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Complete the Following Statement by Means of One of Those Given in Brackets Against Each: If Opposite Angles of a Quadrilateral Are Equal, Then It is Necessarily a - Mathematics

MCQ
Fill in the Blanks

Complete the following statement by means of one of those given in brackets against each:

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

Options

  • parallelogram

  •  rhombus

  •  rectangle

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Solution

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

Reason:

ABCD is a quadrilateral in which  ∠A =∠C and ∠B = ∠D.

We need to show that ABCD is a parallelogram.

In quadrilateral ABCD, we have

 ∠A = ∠C

∠B = ∠D

Therefore,

∠A + ∠B = ∠C + ∠D …… (i)

Since sum of angles of a quadrilateral is  360°

∠A + ∠B = ∠C + ∠D = 360°

From equation (i), we get:

(∠A + ∠B) + ( ∠A + ∠B) = 360°

                  2(∠A +∠B ) = 360°

                      ∠A + ∠B =  180°

Similarly,  ∠C + ∠D = 180°

Now, line AB intersects AD and BC at A and B respectively

Such that  ∠A +∠B = 180°

That is, sum of consecutive interior angles is supplementary.

Therefore, .AD || BC

Similarly, we get AB || DC.

Therefore, ABCD is a parallelogram.

  Is there an error in this question or solution?
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APPEARS IN

RD Sharma Mathematics for Class 9
Chapter 13 Quadrilaterals
Q 14.7 | Page 69
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