   Chapter 6.1, Problem 23E Elementary Geometry For College St...

7th Edition
Alexander + 2 others
ISBN: 9781337614085

Solutions

Chapter
Section Elementary Geometry For College St...

7th Edition
Alexander + 2 others
ISBN: 9781337614085
Textbook Problem

Consider a circle or congruent circles, and explain why each statement is true:a) Congruent arcs have congruent central angles.b) Congruent central angles have congruent arcs.c) Congruent chords have congruent arcs.d) Congruent arcs have congruent chords.e) Congruent central angles have congruent chords.f) Congruent chords have congruent central angles.

To determine

(a)

To Explain:

Consider a circle or congruent circle, the statement congruent arcs have congruent central angles is true.

Explanation

Given:

The arcs are congruent.

Postulate used:

Central angle postulate:

In a circle, the degree of a central angle is equal to the degree measure of its intercepted arc.

Calculation:

Since, in a circle, the degree measure of a central angle is equal to the degree measure of its intercepted arc.

Thus, if two arcs have equal measures, then their associated central angles have same measures.

Consider, the diagram shown below:

Suppose, AB and CD are two arcs and their measure be represented by the x.

By the theorem we obtain,

To determine

(b)

To explain:

Consider a circle or congruent circle, the statement congruent central angles have congruent arcs is true.

To determine

(c)

To explain:

Consider a circle or congruent circle, the statement congruent chords have congruent arcs is true.

To determine

(d)

To explain:

Consider a circle or congruent circle, the statement congruent arcs have congruent chords is true.

To determine

(e)

To explain:

Consider a circle or congruent circle, the statement congruent central angles have congruent chords is true.

To determine

(f)

To explain:

Consider a circle or congruent circle, the statement congruent chords have congruent central angles is true.

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