In circle with the center P, arc EN = 66, m<GPM= 89 and segment GN is a diameter. Find each measure.
measure of the arc GM
measure of the arc GE
measure of the angle EGN
measure of the angle GNM
measure of the angle EGM
Since `bar(GN)` is a diameter, arcGMN is a semicircle with measure `180^@` .
The concepts we need concerning angles and arc measure:
(1) The measure of an arc is equal to the measure of its central angle (the angle whose vertex is at the center of the circle.)
(2) If the vertex of an angle is on the circle (an inscribed angle in this case) the measure of the intercepted arc is twice the measure of the angle.
(a) Since the measure of `/_GPM` is `89^@` , the measure of arcGM is `89^@` since `/_GPM` is a central angle.
(b) Since arcGEN is a semicircle (`bar(GN)` is a diameter) we know that arcGE measures `114^@` (180-66=114)
(c) `/_EGN` is an inscribed angle whose intercepted arc measures 66, so the measure of the angle is `33^@`
(d) `/_GNM` is an inscribed angle whose intercepted arc measures 89, so the measure of the angle is `44.5^@`
(e) `/_EGM` is an inscribed angle. It intercepts arcENM which has measure 66+91=157, so the angle measures `78.5^@`