Video Transcript
Find the measure of angle π΄πΆπ· and the measure of angle π΅π΄πΆ.
We can start this question by highlighting the two different angles that we need to find out, angle π΄πΆπ· and angle π΅π΄πΆ. Looking at our diagram, we can see that the four points π΄, π΅, πΆ, and π· lie on the circumference of the circle. This means that π΄π΅πΆπ· is a cyclic quadrilateral. We can recall that the key property of a cyclic quadrilateral is that opposite angles are supplementary. That means they add to 180 degrees. Weβre given that the measure of angle πΆπ΅π΄ is 67 degrees. So we can work out the measure of angle πΆπ·π΄. Since we have two opposite angles, we can say that the measure of angle πΆπ·π΄ plus 67 degrees equals 180 degrees.
We can rearrange this equation by subtracting 67 from both sides, which gives us the measure of angle πΆπ·π΄ equals 180 degrees minus 67 degrees, which simplifies to 113 degrees. Knowing this angle now will help us to work out our orange angle, angle π΄πΆπ·. Our circle has two congruent arcs given. Therefore, using the fact that congruent arcs have congruent chords, we can mark our triangle πΆπ·π΄ with two congruent sides, which means that this triangle is isosceles and, therefore, has also got two equal angles.
We can define these two equal angles with the letter π₯. We can use the fact that the angles in a triangle add up to 180 degrees to write that 113 degrees plus two π₯ equals 180 degrees. So two π₯ equals 180 degrees subtract 113 degrees, which is 67 degrees. Therefore, π₯ must be equal to 33.5 degrees. Therefore, our first missing angle of the measure of angle π΄πΆπ· is equal to 33.5 degrees. To find our next missing pink angle, angle π΅π΄πΆ, we use another fact about the angles in a circle. We have a line π΄π΅, which is the diameter of a circle creating a semicircle. And the angle in a semicircle is 90 degrees. So our angle π΅πΆπ΄ is 90 degrees.
Using the fact that the angles in a triangle sum to 180 degrees, we can write the measure of angle π΅π΄πΆ plus 67 degrees plus 90 degrees equals 180 degrees. So the measure of angle π΅π΄πΆ plus 157 degrees is equal to 180 degrees. To find the measure of angle π΅π΄πΆ, we subtract 157 from both sides of our equation to give us that the measure of angle π΅π΄πΆ is 23 degrees. Therefore, our final answer is the measure of angle π΄πΆπ· equals 33.5 degrees. And the measure of angle π΅π΄πΆ equals 23 degrees.
