Explore topic-wise InterviewSolutions in .

(d) 109°

In a kite, the two pairs of adjacent sides are equal, i.e., PS = PQ and SR = RQ 

In ΔPSQ, 

∠PSQ =∠PQS = \(\frac{180°-70°}{2}=\frac{110°}{2}\) = 55°

∴ ∠QSR = 90.5° – 55° = 35.5°

⇒ ∠RQS = ∠QSR = 35.5°      (∵RQ = RS) 

In ΔSRQ, 

∠R +∠RQS +∠QSR = 180° 

⇒ ∠R + 35.5° + 35.5° = 180° 

⇒ ∠R = 180° – 71° = 109°.

Correct option is (A) 40°

\(\angle P\;and\;\angle R\) are opposite angles in parallelogram PQRS.

\(\therefore\) \(\angle R=\angle P=40^\circ\)   \((\because\angle P=40^\circ\) (given))

Correct option is  A) 40°

Correct option is (C) 90°

\(\angle P\;\&\;\angle Q\) are consecutive angles in parallelogram ABCD.

\(\therefore\) \(\angle P+\angle Q\) \(=180^\circ\)

\(\Rightarrow\) \(\frac{\angle P}2+\frac{\angle Q}2\) \(=\frac{180^\circ}2\)

\(=90^\circ\)

Correct option is  C) 90°

Correct option is (A) 70°

\(\because\) \(\angle C\;\&\;\angle A\) are opposite angle in parallelogram ABCD.

\(\therefore\) \(\angle C=\angle A\) = \(110^\circ\)

\(\Rightarrow\) \(\angle BCD\) = \(110^\circ\)

Since, \(\angle BCD\;\&\;\angle DCE\) form a linear pair.

\(\therefore\) \(\angle BCD+\angle DCE\) = \(180^\circ\)

\(\Rightarrow\) \(\angle DCE\) \(=180^\circ-\angle BCD\)

\(=180^\circ-110^\circ\)

= \(70^\circ\)

Correct option is  A) 70°

Correct option is (A) 40°

\(\because\) ABCD is a parallelogram.

\(\therefore\) AB || DC

\(\therefore\) \(\angle DCE=\angle ABC\)    \((\because\) \(\angle DCE\;and\;\angle ABC\) are corresponding angles as AB || DE)

\(\Rightarrow\) \(\angle DCE=40^\circ\)   \((\because\) \(\angle ABC=40^\circ)\)

Correct option is A) 40°

i. ∠A = 72° [Given] 

In □ABCD, side BC || side AD and side AB is their transversal. [Given] 

∴ ∠A + ∠B = 180° [Interior angles] 

∴ 72° +∠B = 180° 

∴ ∠B = 180° – 72° = 108°

ii. In ∆BPA and ∆CQD, 

∠BPA ≅ ∠CQD [Each angle is of measure 90°] 

Hypotenuse AB ≅ Hypotenuse DC [Given]

seg BP ≅ seg CQ [Perpendicular distance between two parallel lines] 

∴ ∆BPA ≅ ∆CQD [Hypotenuse side test] 

∴ ∠BAP ≅ ∠CDQ [c. a. c. t.] 

∴ ∠A = ∠D 

∴ ∠D = 72° 

∴ ∠B = 108°, ∠D = 72°

No,

Given measures are AS = 3 cm, SC = 4 cm,CD = 5.4 cm,

DA = 59 cm and AC = 8 cm

Here, we observe that AS + SC = 3 + 4 = 7 cm and AC = 8 cm

i.e. the sum of two sides of a triangle is less than the third side, which is absurd.

Hence, we cannot construct such a quadrilateral.

Correct option is (C) 135°

Sum of interior angles on the same side of a transveral of two parallel lines is \(180^\circ.\)

\(\therefore\) \(\angle A+\angle D=180^\circ\)

\(\Rightarrow\) \(\angle D=180^\circ-\angle A\)

\(=180^\circ-45^\circ\)

\(\Rightarrow\) \(\angle D=135^\circ\)

Correct option is C) 135°

6 cm

Since both the diagonals of a rectangle are equal. Therefore, length of other diagonal is also 6 cm.

square

If in a rectangle, adjacent sides are equal, then it is called a square.

The adjacent sides of a parallelogram are 5 cm and 9 cm. Its perimeter is 28 cm.

We know that, perimeter of Parallelogram = 2 × (sum of lengths of adjacent sides)

= 2 × (5 + 9)

= 2 × 14

= 28 cm

In trapezium ABCD with AB||CD, if ∠A = 100^o, then ∠D =80^o.

We know that, in trapezium adjacent angles of non – parallel sides are supplementary.

∠A + ∠D = 180^o

100^o + ∠D = 180^o

∠D = 180^o – 100^o

∠D = 80^o

supplementary

By property of a parallelogram, we know that, the adjacent angles of a parallelogram are supplementary.

True

Every square is also a parallelogram as it has all the properties of a parallelogram but vice-versa is not true.

(a) In a square, all the sides and all the angles are equal.

Hence, square is an equiangular and equilateral polygon.

opposite

Since the line segment connecting two opposite vertices is called diagonal.

9

Nonagon is a polygon which has 9 sides.

(a) 140^o, 40^o, 20^o, 160^o

We know that sum of interior angles of quadrilaterals is 360^o.

So, 140^o + 40^o + 20^o + 160^o = 360^o

In option (d) has angle sum equal to 360^o, but one angle is 180^o if it is considered then the quadrilateral becomes a triangle.

False

As diagonals of a rhombus are perpendicular to each other but not equal.

one

Concave polygon is a polygon in which at least one interior angle is more than 180°.

False

As all angles of a trapezium are not equal.

False

Since in a trapezium, only one pair of sides is parallel.

True

If opposite angles are equal, it has to be a parallelogram.

True

As a square has all the properties of a trapezium. So, we can say that, every square is a trapezium but vice-versa is not true.

False

Since sum of all the angles of a quadrilateral is 360°.

False

As in a parallelogram, all angles are not right angles, while in a rectangle, all angles are equal and are right angles.

True

Since squares possess all the properties of rectangles. Therefore, we can say that, all squares are rectangles but vice-versa is not true.

False.

Because, the angels of rhombus are not equal to 90^o so all rhombuses are not squares.

False

As kites do not satisfy all the properties of a square.

e.g. In square, all the angles are of 90° but in kite, it is not the case.

True

If diagonals are equal, then it is definitely a rectangle. 

(a) rectangle

We know that, the adjacent angles of a parallelogram are supplementary, i.e. their sum equals 180° and given that both the angles are same. Therefore, each angle will be of measure 90°. .

Hence, the parallelogram is a rectangle.

equal

We know that, in a rectangle, both the diagonals are of equal length.

True

As a square possesses all the properties of a rhombus. So, we can say that, every square is a rhombus but vice-versa is not true.

kite

This is a property of kite, i.e. only one diagonal bisects the other.

False

Since in a rectangle, opposite sides are equal and parallel but in a trapezium, it is not so.

True

As a rectangle satisfies all the properties of a trapezium. So, we can say that, every rectangle is a trapezium but vice-versa is not true.

True

Since rectangles satisfy all ”the”properties” of parallelograms. Therefore, we can say that, all rectangles are parallelograms but vice-versa is not true.

No, it cannot be a rectangle, as in rectangle, both the diagonals are of equal lengths.

(a) 60^o

From the given figure,

∠BAD = 30^o

∠BCD = 30^o … [∵opposite angles of parallelogram are equal]

Now, let us consider the triangle CBD

From angle sum property, ∠DBC + ∠BCD + ∠CDB = 180^o

90^o + 30^o + ∠CDB = 180^o

120^o + ∠CDB = 180^o

∠CDB = 180^o – 120^o

∠CDB = 60^o

∴∠BEC = 60^o, because opposite angles of parallelogram are equal.

(b) The diagonals of a kite are perpendicular to each other.

In Rhombus ABCD,

AO = OP + PA = 2+2 = 4 units.

And, OB = OQ + QB = 2 + 1 = 2 units.

Now,

We know that the diagonals of the rhombus bisect at 90⁰

∴ In ΔOAB,

AB² = OA² + OB²

⇒ AB² = 4² + 3²

⇒ AB = √25

⇒ AB = 5 units.

Now, AB is also the diameter of the semi-circle.

∴ radius of the circle will be 2.5 units.

Let the length of other non-consecutive side be x cm.

Then, we have, perimeter of playground = 23 + 23+ x + x

=> 106 = 2 (23+ x)

=>46 + 2x = 106 2x = 106 – 46

=>2x = 60

=>x = 30 m

Hence, the lengths of other three sides are 23m, 30m and 30m. As a kite has two pairs of equal consecutive sides.

Sticks can be treated as the diagonals of a quadrilateral.

Now, since the diagonals (sticks) are bisecting each other at right angles, therefore the shape formed by joining their end points will be a rhombus.

Sticks can be taken as the diagonals of a quadrilateral.

Now, since they are bisecting each other, therefore the shape formed by joining their end points will be a parallelogram.

Hence, it may be a rectangle or a square depending on the angle between the sticks.

Option : (A)

∠P = 100°, ∠Q = 80°, ∠R = 100°

No, diagonals of a parallelogram bisect each other.

According to figure,

In ΔOCD,

∠ODC + ∠OCD + ∠COD = 180°

⇒ 25° + 25° + ∠COD = 180°

⇒ ∠COD = 130°

Acute angle i.e. ∠DOA between the diagonals 

= 180° – ∠DOC 

= 180° – 130° = 50°

This statement is not true as diagonal of a rectangle are equal but not perpendicular.

Answer is (B) 50°

Answer is (C) diagonals of PQRS are perpendicular