Question

A circle is shown. 2 chords intersect to form 4 angles. From top left, clockwise, the angles are: 3, 2, 1, 4. Angle 3 intersects an arc with measure 54 degrees. Angle 1 intersects an arc with measure 112 degrees.
Marcel is designing a circular necklace that will consist of 4 sections, each with a different color of plastic. Determine how he needs to cut the plastic by finding the measures of the angles.

m∠1 =
°

m∠2 =
°

Answer 1

The measure of an angle formed by two chords that intersect inside a circle is half the sum of the measures of the arcs intercepted by that angle and its vertical opposite.

Angle 1 and angle 3 are vertical angles, so
m∠1 = m∠3 = 1/2(arc of 54° + arc of 112°) = 1/2(166°) = 83°.

Adjacent angles are supplementary, so
m∠2 = 180° − m∠1 = 180° − 83° = 97°.

m∠1 = 83°
m∠2 = 97°