Question

In the figure, m[arc AXC] =110
m(arc DYE=50
Find ABC
​

Answer 1

Given:

m(arc AXC) = 110°

m(arc DYE) = 50°

To find:

∠ABC

Solution:

As per the Angle Outside the Circle Theorem, we have

If a tangent and a secant or two tangents or two secants intersect anywhere outside a circle then the measure of the angle formed is half the difference between the measures of the intercepted arcs.

Here, we have

AB and BC are two secants

arc AXC and arc DYE are the two intercepted arcs

∠ABC is the angle formed outside the circle

Now,

By using the above theorem, we get

$\angle ABC = \frac{1}{2} \times [m (arc \:AXC) - m(arc \:DYE)]$

on substituting m(arc AXC) = 110° and m(arc DYE) = 50°

$\implies \angle ABC = \frac{1}{2} \times [110\° - 50\°]$

$\implies \angle ABC = \frac{1}{2} \times 60\°$

$\implies \bold{\angle ABC = 30\°}$

Thus, the measure of ∠ABC is → 30°.

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