Question

In the given figure, AB, BC and AC touch the circle at L, M and N respectively. If ∠B = 70° and ∠C = 60°, then the measure of ∠LON is:-
   (a) 50°
   (b)110°
   (c)120°
   (d) 130°

Answer 1

HINT:- To solve this question, we must be familiar with the term Tangent.
TANGENT: A tangent line of a circle is a line that touches the boundary of the circle or circumference exactly at one point. Also, it never enters the interior of a circle.

Complete step-by-step solution -
Given: ABC is a triangle and LMN is a circle.
              AB and AC are the tangents of the circle LMN.
              O is the center point of the circle LMN.
              ∠B = 70° and ∠C = 60°
To find: Measure of angle ∠LON.

We know that ∠B = 70° and ∠C = 60°
So, ∠A + ∠B + ∠C = 180° (Angle sum property of triangles)
∠A + 70° + 60° = 180°
∠A + 130° = 180°
∠A = 180° - 130°
∠A = 50°
We know that AB and AC are tangents of the circle LMN and these two tangents are meeting at point A. Therefore, A is known as the intersecting point of the tangents.
In such a situation, ∠BAC and ∠LON are supplementary, i.e. ∠BAC + ∠LON = 180°.
We know that the angle BAC = 50°.
Therefore, 50° + ∠LON = 180°
∠LON = 180° - 50°
Therefore, ∠LON = 130°
Hence, the measure of the angle ∠LON is 130°.

NOTE:-
What are Supplementary Angles:-
Two Angles are Supplementary when they add up to 180°, in other words, when the sum of two angles is 180°, then the two angles are said to be supplementary.
What are Complementary Angles:-
Two angles are Complementary when they add up to 90°, in other words, when the sum of two angles is 90°, then the two angles are said to be complementary.