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

# Question Video: Finding the Measure of an Inscribed Angle by Using the Properties of Tangents to the Circle Mathematics • Third Year of Preparatory School

## Join Nagwa Classes

Attend live Mathematics sessions on Nagwa Classes to learn more about this topic from an expert teacher!

Given that line π΄π΅ is a tangent to a circle with center π and πβ π΄π΅π = 49Β°, determine πβ π΄π·π΅.

04:39

### Video Transcript

Given that line π΄π΅ is a tangent to a circle with center π and the measure of angle π΄π΅π is equal to 49 degrees, determine the measure of angle π΄π·π΅.

Angle π΄π·π΅ is the angle formed when we travel from π΄ to π· to π΅, so itβs the angle marked in orange in the diagram. Angle π΄π΅π is the angle formed when we travel from π΄ to π΅ to π. Itβs the angle now marked in pink on the diagram with a measure of 49 degrees.

From the information given, we arenβt able to calculate the measure of angle π΄π·π΅ directly. Weβre going to need to find the measures of some other angles in the figure first. The other key piece of information given in the question is that line π΄π΅ is a tangent to the circle with center π. And a key property about tangents of circles is that a tangent to a circle is perpendicular to the radius at the point of contact.

The point where the tangent meets the circle is point π΄, and the radius here is the line segment π΄π. So, we know that the measure of angle π΅π΄π is 90 degrees. Now we know one more angle within the figure. However, we still arenβt able to calculate the measure of angle π΄π·π΅ directly, so letβs see what other angles we can work out.

We have a triangle. In fact, triangle π΄ππ΅ is a right triangle, and we know two of its angles, the right angle and the angle measuring 49 degrees. So, using the fact that the angle measures in a triangle sum to 180 degrees, we can calculate the third angle in this triangle. We do this by writing an equation which states that the measure of angle π΄ππ΅ plus 49 degrees plus 90 degrees is equal to 180 degrees. 49 plus 90 is 139, and subtracting this from 180, we find the measure of angle π΄ππ΅ is 41 degrees. So we now know another angle in our diagram.

We still donβt have enough information to calculate the measure of an angle π΄π·π΅, but we can now calculate a different angle, angle π΄ππ·. We know that the angle measures on a straight line sum to 180 degrees. So, the measure of angle π΄ππ· plus the angle measure weβve just calculated, 41 degrees, must equal 180 degrees. The measure of angle π΄ππ· is therefore equal to 180 degrees minus 41 degrees. Thatβs 139 degrees. Now we found almost all the angles in the figure, but still not the one we were looking for.

The final step is to consider triangle π΄ππ·, in which we know one angle is 139 degrees. We recognize both line segment ππ΄ and line segment ππ· as radii of the same circle with center π. Therefore, they have the same length. It follows that triangle ππ·π΄ is an isosceles triangle, and this means that angles π·π΄π and π΄π·π have equal measure. We can therefore find the measure of each angle by subtracting the measure of the third angle, 139 degrees, from the total angle sum in a triangle, 180 degrees, and then splitting the remainder in half. Doing so gives each of these angles a measure of 20.5 degrees.

Now we know that angle π΄π·π is in fact the same as angle π΄π·π΅. They both refer to the angle highlighted here in pink. And so, weβve completed the problem. By using some of the more basic facts of angles in triangles and angles in straight lines and the key property that the tangent to a circle is perpendicular to the radius at the point of contact, we found that the measure of angle π΄π·π΅ is 20.5 degrees.

## Join Nagwa Classes

Attend live sessions on Nagwa Classes to boost your learning with guidance and advice from an expert teacher!

• Interactive Sessions
• Chat & Messaging
• Realistic Exam Questions