Question

A, B, and C are points on the circumference of a circle with center O. Line DAE is a tangent to the circle at point A. Angle BAE measures 53 degrees, and angle CBO measures 32 degrees. Find the measure of angle CAO.

Answer 1

To find the measure of angle CAO, subtract angle BAE from 90 degrees, and then subtract angle BAO from angle CBO. Add 180 degrees to get the positive measure of angle CAO.

Step-by-step explanation:

To find the measure of angle CAO, we can use the fact that the sum of the angles in a triangle is equal to 180 degrees. First, we know that angle BAE is 53 degrees. Since DA is a tangent to the circle, angle BAO is a right angle. Therefore, angle BAO is 90 degrees.

Since angles BAO and BAE are adjacent angles, we can find angle BAO by subtracting angle BAE from 90 degrees: 90 degrees - 53 degrees = 37 degrees.

Now, we can find angle CAO by subtracting angle BAO from angle CBO: 32 degrees - 37 degrees = -5 degrees. However, since angles cannot have negative measures, we add 180 degrees to get the positive measure of angle CAO: -5 degrees + 180 degrees = 175 degrees.