   Chapter 6.1, Problem 40E 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

Use a paragraph proof to complete this exercise. Given: ⊙ O with chords A B ¯ and B C ¯ , radii A O ¯ and O C ¯ Prove: m ∠ A B C < m ∠ A O C To determine

To prove:

In a circle with centre O, AB¯ and BC¯ are the chords and AO¯,OC¯ are radii, then mABC<mAOC.

Explanation

Given:

In a circle with centre O, AB¯ and BC¯ are the chords and AO¯,OC¯ are radii.

Theorem used:

The measure of an inscribed angle of a circle is one-half the measure of its intercepted arc.

Central angle postulate:

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

Proof:

Consider a circle with center O.

Let AB¯ and BC¯ be the chords and let AO¯,OC¯ be radii of the circle.

From the figure, we see that AOC is the central angle.

By central angle postulate, mAOC=mAC.

From the figure we see that ABC is the inscribed angle with the intercepted arc AC

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