11.1.2 Arcs and Chords (continued)
Example 2 Using the Arc Addition Postulate
The measure of an arc that is formed by two adjacent arcs (arcs that share an endpoint) is the sum of the measure of the two arcs.
The measure of an arc is determined in this example. The measures of two central angles in the circle are given. Segment AD and segment BE are given as diameters. m CD is 90 ° , since this is the measure of its central angle. m AFB = m EFD by the Vertical Angles Theorem. So m CDE is 90 ° + 15 ° = 105 ° by the Arc Addition Postulate.
Example 3 Applying Congruent Angles, Arcs, and Chords
In a circle, or in congruent circles: (1) Congruent central angles have congruent chords. (2) Congruent chords have congruent arcs. (3) Congruent arcs have congruent central angles.
The measure of PQ is determined in this example. It is given that PQ ≅ RS . The lengths of PQ and RS are given as algebraic expressions. Since congruent chords have congruent arcs, = PQ RS m m . Substitute the algebraic expressions given for the lengths of each arc into the equation. Solve for x . Substitute x = 20 back into the expression for the length of PQ . Solving yields = PQ m 100 .
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