Question

Given,O is the center of the circle, if angle OAB = $20^\circ $ and angle OCB = $55^\circ $, then what is the value of angle AOC?

1. $40^\circ $
2. $50^\circ $
3. $60^\circ $
4. $70^\circ $

Answer 1

Hint: OB and OC are the radius of the circle. So, triangle OBC will be an isosceles triangle. Angles on base of an isosceles triangle are equal. So, we will get two angles of the triangle. After that we will use the angle sum property of the triangle to find out the third angle of the triangle. Then, again using angle sum property we will find angle AOB and then subtract angle BOC from angle AOB.

Complete step by step answer:
Given: angle OCB = 55 degree
Angle OAB = 20 degree
OC and OB are the radius of the triangle. So, both will be equal.
OC = OB
Triangle whose two sides are equal is an isosceles triangle. So, triangle OCB is an isosceles triangle. Angles on base of an Isosceles triangle are also equal. So, angle OCB and angle OBC will be equal.
$\angle OCB = \angle OBC$$ = 55^\circ $
Now, we have the value of two angles of the triangle OCB and we have to find the third angle.
We will find the third angle by using the Angle sum property of the triangle.
$\angle OCB + \angle OBC + \angle BOC = 180^\circ $
Putting the values in above equation:
$55^\circ + 55^\circ + \angle BOC = 180^\circ $
$\angle BOC = 180^\circ - 110^\circ $
$\angle BOC = 70^\circ $
Now, again using the angle sum property of the triangle we will find the angle AOB.
In triangle AOB, we know that angle OAB = angle OBA = $20^\circ $.
$\angle OAB + \angle OBA + \angle AOB = 180^\circ $
$20^\circ + 20^\circ + \angle AOB = 180^\circ $
$\angle AOB = 180^\circ - 40^\circ $
$\angle AOB = 140^\circ $
Now, we have to find the angle AOC. so, we will subtract the value of angle BOC from the value of angle AOB.
$\angle AOC = \angle AOB - \angle BOC$
$\angle AOC = 140^\circ - 70^\circ $
$\angle AOC = 70^\circ $
The value of angle AOC is $70^\circ $.

So, the correct answer is “Option 4”.

Note:
An Isosceles triangle is a triangle which has two equal sides. Also, the two angles opposite to the two equal sides are equal. In other words, we can say that “An isosceles triangle is a triangle which has two congruent sides”.
A triangle is a closed figure formed by the three line segments, consisting of interior as well as exterior angles. An interior angle is an angle formed between two adjacent sides of a triangle, whereas an exterior angle is an angle formed between a side of the triangle and an adjacent side extending outward. As per the angle sum property, the sum of all three angles(interior) of a triangle is 180 degrees.