Question Video: Finding the Angle of a Point outside a Circle given the Angle of the opposite Arc | Nagwa Question Video: Finding the Angle of a Point outside a Circle given the Angle of the opposite Arc | Nagwa

# Question Video: Finding the Angle of a Point outside a Circle given the Angle of the opposite Arc Mathematics • Third Year of Preparatory School

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Find π₯.

01:56

### Video Transcript

Find π₯.

In this question, weβre asked to find the value of π₯. And we can see in our diagram that π₯ is the angle between two secant lines which intersect outside of our circle. And we can find the measure of π₯ by recalling the following fact. The angle between two secant lines in a circle which intersect outside of a circle is one-half the positive difference of the measures of the arcs intercepted by the sides of the angle.

To apply this property, letβs do this step by step. First, letβs mark the sides of the angle of π₯. We can see that π₯ is the angle between the lines π΄πΆ and π΄πΈ. So the two sides of our angle are the line segment π΄πΆ and the line segment π΄πΈ. Next, we need to find the measures of the arcs intercepted by the two sides of our angle. The first side of our angle intersects the circle at the point π΅, and the second side of our angle intersects the circle at the point π·. So one of the arcs weβre going to use is the arc from π΅ to π·. Similarly, the first side of our angle intercepts the circle at the point πΆ, and the second side of our angle intercepts the circle at the point πΈ. So the other arc weβre interested in is the arc from πΆ to πΈ.

Finally, the measure of our angle will be one-half the positive difference between the measures of these two arcs. And since the arc from πΆ to πΈ is bigger than the arc from π΅ to π·, this gives us the following result. π₯ will be equal to one-half multiplied by the measure of arc πΆπΈ minus the measure of arc π΅π·. And weβre given both of these values in the diagram. The measure of arc πΆπΈ is 132 degrees, and the measure of arc π΅π· is 36 degrees. So we substitute these values into our formula. We get that π₯ is equal to one-half multiplied by 132 degrees minus 36 degrees. And we can then evaluate this expression. 132 minus 36 is equal to 96. And if we multiply this by one-half, we get 48. Therefore, π₯ is equal to 48 degrees.

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