# Question Video: Finding the Measure of an Angle Using the Properties of Triangles and Tangents to a Circle Mathematics

In the given figure, line π΄π΅ is a tangent to a circle with center π at π΅, line segment πΆπ· is a diameter, πβ π΅π΄π = π₯, and πβ ππ·π΅ = 2π₯ β 55. Find the value of π₯ in degrees.

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### Video Transcript

In the given figure, line π΄π΅ is a tangent to a circle with center π at π΅, line segment πΆπ· is a diameter, the measure of angle π΅π΄π equals π₯, and the measure of angle ππ·π΅ equals two π₯ minus 55. Find the value of π₯ in degrees.

We are told in the question that the line π΄π΅ is a tangent to the circle at point π΅. We know that a tangent meets the radius at the point of contact at right angles. The measure of angle π΄π΅π is therefore equal to 90 degrees. As all radii are equal in length, we know that the line segments ππ· and ππ΅ have the same length. This means that triangle ππ·π΅ is isosceles. This in turn means that the measures of angles ππ·π΅ and ππ΅π· are equal. And they are equal to the expression two π₯ minus 55. We are also told in the question that the measure of angle π΅π΄π is equal to π₯.

We will now consider triangle π΄π΅π· and the fact that angles in a triangle sum to 180 degrees. This means that the measures of the angles π΄π·π΅, π΄π΅π·, and π΅π΄π· must equal 180 degrees. Angle π΄π·π΅ is equal to two π₯ minus 55, angle π΄π΅π· is equal to two π₯ minus 55 plus 90, and angle π΅π΄π· is equal to π₯.

This gives us the following equation. Two π₯ minus 55 plus two π₯ minus 55 plus 90 plus π₯ is equal to 180. Collecting like terms on the left-hand side gives us five π₯ minus 20, and this is equal to 180. We can then add 20 to both sides. Finally, dividing through by five gives us π₯ is equal to 40. The value of π₯ in degrees is 40 degrees.