Question Video: Finding the Angle of Intersection of Two Secants Using the Two Secants Angle Theorem | Nagwa Question Video: Finding the Angle of Intersection of Two Secants Using the Two Secants Angle Theorem | Nagwa

# Question Video: Finding the Angle of Intersection of Two Secants Using the Two Secants Angle Theorem Mathematics • First Year of Secondary School

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The point π΄ is outside a circle with center π. The line between π΄ and πΆ intersects the circle at π΅ and πΆ, and the line between π΄ and πΈ meets the circle at points π· and πΈ. Given that πβ πΆππΈ = 130Β°β and πβ π΅ππ· = 56Β°β, find πβ π΄.

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### Video Transcript

The point π΄ is outside a circle with center π. The line between π΄ and πΆ intersects the circle at π΅ and πΆ, and the line between π΄ and πΈ meets the circle at points π· and πΈ. Given that the measure of angle πΆππΈ equals 130 degrees and the measure of angle π΅ππ· is equal to 56 degrees, find the measure of angle π΄.

Weβve been given a description of our circle and lines, but we havenβt been given an image. The best place to start here is with sketching. We know we have a circle and that point π΄ is outside the circle. We also know that there is a line between π΄ and πΆ; thereβs a line with endpoints π΄ and πΆ. And this line intersects the circle at π΅ and πΆ. If we draw a line from π΄ to the circle, we know that the endpoints of this line were π΄ and πΆ, and the other intersection along the circle was point π΅.

Similarly, we have a line between π΄ and πΈ that meets the circle at π· and πΈ. Weβll draw another line from point π΄. The endpoint is πΈ and its other intersection point is π·. The circle has a center π. Weβve been told the measure of angle πΆππΈ, which would be this angle, is 130 degrees. And weβve been told the measure of angle π΅ππ·, which would be this angle, and that measures 56 degrees. And we want to know the measure of angle π΄.

Now, of course, when we look at our sketch, we know that these angles are a little bit off. But this sketch gives us enough information to figure out how weβre going to try and solve for the measure of angle π΄. Because we have two lines intersecting outside of a circle, then the angle created by the two lines intersecting outside the circle is half the positive difference between the intercepted arcs.

And that means the measure of angle π΄ is the angle created outside the circle. Itβs going to be one-half the measure of arc πΆπΈ minus the measure of arc π·π΅. Arc πΆπΈ measures 130 degrees; arc π΅π· measures 56 degrees. 130 minus 56 equal 74, and half of 74 is 37. And so, a circle under these conditions will have the angle measure π΄ equal to 37 degrees.

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