[5 marks sum]

Question 15 Marks

Show that the bisectors of the angles of a parallelogram enclose a rectangle.

Answer

$\begin{array}{l} \text { [Hint. } \angle A+\angle D=180 \Rightarrow \angle A+\angle D=90 \\ \Rightarrow \angle A P D=90 \Rightarrow \angle S P Q=90 . \\ \text { Similarly, } \angle S R Q=90, \angle P Q R=90 \text { and } \angle P S R=90 .]\end{array}$

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Question 25 Marks

In the given figure, ABCD is a parallelogram and X, Y are points on diagonal BD such that DX = BY Prove that CXAY is a parallelogram.

Answer

[Hint. Join AC, meeting BD at $O$.
Diagonals of a llgm bisect each other. So, $O A=O C$ and $O D=O B$.
Also, $O D-D X=O B-B Y \Rightarrow O X=O Y$.
$\therefore$ CXAY is a quadrilateral whose diagonals bisect each other.]

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Question 35 Marks

In the adjoining figure, ABCD is a parallelogram.
Line segments AX and CY bisect $\angle A$ and $\angle C$ respectively.
Prove that :
(i) $\triangle ADX \cong \triangle CBY$
(ii) $AX = CY$
(iii) $AX \| CY$
(iv) AYCX is a parallelogram.

Answer

[Hint.
(i) $\angle D=\angle B, \angle D A X=\angle B C Y\left(\frac{1}{2} \angle A=\frac{1}{2} \angle C\right), A D=B C$.
(ii) $\angle 1=\angle 2\left(\frac{1}{2} \angle A=\frac{1}{2} \angle C\right)$, and $\angle 2=\angle 3$ (alt. int. $\left.\angle s \right) \Rightarrow \angle 1=\angle 3$.]

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Question 45 Marks

In the given figure, ABCD is a rhombus and $\triangle EDC$ is equilateral.
If $\angle BAD =78$, calculate :
(i) $\angle CBE$
(ii) $\angle DBE$.

Answer

$\angle CBE =21, \angle DBE =30$
[Hint. $\angle E C B=(78+60)$ and $C E=C B$.\[\begin{array}{l}A B=A D \Rightarrow \angle A B D=\angle A D B=51 \\\angle A B C=(180-78)=102 .]\end{array}\]

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Question 55 Marks

In the given figure, ABCD is an isosceles trapezium in which $\angle CDA =2 x$ and $\angle BAD =3 x$.
Find all the angles of the trapezium.

Answer

$\angle A =108, \angle C =72, \angle D =72, \angle B=108$
$\left[\right.$ Hint. $\angle C D A+\angle B A D=180^{\circ}$
$A B C D$ being an isosceles trapezium, we have $\angle A =\angle B$ and $\angle D=\angle C$ ]

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Question 65 Marks

Rajbeer is a farmer. He has a plot of land in the shape of a quadrilateral ABCD as shown in the figure. In ABCD, $A B \| C D$ and AD || BC. He divided the field into two parts, viz, triangle BCE and trapezium CDAE by making an embankment CE such that CE = AD.

Based on the above information, answer the following questions:
Q.1. ABCD is a:
(a) Rectangle (b) Parallelogram (c) Square (d) Trapezium
Q.2. $\angle A$ is equal to:
(a) $\angle D$ (b) $\angle B$ (c) $\angle C$ (d) $\angle E$
Q.3. $\angle DCE$ is equal to:
(a) $\angle E$ (b) $\angle A$ (c) $\angle D$ (d) $\angle B$
Q.4. Which of the following is correct?
(a) $AEC \cong EAD$ (b) $ACE \cong AED$ (c) $CEA \cong EAD$ (d) none of these
Q.5. Diagonal DE is equal to:
(a) diagonal DB (b) diagonal AC (c) DC (d) BC

Answer

1. (b) 2. (d) 3. (c) 4. (a) 5. (b)

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Question 75 Marks

Construct a quadrilateral PQRS in which $PQ =4.5 cm, QR =5.6 cm, RS =4.1 cm$, $\angle Q=60^{\circ}$ and $\angle R =120^{\circ}$.

Answer

self

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Question 85 Marks

Construct a quadrilateral UVWZ in which $UV =6 cm, VW =5.3 cm, UZ =5 cm$, $\angle U =60^{\circ}$ and $\angle V =75^{\circ}$.

Answer

self

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Question 95 Marks

Construct a quadrilateral ABCD in which $AB =4.5 cm, BC =5.2 cm, CD =5 cm$, $DA =4.7 cm$ and $\angle ABC =75^{\circ}$.

Answer

self

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Question 105 Marks

Construct a quadrilateral PQRS in which $PQ =4 cm, \angle P =90^{\circ}, QR =4.3 cm$, $RS =3.6 cm$ and $SP =3.2 cm$.

Answer

self

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Question 115 Marks

Case Study : Rajbeer is a farmer. He has a plot of land in the shape of a quadrilateral ABCD as shown in the figure. In $A B C D, A B \| C D$ and $A D \| B C$. He divided the field into two parts, viz, triangle BCE and trapezium CDAE by making an embankment CE such that $CE = AD$.
Based on the above information, answer the following questions :
1. ABCD is a :
(a) Rectangle
(b) Parallelogram
(c) Square
(d) Trapezium
2. $\angle A$ is equal to:
(a) $\angle D$
(b) $\angle B$
(c) $\angle C$
(d) $\angle E$
3. $\angle DCE$ is equal to:
(a) $\angle E$
(b) $\angle A$
(c) $\angle D$
(d) $\angle B$
4. Which of the following is correct?
(a) $\triangle AEC \cong \triangle EAD$
(b) $\triangle ACE \cong \triangle AED$
(c) $\triangle CEA \cong \triangle EAD$
(d) none of these
5. Diagonal DE is equal to:
(a) diagonal DB
(b) diagonal AC
(c) DC
(d) BC

Answer

(1. b), (2. d), (3. c), (4. a), (5. b)

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MATHEMATICS [5 marks sum]

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