.

Answer

Let each equal angle be x°
x + x + x + 69° = 360°

3x = 360°- 69
3x = 291
x = 97°
Each, equal angle = 97°

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Question 63 Marks

In the given figure : ∠b = 2a + 15 and ∠c = 3a + 5; find the values of b and c.

Answer

∵ Sum of angles of quadrilateral = 360°
70° + a + 2a + 15 + 3a + 5 = 360°
6a + 90° = 360°
6a = 270°
a = 45°
∴ b = 2a + 15 = 2 x 45 + 15 = 105°
c = 3a + 5 = 3 x 45 + 5 = 140°
Hence ∠b and ∠c are 105° and 140°

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Question 73 Marks

Two angles of a quadrilateral are 68° and 76°. If the other two angles are in the ratio 5 : 7; find the measure of each of them.

Answer

Two angles are 68° and 76°
Let other two angles be 5x and 7x
68° + 76°+ 5x + 7x = 360°
12x + 144° = 360°
12x = 360° – 144°
12x = 216°
x = 18°
angles are 5x and 7x
i.e. 5 x 18° and 7 x 18° i.e. 90° and 126°

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Question 83 Marks

Use the information given in the following figure to find the value of x.

Answer

Take A, B, C, D as the vertices of Quadrilateral and BA is produced to E (say).
Since ∠EAD = 70°
∴ ∠DAB = 180° – 70°= 110° [EAB is a straight line and AD stands on it ∠EAD+ ∠DAB = 180°]
∴ 110° + 80° + 56° + 3x – 6° = 360°
[∵ sum of interior angles of a quadrilateral = 360°]
∴ 3x = 360° – 110° – 80° – 56° + 6°
3x = 360° – 240° = 120°
∴ x = 40°

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Question 93 Marks

Two angles of a quadrilateral are 89° and 113°. If the other two angles are equal; find the equal angles.

Answer

Let the other angle = x°
According to given,
89° + 113° + x° + x° = 360°
2x° = 360° – 202°
2x° = 158°
$x^{\circ}=\frac{158}{2}$
x^{\circ}=\frac{158}{2}

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MATHS [3 marks sum]

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