Question

In triangle ABC, AB=AC and AD is the bisector of angle A, PROVE THAT ) Traingle ABD is congruent to triangle ADC ii) AD is perpendicular to BC ​

Answer 1

Step-by-step explanation:

Given:-

• In ∆ABC AB = AC.
• AD is bisecting the angle A.

To prove:-

• ∆ABD is congruent to ∆ACD
• AD is Perpendicular to BC.

Prove:-

We know if any line bisect angle in two angles. Then angles are equal.

Here, AD is bisector of angle A.

So, ∠BAD = ∠CAD ----------(i)

In ∆ABD and ∆ACD.

✤ AB = AD [Given]

✤ ∠BAD = ∠CAD [By equation (i)]

✤ AD = AD [Common]

By SAS congruency,

• ∆ABD ≅ ∆ACD

Hence, Proved!!

__________________________________

Now,

By, CPCT

• ∠ADB = ∠ADC

Let,

• ∠ADB = ∠ADC = x

We know,

Sum of all angles forms on straight line is equal to 180°. We also say this statement be linear pair.

So,

$\sf \leadsto \angle ADB + \angle ADC = 180\degree$

$\sf \leadsto x + x = 180\degree$

$\sf \leadsto 2x = 180\degree$

$\sf \leadsto x = \dfrac{180\degree}{2}$

$\leadsto \boxed{\sf \bold{x = 90\degree}}$

So,

∠ADC = 90°

• We know, perpendicular means 90°.

Thus, AD is Perpendicular on BC.

Hence, Proved!!

Answer 2

$\huge{\boxed{\bold{Question}}}$

In triangle ABC, AB=AC and AD is the bisector of angle A, PROVE THAT ) Traingle ABD is congruent to triangle ADC ii) AD is perpendicular to BC

$\huge{\boxed{\bold{Answer}}}$

In ∆ABC AB = AC.

AD is bisecting the angle A.

To prove:-

∆ABD is congruent to ∆ACD

AD is Perpendicular to BC.

Prove:-

We know if any line bisect angle in two angles. Then angles are equal.

Here, AD is bisector of angle A.

So, ∠BAD = ∠CAD ----------(i)

In ∆ABD and ∆ACD.

✤ AB = AD [Given]

✤ ∠BAD = ∠CAD [By equation (i)]

✤ AD = AD [Common]

By SAS congruency,

• ∆ABD ≅ ∆ACD

Hence, Proved!!

__________________________________

Now,

By, CPCT

• ∠ADB = ∠ADC

Let,

• ∠ADB = ∠ADC = x

We know,

Sum of all angles forms on straight line is equal to 180°. We also say this statement be linear pair.

So,

⇝∠ADB+∠ADC=180°

⇝x+x=180°

⇝2x=180°

⇝x= 2/180°

⇝ x=90°

So,

∠ADC = 90°

We know, perpendicular means 90°.

Thus, AD is Perpendicular on BC.

Hence, Proved!!