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Question 11 Mark

Complete the following statements by means of one of those given in brackets against each:
If one pair of opposite sides are equal and parallel, then the figure is _____________. (parallelogram, rectangle, trapezium).

Answer

If one pair of opposite sides are equal and parallel, then the figure is parallelogram.Explanation:

In $\triangle\text{ABC}$ and $\triangle\text{CDA},$ AB = DC (Given) AC = AC (Common)$\angle\text{BAC}=\angle\text{DCA}$ (Because AB || CD, Alternate interior angle are equal)
So, by SAS Congruence rule, we have $\triangle\text{ABC}\cong\triangle\text{CDA}$ Also,$\angle\text{BCA}=\angle\text{DAC}$ (Corresponding parts of congruent triangles are equal)
But, these are alternate interior angles, which are equal. AD || BC Thus, AB || CD and AD || BC. Hence, quadrilateral ABCD is parallelogram.

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Question 21 Mark

Complete the following statements by means of one of those given in brackets against each:
If in a quadrilateral only one pair of opposite sides are parallel, the quadrilateral is _____________. (square, rectangle, trapezium).

Answer

If in a quadrilateral only one pair of opposite sides are parallel, the quadrilateral is trapezium.

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Question 31 Mark

Fill in the blanks to make the following statements correct:
The figure formed by joining the mid-points of consecutive sides of a quadrilateral is ____________.

Answer

The figure formed by joining the mid-points of consecutive sides of a quadrilateral is Parallelogram.

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Question 41 Mark

Fill in the blanks to make the following statements correct:
The triangle formed by joining the mid-points of the sides of an isosceles triangle is ____________.

Answer

The triangle formed by joining the mid-points of the sides of an isosceles triangle is Isosceles.

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Question 51 Mark

Complete the following statements by means of one of those given in brackets against each:
If both pairs of opposite sides of a quadrilateral are equal, then it is necessarily a _________. (rectangle, parallelogram, rhombus).

Answer

If both pairs of opposite sides of a quadrilateral are equal, then it is necessarily a parallelogram. Explanation:

ABCD is a quadrilateral in which AB = CD and BC = DA. We need to show that ABCD is a parallelogram. In $\triangle\text{ACB}$ and $\triangle\text{CAD},$ we have AC = CA (Common) CB = AD (Given) AB = CD (Given) So, by SSS criterion of congruence, we have$\triangle\text{ACB}\cong\triangle\text{CAD}$
By corresponding parts of congruent triangles property.$\angle\text{CAB}=\angle\text{ACD}\ ....(\text{i})$
And $\angle\text{ACB}=\angle\text{CAD}$ Now lines AC intersects AB and DC at A and C,such that$\angle\text{CAB}=\angle\text{ACD}$ [From (i)]
That is, alternate interior angles are equal. Therefore, AB || DC. Similarly, AD || BC. Therefore, ABCD is a parallelogram.

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Question 61 Mark

Complete the following statements by means of one of those given in brackets against each:
If one angle of a parallelogram is a right angle, then it is necessarily a _____________. (rectangle, square, rhombus).

Answer

If one angle of a parallelogram is a right angle, then it is necessarily a rectangle. Explanation:

We have, $\angle\text{A}=90^\circ$ In a parallelogram, opposite angles are equal. Therefore,$\angle\text{C}=90^\circ$
Similarly,$\angle\text{A}+\angle\text{D}=180^\circ$
$90^\circ+\angle\text{D}=180^\circ$
$\angle\text{D}=90^\circ$
Also, $\angle\text{B}=90^\circ$ Thus, a parallelogram with all the angles being right angle and opposite sides being equal is a rectangle.

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Question 71 Mark

Complete the following statements by means of one of those given in brackets against each:
A line drawn from the mid-point of one side of a triangle _____________ another side intersects the third side at its mid-point. (perpendicular to parallel to, to meet).

Answer

A line drawn from the mid-point of one side of a triangle parallel to another side intersects the third side at its mid-point.
Explanation:
This is a theorem.

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Question 81 Mark

Complete the following statements by means of one of those given in brackets against each:
If opposite angles of a quadrilateral are equal, then it is necessarily a _________. (parallelogram, rhombus, rectangle).

Answer

If opposite angles of a quadrilateral are equal, then it is necessarily a parallelogram. Explanation:

ABCD is a quadrilateral in which $\angle\text{A}=\angle\text{C}$ and $\angle\text{B}=\angle\text{D}.$ We need to show that ABCD is a parallelogram. In quadrilateral ABCD, we have$\angle\text{A}=\angle\text{C}$
$\angle\text{B}=\angle\text{D}$
Therefore,$\angle\text{A}+\angle\text{B}=\angle\text{C}+\angle\text{D}\ ....(\text{i})$
Since sum of angles of a quadrilateral is 360º$\angle\text{A}+\angle\text{B}+\angle\text{C}+\angle\text{D}=360^\circ$
From equation (i), we get:$(\angle\text{A}+\angle\text{B})+(\angle\text{A}+\angle\text{B})=360^\circ$
$2(\angle\text{A}+\angle\text{B})=360^\circ$
$\angle\text{A}+\angle\text{B}=180^\circ$
Similarly, $\angle\text{C}+\angle\text{D}=180^\circ$ Now, line AB intersects AD and BC at A and B respectively Such that $\angle\text{A}+\angle\text{B}=180^\circ$ That is, sum of consecutive interior angles is supplementary. Therefore, AD || BC. Similarly, we get AB || DC. Therefore, ABCD is a parallelogram.

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Question 91 Mark

Fill in the blanks to make the following statements correct:
The triangle formed by joining the mid-points of the sides of a right triangle is ____________.

Answer

The triangle formed by joining the mid-points of the sides of a right triangle is Right triangle.

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Question 101 Mark

Complete the following statements by means of one of those given in brackets against each:
Consecutive angles of a parallelogram are _____________. (supplementary, complementary).

Answer

Consecutive angles of a parallelogram are supplementary.
Explanation:
Let ABCD be the given parallelogram.

Thus, AB || DC.
Therefore, $\angle\text{A}+\angle\text{D}=180^\circ$
Consecutive angles $\angle\text{A}$ and $\angle\text{D}$ are supplementary.

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Question 111 Mark

Complete the following statements by means of one of those given in brackets against each:
If consecutive sides of a parallelogram are equal, then it is necessarily a _________. (kite, rhombus, square).

Answer

If consecutive sides of a parallelogram are equal, then it is necessarily a rhombus.
Explanation:

We have ABCD, a parallelogram with AB = BC.
Since ABCD is a parallelogram.
Thus, AB = CD
And BC = AD
But, AB = BC
Therefore,all four sides of the parallelogram are equal, then it is a rhombus.

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