What type of a quadrilateral is one in which the opposite sides are equal? Justify your answer. Class 8

What type of a quadrilateral is one in which the opposite sides are equal? Justify your answer. Class 8

Question 1.

What type of a quadrilateral is one in which the opposite sides are equal? Justify your answer. Class 8

[Hint : Draw a diagonal and check for congruent triangles.]

Solution:

Draw diagonal AC.

In ∆ADC and ∆CBA,

AD = BC (Given)

CD = AB (Given)

and AC = AC (Common)

So, ∆ADC ≅ ∆CBA (By SSS)

Hence, ∠DCA = ∠BAC

and ∠DAC = ∠BCA

From ∠DCA = ∠BAC, we get,

AB || CD

and from ∠DAC = ∠BCA, we get,

AD || BC.

Thus, ABCD is a parallelogram.

Question 2.

Will the sum of the angles in a quadrilateral such as the following one also be 360°? Find the answer using geometric reasoning as well as by constructing this figure and measuring. Class 8

Solution:

Construction : Join BD.

Now, ∠A + ∠ABD + ∠ADB = 180°

and ∠C + ∠CBD + ∠CDB = 180°

(Angle sum property of a triangle)

So, ∠A + ∠ABD + ∠ADB + ∠C + ∠CBD + ∠CDB = 180° + 180°.

⇒ ∠A + ∠B + ∠C + ∠D = 360°.

Note : Verify it by construction also.