Question

In the figure to the right, the isosceles ΔABC with a base AC and measure of ∠B equal to 36° has angle bisector AD drawn through it. Prove that triangles ∆CDA and ∆ADB are isosceles

Answer 1

Refer to the attached image.

Given : Triangle ABC is isosceles, measure of ∠B equal to 36° and AD is an angle bisector.

To prove: Triangle CDA and ADB are isosceles

Proof:

Since triangle ABC is isosceles,

therefore AB=BC

Now, $\angle A=\angle C$

(Angles opposite to the equal opposite sides are always equal)

Let $\angle A=\angle C = x$

Therefore, by angle sum property which states

"The sum of all the angles of a triangle is 180 degrees"

$\angle A+\angle B+\angle C=180^{\circ}$

$x+36^{\circ}+x=180^{\circ}$

$2x+36^{\circ}=180^{\circ}$

$2x=180^{\circ}-36^{\circ}$

$2x= 72^{\circ}$

$x=36^{\circ}$

Hence, $\angle A=\angle C=72^{\circ}$

Since, AD is an angle bisector.

Therefore, it divides angle A into two equal parts.

Therefore, $\angle BAD=\angle DAC=36^{\circ}$

Now, consider triangle ABD,

here since $\angle ABD=\angle BAD=36^{\circ}$

Therefore, AD = BD

"By the converse of the base angles theorem, which states that if two angles of a triangle are congruent, then sides opposite those angles are congruent."

Therefore, Triangle ABD is isosceles triangle.

Similarly consider triangle ACD,

By angle sum property,

$\angle ADC+\angle DCA+\angle CAD=180^{\circ}$

$36^{\circ} +72^{\circ} + \angle ADC = 180^{\circ}$

$\angle ADC = 72^{\circ}$

Therefore, $\angle ADC=\angle ACD=72^{\circ}$

Therefore, AC = CD

"By the converse of the base angles theorem, which states that if two angles of a triangle are congruent, then sides opposite those angles are congruent."

Therefore, Triangle ADC is isosceles triangle.

Answer 2

If the two sides or two angles are congruent in a triangle then the triangle is known as the isosceles triangle. Both triangles ∆CDA and ∆ADB are isosceles.

What is the triangle?

Triangle is a polygon that has three sides and three angles. The sum of the angle of the triangle is 180 degrees.

In the figure to the right, the isosceles ΔABC with a base AC and measure of ∠B equal to 36° has angle bisector AD drawn through it.

Since the ΔABC is an isosceles triangle, then the side AB and BC will be equal and their angles too.

AB = BC

∠A = ∠C = x

We know that the sum of angles of the triangle is 180 degrees. Then we have

∠A + ∠B + ∠C = 180

      x + 36 + x = 180

                  2x = 144

                    x = 72

Then the value of the angles ∠A and ∠C are 72 degrees.

Since AD is an angle bisector, then we have

∠BAD = ∠DAC = 36°

Then consider the triangle ΔABD, then we have

∠BAD = ∠ABD = 36°

And  AD = BD

The reverse of the base angles theorem argues that if two triangle angles are congruent, then the edges opposite both angles are congruent.

Similarly, in triangle ADB

∠ADC will be 72°

Then we have

∠ADC = ∠ACD = 72°

Then the triangle is an isosceles triangle.

More about the triangle link is given below.

https://brainly.com/question/25813512

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