Question

If the bisector of vertical angle of a triangle is perpendicular to the base of the triangle, then the triangle is:​

Answer 1

Answer:

Hi I will answer your Question

Step-by-step explanation:

Given △ABC, AD is a bisector of ∠A which meets base BC at D such that BD = DC.

Produce AD to meet E such that AD = ED.

Now, in △ABD and △DEC

BD=DC ...... [Given]

AD=DE ........ [By construction]

∠ADB=∠EDC ..... [Vertically opposite angles]

∴ △ABD ≅△EDC [∵SAS congruence ]

⟹ AB=EC and ∠BAD=∠DEC ..... [CPCT]

Also, ∠BAD=∠DAC

⟹ ∠DAC=∠DEC

⟹ In △ACE, ∠AEC=∠CAE

⟹ AC=CE ........ [Sides opposite to equal angles]

⟹ AB=AC

Hence, △ABC is isosceles.

       HENCE, VERIFIED

Answer 2

If the bisector of the vertical angle of a triangle is perpendicular to the base of the triangle, then the triangle is an isosceles triangle.

Step-by-step explanation:

• Let us consider a triangle ABC in which AD is the bisector of ∠A meeting BC in D such that BD=CD
• With reference to the image, AD is produced to E such that AD=DE and connect EC.
• In ΔADB and ΔEDC, we have AD=DE

                        ∠ADB = ∠CDE (∵ Vertically opposite angles are equal)  

• By using the SAS criterion of congruence,

                         ΔADB ≅ ΔEDC  

                      ⇒ AB = EC ------- ($1$)  

                      ⇒ ∠BAD = ∠CAD  

                      ⇒ ∠CAD = ∠CED  

Therefore, AC = EC and AC = AB

In an isosceles triangle, at least two sides of the triangle are equal and we have AB = AC

Hence ΔABC  is an isosceles triangle.