WEBVTT
00:00:02.120 --> 00:00:07.110
Find the measure of angle π΄πΆπ· and the measure of angle π΅π΄πΆ.
00:00:08.250 --> 00:00:16.600
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.
00:00:17.370 --> 00:00:24.140
Looking at our diagram, we can see that the four points π΄, π΅, πΆ, and π· lie on the circumference of the circle.
00:00:24.490 --> 00:00:28.430
This means that π΄π΅πΆπ· is a cyclic quadrilateral.
00:00:28.750 --> 00:00:34.780
We can recall that the key property of a cyclic quadrilateral is that opposite angles are supplementary.
00:00:34.920 --> 00:00:37.700
That means they add to 180 degrees.
00:00:38.100 --> 00:00:42.360
Weβre given that the measure of angle πΆπ΅π΄ is 67 degrees.
00:00:42.580 --> 00:00:45.800
So we can work out the measure of angle πΆπ·π΄.
00:00:46.170 --> 00:00:55.290
Since we have two opposite angles, we can say that the measure of angle πΆπ·π΄ plus 67 degrees equals 180 degrees.
00:00:55.700 --> 00:01:10.040
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.
00:01:10.460 --> 00:01:16.550
Knowing this angle now will help us to work out our orange angle, angle π΄πΆπ·.
00:01:16.960 --> 00:01:20.080
Our circle has two congruent arcs given.
00:01:20.480 --> 00:01:35.470
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.
00:01:35.840 --> 00:01:39.460
We can define these two equal angles with the letter π₯.
00:01:39.950 --> 00:01:52.330
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.
00:01:52.590 --> 00:02:00.250
So two π₯ equals 180 degrees subtract 113 degrees, which is 67 degrees.
00:02:00.680 --> 00:02:04.780
Therefore, π₯ must be equal to 33.5 degrees.
00:02:05.310 --> 00:02:12.200
Therefore, our first missing angle of the measure of angle π΄πΆπ· is equal to 33.5 degrees.
00:02:12.900 --> 00:02:19.930
To find our next missing pink angle, angle π΅π΄πΆ, we use another fact about the angles in a circle.
00:02:20.090 --> 00:02:25.750
We have a line π΄π΅, which is the diameter of a circle creating a semicircle.
00:02:25.810 --> 00:02:28.990
And the angle in a semicircle is 90 degrees.
00:02:29.170 --> 00:02:32.450
So our angle π΅πΆπ΄ is 90 degrees.
00:02:32.930 --> 00:02:45.830
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.
00:02:46.090 --> 00:02:52.380
So the measure of angle π΅π΄πΆ plus 157 degrees is equal to 180 degrees.
00:02:52.750 --> 00:03:03.230
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.
00:03:03.660 --> 00:03:09.530
Therefore, our final answer is the measure of angle π΄πΆπ· equals 33.5 degrees.
00:03:09.790 --> 00:03:13.600
And the measure of angle π΅π΄πΆ equals 23 degrees.