Ex.8.1 Q11 Quadrilaterals Solution - NCERT Maths Class 9

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Question

In \(\Delta ABC \,\rm{and} \,\Delta DEF\), \(AB \;|| \;DE\),

\(BC = EF\) and \(BC\: || \:EF\). Vertices \(A\), \(B\) and \(C\) are joined to vertices \(D\), \(E\) and \(F\) respectively (see the given figure). Show that

(i) Quadrilateral \(ABED\) is a parallelogram

(ii) Quadrilateral \(BEFC\) is a parallelogram 

(iii) \(\begin{align} &{ \mathrm{AD} \| \mathrm{CF} \text { and } \mathrm{AD}=\mathrm{CF}}  \end{align}\)

(iv) Quadrilateral \(ACFD\) is a parallelogram

(v) \(AC = DF\)

(vi) \(\begin{align}   &{ \triangle \mathrm{ABC} \cong \triangle \mathrm{DEF} \text { . }}\end{align}\)

 

 Video Solution
Quadrilaterals
Ex 8.1 | Question 11

Text Solution

 

What is known?

In \(\Delta ABC\text{ }and\text{ }\Delta DEF,\) 

\(AB = DE, AB || DE, BC = EF\) and \(BC || EF.\)

What is unknown?

How we can show that

(i) Quadrilateral \(ABED\) is a parallelogram

(ii) Quadrilateral \(BEFC\) is a parallelogram

(iii) \(AD || CF\) and \(AD = CF\)

(iv) Quadrilateral \(ACFD\) is a parallelogram

(v) \(AC = DF\)

(vi) \(\Delta ABC\cong \Delta DEF.\)

Reasoning:

We can use the fact that in a quadrilateral if one pair of opposite sides are parallel and equal to each other then it will be a parallelogram and converse is also true. Also by using suitable congruence criterion we can show triangles congruent then we can say corresponding parts of congruent triangles will be equal.

Steps:

(i) It is given that \(AB = DE\) and \(AB \;|| \;DE\).

If one pair of opposite sides of a quadrilateral are equal and parallel to each other, then it will be a parallelogram.

Therefore, quadrilateral \(ABED\) is a

parallelogram.

(ii) Again, \(BC = EF\) and \(BC\: || \:EF\)

Therefore, quadrilateral \(BCFE\)

is a parallelogram.

(iii) As we had observed that \(ABED\) and \(BEFC\) are parallelograms, therefore

\(AD = BE\) and \(AD \;|| \;BE\)

(Opposite sides of a parallelogram are equal and parallel)

And, \(BE = CF\) and \(BE\; || \;CF\)

(Opposite sides of a parallelogram are equal and parallel)

\(\therefore\)  \(AD = CF\) and \(AD \;|| \;CF\)

(iv) As we had observed that one pair of opposite sides (\(AD\) and \(CF\)) of quadrilateral \(ACFD\) are equal and parallel to each other, therefore, it is a parallelogram.

(v) As \(ACFD\) is a parallelogram, therefore, the pair of opposite sides will be equal and parallel to each other.

\(\therefore\)  \(AC\; || \;DF\) and \(AC = DF\)

(vi) \(\begin{align} { \triangle \mathrm{ABC} \text { and } \triangle \mathrm{DEF} \text { , }}  \end{align}\)

\[\begin{align} \mathrm{AB} &=\mathrm{DE}(\text {Given}) \\ \mathrm{BC} &=\mathrm{EF}(\text { Given}) \\ \mathrm{AC} &=\mathrm{DF}\\(\mathrm{ACFD} &\text { is a parallelogram) }\\ \therefore \triangle \mathrm{ABC} &\cong \Delta \mathrm{DEF}\\\text{(By  SSS} &\text { congruence rule })\end{align}\]

 Video Solution
Quadrilaterals
Ex 8.1 | Question 11