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

The point 𝐴 is outside a circle with center 𝑀. The line between 𝐴 and 𝐢 intersects the circle at 𝐡 and 𝐢, and the line between 𝐴 and 𝐸 meets the circle at points 𝐷 and 𝐸. Given that π‘šβˆ πΆπ‘€πΈ = 130Β°β€Ž and π‘šβˆ π΅π‘€π· = 56Β°β€Ž, find π‘šβˆ π΄.

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Video Transcript

The point 𝐴 is outside a circle with center 𝑀. The line between 𝐴 and 𝐢 intersects the circle at 𝐡 and 𝐢, and the line between 𝐴 and 𝐸 meets the circle at points 𝐷 and 𝐸. Given that the measure of angle 𝐢𝑀𝐸 equals 130 degrees and the measure of angle 𝐡𝑀𝐷 is equal to 56 degrees, find the measure of angle 𝐴.

We’ve been given a description of our circle and lines, but we haven’t been given an image. The best place to start here is with sketching. We know we have a circle and that point 𝐴 is outside the circle. We also know that there is a line between 𝐴 and 𝐢; there’s a line with endpoints 𝐴 and 𝐢. And this line intersects the circle at 𝐡 and 𝐢. If we draw a line from 𝐴 to the circle, we know that the endpoints of this line were 𝐴 and 𝐢, and the other intersection along the circle was point 𝐡.

Similarly, we have a line between 𝐴 and 𝐸 that meets the circle at 𝐷 and 𝐸. We’ll draw another line from point 𝐴. The endpoint is 𝐸 and its other intersection point is 𝐷. The circle has a center 𝑀. We’ve been told the measure of angle 𝐢𝑀𝐸, which would be this angle, is 130 degrees. And we’ve been told the measure of angle 𝐡𝑀𝐷, which would be this angle, and that measures 56 degrees. And we want to know the measure of angle 𝐴.

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 we’re going to try and solve for the measure of angle 𝐴. 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 𝐴 is the angle created outside the circle. It’s going to be one-half the measure of arc 𝐢𝐸 minus the measure of arc 𝐷𝐡. Arc 𝐢𝐸 measures 130 degrees; arc 𝐡𝐷 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 𝐴 equal to 37 degrees.

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