Discuss how an inscribed angle is related to its intercepted arc. How is this different from a central angles relationship to its intercepted arc? What does this mean is true about the opposite angles of a quadrilateral inscribed in a circle?

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02-05-2024
Mathematics

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Discuss how an inscribed angle is related to its intercepted arc. How is this different from a central angles relationship to its intercepted arc? What does this mean is true about the opposite angles of a quadrilateral inscribed in a circle?

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selim272 selim272 · Answer 1 · 2024-05-02T10:13:31-04:00

Step-by-step explanation:

Inscribed angles and central angles are related to their intercepted arcs in a circle, but in different ways:

Inscribed Angle: The measure of an inscribed angle is equal to half the measure of its intercepted arc.

Central Angle: The measure of a central angle is equal to the measure of its intercepted arc.

This difference arises because inscribed angles only "see" a portion of the circle, while central angles encompass the entire intercepted arc.

In a quadrilateral inscribed in a circle, the opposite angles are supplementary (add up to 180°). This means that their intercepted arcs together form the entire circle, so their combined measure is 360°. Since an inscribed angle measures half its intercepted arc, each opposite angle must have a measure of half the circle's arc measurement, which is 180°.