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G-C.3 Core M2-031-R09-V03

Construct inscribed/circumscribed circles of triangles and prove angle properties of cyclic quadrilaterals.

Relate inscribed angles to intercepted arcs

Problem

In cyclic quadrilateral \(ABCD\), \(\angle~ABC\) intercepts \(\text{arc}~ADC\) measuring \(210°\). Calculate \(m\angle~ABC\) from its intercepted arc, then use the cyclic-quadrilateral opposite-angle relationship to calculate \(m\angle~ADC\).

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m∠ABC = 105° and m∠ADC = 105° because opposite angles intercept congruent arcs.

m∠ABC = 210° and m∠ADC = 150° because each angle equals its intercepted arc.

m∠ABC = 105° and m∠ADC = 75°; opposite angles of a cyclic quadrilateral sum to 180°.

m∠ABC = 75° and m∠ADC = 105° by using the other arc ABC for ∠ABC.

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Curriculum context

Course: Math II
Standard: G-C.3
Category: Geometry
Domain: Circles
Objective: Construct inscribed/circumscribed circles of triangles and prove angle properties of cyclic quadrilaterals.
Problem type: Relate inscribed angles to intercepted arcs