Find the measure of angle 𝐴𝐶𝐷 and the measure of angle 𝐵𝐴𝐶.
We can start this question by highlighting the two different angles that we need to find out, angle 𝐴𝐶𝐷 and angle 𝐵𝐴𝐶. Looking at our diagram, we can see that the four points 𝐴, 𝐵, 𝐶, and 𝐷 lie on the circumference of the circle. This means that 𝐴𝐵𝐶𝐷 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. We’re given that the measure of angle 𝐶𝐵𝐴 is 67 degrees. So we can work out the measure of angle 𝐶𝐷𝐴. Since we have two opposite angles, we can say that the measure of angle 𝐶𝐷𝐴 plus 67 degrees equals 180 degrees.
We can rearrange this equation by subtracting 67 from both sides, which gives us the measure of angle 𝐶𝐷𝐴 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 𝐴𝐶𝐷. Our circle has two congruent arcs given. Therefore, using the fact that congruent arcs have congruent chords, we can mark our triangle 𝐶𝐷𝐴 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 𝑥. We can use the fact that the angles in a triangle add up to 180 degrees to write that 113 degrees plus two 𝑥 equals 180 degrees. So two 𝑥 equals 180 degrees subtract 113 degrees, which is 67 degrees. Therefore, 𝑥 must be equal to 33.5 degrees. Therefore, our first missing angle of the measure of angle 𝐴𝐶𝐷 is equal to 33.5 degrees. To find our next missing pink angle, angle 𝐵𝐴𝐶, we use another fact about the angles in a circle. We have a line 𝐴𝐵, which is the diameter of a circle creating a semicircle. And the angle in a semicircle is 90 degrees. So our angle 𝐵𝐶𝐴 is 90 degrees.
Using the fact that the angles in a triangle sum to 180 degrees, we can write the measure of angle 𝐵𝐴𝐶 plus 67 degrees plus 90 degrees equals 180 degrees. So the measure of angle 𝐵𝐴𝐶 plus 157 degrees is equal to 180 degrees. To find the measure of angle 𝐵𝐴𝐶, we subtract 157 from both sides of our equation to give us that the measure of angle 𝐵𝐴𝐶 is 23 degrees. Therefore, our final answer is the measure of angle 𝐴𝐶𝐷 equals 33.5 degrees. And the measure of angle 𝐵𝐴𝐶 equals 23 degrees.