Question

Triangle abc is an isosceles triangle in which ab ac ad bisects exterior angle

Answer 1

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student-nameFatima Afaq asked in Math

abc is an isosceles triangle in which ab=ac.ad bisects exterior angle pac and cd is parallel to ab.show that angle dac=angle bca and show that abcd is a parallelogram

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student-nameVimala answered this

in Math, Class

Let us observe the following figure.

Since ABC is an isosceles triangle we have AB = AC.

AD is the bisector of externaland CD is parallel to AB.

In a triangle an exterior angle is equal to the sum of two opposite interior angles.

Thus in triangle ABC, we have

Since AC intersects lines AD and BC at A and C respectively such that, alternate interior angles are equal.

Therefore, AD is parallel to BC.

Given that CD is parallel to AB.

Thus, ABCD is a parallelogram.

Answer 2

Answer:

Given : ABC is an isosceles triagle in which AB=AC.AD bisects exterior angle QAC and CD∣∣BA.

To show : 

(i)∠DAC=∠BCA

(ii) ABCDABCD is a parallelogram

Proof :

(i)

∠ABC=∠BCA=y (let) (Because triangle ABC is an isosceles triangle)

∠QAD=∠DAC=x (let) (Given)

∠DCA=∠BAC=z (let) (Alternate interior angles)

And we know that an exterior angle of a triangle is equal to the sum of the opposite interior angles.

So,

∠QAD+∠DAC=∠ABC+∠BCA

x+x=y+y

2x=2y

x=y

∠DAC=∠BCA (hence proved)

Now because,

∠DAC=∠BCA (proved above)

Therefore , AD∣∣BC

And CD|∣BA (Given)

Since opposite sides of quadrilateral ABCD are parallel therefore ABCD is a parallelogram.