Question Video: Finding the Measure of an Angle given the Measure of an Inscribed Angle by Using the Properties of Tangents to the Circle Mathematics

Given that π‘šβˆ π΅πΈπΆ = 31Β°β€Ž, find π‘šβˆ πΆ and π‘šβˆ π΅π·π΄.

03:17

Video Transcript

Given that the measure of angle 𝐡𝐸𝐢 is 31 degrees, find the measure of angle 𝐢 and the measure of angle 𝐡𝐷𝐴.

We’ve been given that angle 𝐡𝐸𝐢 is 31 degrees. That’s our first piece of information. We wanna go ahead and list out the other things we know based on the diagram. We can say that line segment 𝑁𝐹, line segment 𝑁𝐴, line segment 𝑁𝐸, and line segment 𝑁𝐡 are all radii of the circle 𝑁. And we can say that ray 𝐢𝐡 is tangent to this circle at the point 𝐡. We want to know the measure of angle 𝐢, which is here, and the measure of angle 𝐡𝐷𝐴, which is here.

We wanna use our given statements and then draw some conclusions. First of all, we can say that all of the radii will be equal in length to one another because we know the definition of a radius. We can also say that triangle 𝐴𝑁𝐹 and triangle 𝑁𝐸𝐡 are isosceles triangles because they both have two sides of the triangle that are equal in length. We can also say that the measure of angle 𝐴𝑁𝐹 will be equal to the measure of angle 𝐡𝑁𝐸 because they’re vertical angles. And that means we can say that triangle 𝐡𝑁𝐸 is congruent to triangle 𝐴𝑁𝐹 because they have a side, an angle, and a side that are congruent. And since these triangles are congruent, all of these angles will be equal to one another. They will be congruent angles, all measuring in at 31 degrees.

Something else we can now identify is that the measure of angle 𝐴𝐡𝐢 is 90 degrees because of tangent line properties. So far, we’ve identified a lot of the angles, but not the ones we need to find. We now wanna focus in on the triangle created from points 𝐸, 𝐢, and 𝐡. The angle 𝐢𝐡𝐸 is created from a 90-degree angle and a 31-degree angle. That means the angle 𝐢𝐡𝐸 is 121 degrees. The measure of angle 𝐢 plus 121 degrees plus 31 degrees must equal 180 degrees. 121 plus 31 equals 152. We subtract that value from both sides of the equation. And we get that the measure of angle 𝐢 is 28 degrees.

To find the measure of angle 𝐡𝐷𝐴, we need to follow a really similar procedure. We’re gonna focus on the triangle 𝐴𝐡𝐷, which has a right angle and a 31-degree angle. In this case, the measure of angle 𝐡𝐷𝐴 plus 90 degrees plus 31 degrees will equal 180 degrees. 90 plus 31 is 121. So we subtract 121 from both sides of the equation, which tells us that the measure of angle 𝐡𝐷𝐴 is 59 degrees. And we found both of these answers by knowing that the sum of the interior angles in a triangle must be equal to 180 degrees.

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