Answer

Verified

357.9k+ views

**Hint:**The given question is based on an isosceles triangle. An isosceles triangle is a triangle with (at least) two equal sides. In this question we want to draw an isosceles triangle by the given properties and want to prove that \[OB = OC\] and \[AO\] bisects \[\angle A\].

**Complete step by step solution:**

In this problem,

In an isosceles triangle, \[ABC\] , with \[AB = AC\] the bisectors of \[\angle B\] and \[\angle C\] intersect at each other. Join \[A\] to \[O\].

Here, we need to show that,

(i) \[OB = OC\]

Let us draw an isosceles triangle \[ABC\],

Given,

In \[\Delta ABC\], we have

\[AB = AC\]

\[OB\]is the bisector of \[\angle B\]

So, \[\angle ABO = \angle OBC = \dfrac{1}{2}\angle B\] ………………………………. (1)

\[OC\] is the bisector of \[\angle C\]

So,\[\angle ACO = \angle OCB = \dfrac{1}{2}\angle C\] …………………………………. (2)

To prove: \[OB = OC\]

Proof:

Given , \[AB = AC\]

Angles opposite to equal sides are equal

\[\angle ACB = \angle ABC\]

\[\dfrac{1}{2}\angle ACB = \dfrac{1}{2}\angle ABC\]

By equation (1) and (2), we get

\[\angle OCB = \angle OBC\]

Hence,

Sides opposite to equal angles are equal,

\[OB = OC\]

**Hence proved.**

(ii) \[AO\] bisects \[\angle A\]

To prove: \[AO\] bisects \[\angle A\]

i.e., \[\angle OAB = \angle OAC\]

Proof:

We already proved that \[OB = OC\]

Also in \[\Delta ABC\] and \[\Delta ACO\]we have,

Given, \[AB = AC\] (given)

\[AO = AO\] (common)

From the equation (1) \[OB = OC\]

Therefore, by SAS (Side Angle Side) Congruence rule,

\[\Delta ABO \cong \Delta ACO\]

Since corresponding parts of congruent triangles are equal,

\[\angle OAB = \angle OAC\]

\[AO\] bisects \[\angle A\]

**Hence proved.**

**Note:**

1) An isosceles triangle therefore has both two equal sides and two equal angles.

2) Angles opposite to equal sides are equal.

3) Sides opposite to equal angles are equal.

4) If any two sides and angle included between the sides of one triangle are equivalent to the corresponding two sides and the angle between the sides of the second triangle, then the two triangles are said to be congruent by SAS rule.

5) Two triangles are congruent if their corresponding sides are equal in length, and their corresponding angles are equal in measure.

Recently Updated Pages

When people say No pun intended what does that mea class 8 english CBSE

Name the states which share their boundary with Indias class 9 social science CBSE

Give an account of the Northern Plains of India class 9 social science CBSE

Change the following sentences into negative and interrogative class 10 english CBSE

Advantages and disadvantages of science

10 examples of friction in our daily life

Trending doubts

Which are the Top 10 Largest Countries of the World?

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths

Who was the first to raise the slogan Inquilab Zindabad class 8 social science CBSE

Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE

One cusec is equal to how many liters class 8 maths CBSE

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

A resolution declaring Purna Swaraj was passed in the class 8 social science CBSE