Question

if the bisector of the vertical angle of a traingle bisects the base of the traingle, then prove that traingle is an isosceles traingle.​

Answer 1

Answer:

angle ADB = ABC = 180/2 = 90°

Step-by-step explanation:

Given:

∆ ABC is an isoscelos triangle AD bisects

To prove AD bisects BC at right angle

Prove

Consider ∆ ABD and ∆ADC

(I) AB = AC (Given)

(ii) angle BAD = angle DAC

(iii) AD = AD (common)

By s a s rule ∆ ABD ~= ∆ ADB

By corresponding part of congruent triangles (CPCT)

In ∆ ABD and ∆ ADC, BC = DC, angle BOD = angle DAC = 90°

BO = DC

angle ADB = angle ABC = 180/2 = 90°

Answer 2

★Given -

• ∠BAD = ∠CAD
• BD = CD

★To prove -

• ∆ABC is isosceles.

★Construction -

• Produce AD to E such that AD = DE.
• Join CE.

★Proof -

In ∆ABD and ∆CED,

AD = ED (construction)

∠ADB = ∠EDC (vertically opposite angles)

BD = CD (given)

Hence, ∆ABD ≅ ∆CE (SAS)

AB = CE (CPCT)----------(1)

∠BAD = ∠CED (CPCT)

But, ∠BAD = ∠CAD

∴∠CED = ∠CAD

=> AC = EC (sides opposite to equal angles)-------(2)

From (1) and (2),

AB = AC

∴ ∆ABC is isosceles.

★More to know -

• An isosceles triangle is a triangle with two equal sides.