Question

ABC is an isosceles triangle in which AB=AC.
AD bisects exterior angle QAC and CD parllel to BA
as shown in the figure, Show that
1.DAC = BCA
2.ABCD is a parallelogram​

Answer 1

AnswEr :

$\bf{\green{\underline{\underline{\bf{Given\::}}}}}$

ABC is an Isosceles triangle in which AB = AC. AD bisects exterior angle QAC and CD parallel to BA as shown in the figure.

$\bf{\green{\underline{\underline{\bf{To\:find\::}}}}}$

1. DAC = BCA
2. ABCD is a parallelogram.

$\bf{\green{\underline{\underline{\bf{Explanation\::}}}}}$

1.DAC = BCA

$\sf{AD\:bisects\:\angle PAC\:\:\:\:\underbrace{\bf{Given}}}\\\\\\\angle PAD=\angle DAC...........(1)}}$

$\bf{\red{\underline{\bf{In\:\triangle ABC\::}}}}$

$\sf{\angle B=\angle C\:\:\:\:\underbrace{\bf{Isosceles\:\triangle\:(AB=AC) }}}}}\\\\\sf{Exterior\: \angle PAC=\angle B+\angle C\:\:\:\: \underbrace{\bf{Exterior\:angle\:theorem}}}}}\\\\\sf{\angle PAC=2\angle C}\\\\\sf{\angle PAD+\angle DAC=2\angle ACB}\\\\\sf{\angle DAC+\angle DAC=2\angle ACB\:\:\:\:\underbrace{\bf{\big[from(1)\big]}}}}\\\\\sf{\cancel{2}\angle DAC=\cancel{2}\angle ACB}\\\\\sf{\red{\angle DAC=\angle ACB}}\:\:\:\:\bf{\big[proved\big]}}$

2.ABCD is a parallelogram..

According to the figure for lines BC, AD with transversal AC.

$\sf{\angle DAC=\angle BCA\:are\:alternate\:Interior\:angle\:and\:they\:are\:equal.}\\So,\\\bf{BC||AD}}$

$\bf{\red{\underline{\bf{In\:ABCD\::}}}}}$

$\sf{CD\:parallel\:BA\:\:\:\:\underbrace{\bf{Given}}}}\\\\\sf{BC||AD\:\:\&\:\:CD||BA}}$

∴ Both pairs of opposite sides of quadrilateral ABCD are parallel.

Thus,

$\bf{\red{\underline{\sf{ABCD\:is\:a\:parallelogram.}}}}$

Answer 2

Answer:

Step-by-step explanation:

∠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)

(ii)

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.