Question Video: Finding the Measure of an Angle inside a Triangle given the Measure of the Two Other Angles Using the Properties of a Circle’s Tangents | Nagwa Question Video: Finding the Measure of an Angle inside a Triangle given the Measure of the Two Other Angles Using the Properties of a Circle’s Tangents | Nagwa

# Question Video: Finding the Measure of an Angle inside a Triangle given the Measure of the Two Other Angles Using the Properties of a Circleβs Tangents Mathematics • Third Year of Preparatory School

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Given that the line π΄π΅ is a tangent to the circle with center π and πβ ππ΅πΉ = 123Β°, πβ π΄ππ΅.

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### Video Transcript

Given that the line π΄π΅ is a tangent to the circle with center π and the measure of angle ππ΅πΉ is 123 degrees, determine the measure of angle π΄ππ΅.

Angle π΄ππ΅ is the angle formed when we travel from π΄ to π and then to π΅. Angle ππ΅πΉ is the angle made when we travel from π to π΅ to πΉ, and weβre told that its measure is 123 degrees. We can see that the angle weβre looking to find, π΄ππ΅, is contained within a triangle. If we can work out the other two angles in this triangle, we can use the fact that the sum of the angles in any triangle is 180 degrees to find the angle weβre looking for.

First, letβs consider the angle ππ΅π΄. One of our most basic angle facts is that the angles on a straight line sum to 180 degrees. And this angle is on a straight line with the angle weβve already marked as being 123 degrees. So we can say that the measures of angles ππ΅πΉ and ππ΅π΄ sum to 180 degrees. We can substitute 123 degrees for the measure of angle ππ΅πΉ into our equation. And we now have an equation we can solve to find the measure of angle ππ΅π΄. To do this, we need to subtract 123 degrees from both sides of the equation. And doing so gives us that the measure of angle ππ΅π΄ is equal to 57 degrees. So weβve found one of the angles in triangle ππ΅π΄. Can we find another one? What about angle ππ΄π΅?

Well, this is the angle formed where a tangent to the circle, thatβs the line π΄π΅, meets the radius of the circle, π΄π. And we know that a tangent to a circle is perpendicular to the radius at the point of contact. So we know that angle ππ΄π΅ is a right angle, so it has measure 90 degrees.

Weβve therefore found two of the angles in triangle ππ΅π΄. And using the angle sum in a triangle, we can find the third. Since the angles in a triangle sum to 180 degrees, we have the equation the measure of angle π΄ππ΅ plus 90 degrees plus 57 degrees equals 180 degrees. Subtracting 90 degrees and 57 degrees from both sides then, we have 180 minus 90 minus 57 degrees on our right-hand side. And this is equal to 33 degrees. And so the measure of angle π΄ππ΅ equals 33 degrees.

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