Inscribed Angles Examples

Given that ∠ACB is inscribed in a circle containing points A and B, and that m∠ACB = 23°, what is the measure of arc AB?

Here we have an inscribed angle intercepting an arc, so we can bust out the Inscribed Angle Theorem. We know the measure of arc mAB equals 2 × m∠ACB. Substituting the measure of the angle, we get mAB = 2 × 23° = 46°.

Now suppose m∠ACB = 79°. What is the measure of arc AB?

We still have an inscribed angle intercepting an arc, only with a new measurement. We know that mAB = 2 × m∠ACB, so let's substitute 79° and calculate the measure of the arc. If we do, we get mAB = 2 × 79° = 158°.

In the figure below, points A, B, and C are on ⊙O and m∠AOB = 138°. What is m∠ACB?

By definition of arc measure, the measure of arc AB is 138°. Since ∠ACB is an inscribed angle intercepting arc AB, we can find m∠ACB using the Inscribed Angle Theorem. If we substitute 138° for mAB, we should end up with 69° = m∠ACB.

Refer to the same figure as the previous example, but suppose instead that m∠AOB = 55°. What is m∠ACB?

By definition of arc measure, the measure of arc AB is 55°. We know ∠ACB is an inscribed angle intercepting arc AB, so finding m∠ACB shouldn't be a problem. Dividing 55° by 2 gives us m∠ACB = 27.5°.

In the figure below, points A, B, C, and D are on the circle. The measure of arc AB = 80° and the measure of arc BC is 90°. What is m∠ADC?

To find mABC, we can add arcs AB and BC to get mABC = 80° + 90° = 170°. Now we see that ∠ADC is an inscribed angle intercepting arc ABC, so we can use the Inscribed Angle Theorem and the measure of ABC to find m∠ADC. If we divide 170° by 2, we get m∠ADC = 85°.

In the figure below, ΔABC is inscribed in the circle (meaning that each angle of ΔABC is inscribed in the circle). If m∠A = 70° and m∠C = 50°, what is the measure of arc AC?

Notice that arc AC subtends the inscribed ∠B. We can find m∠B with the Angle Sum Theorem for triangles. We know that ∠A, ∠B, and ∠C all have to add up to 180°, so we can find ∠B by subtracting 180° – 70° – 50° = 60°. Now we can use the Inscribed Angle Theorem. This gives us mAC = 2 × m∠B = 2 × 60° = 120°.