Question

Find the measure of each arc. For each arc, write two or more complete sentences explaining which theorem or postulate you used to find your answer. Include your equations and calculations in your final answer.

m arcQS

Answer 1

m arcRP = 125°

m arcQS = 125°

m arcSP = 55°

m arcQR = 55°

Explanation:

An arc is the curve between two points on the circumference of a circle.

An arc measure equals its corresponding central angle measure.

Vertical Angles Theorem: When two straight lines intersect, the opposite vertical angles are congruent.

Angles on a Straight Line Theorem: The sum of angles formed on a straight line is equal to 180°.

Arc RP

As m∠ROP = 125°

Therefore, m arcRP = 125°

Given the measure of angle ROP is 125°, and an arc measure equals its corresponding central angle measure, the measure of arc RP is 125°.

Arc QS

Using the Vertical Angles Theorem:

⇒ m∠QOS = 125°

Therefore, m arcQS = 125°

Using the Vertical Angles Theorem, the measure of angle QOS is equal to the measure of angle ROP, so the measure of angle QOS is 125°. As an arc measure equals its corresponding central angle measure, the measure of arc QS is 125°.

Arc SP

Using the Angles on a Straight Line Theorem:

⇒ m∠SOP = 180° - 125° = 55°

Therefore, m arcSP = 55°

Using the Angles on a Straight Line Theorem, the measure of angle SOP is 55°. As an arc measure equals its corresponding central angle measure, the measure of arc SP is 55°.

Arc QR

Using the Angles on a Straight Line Theorem:

⇒ m∠QOR = 180° - 125° = 55°

Therefore, m arcQR = 55°

Using the Angles on a Straight Line Theorem, the measure of angle QOR is 55°. As an arc measure equals its corresponding central angle measure, the measure of arc QR is 55°.