   Chapter 6.4, Problem 4E 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

In Exercises 1 to 8, use the figure provided.If m C D ⌢ < m A B ⌢ , write an inequality that compares m ∠ A and m ∠ C . To determine

To write:

An inequality that compares mA and mC.

Explanation

Given:

mCD<mAB and the figure given below,

Theorem:

In a circle (or in congruent circles) containing two unequal arcs, the larger arc corresponds to the larger central angle.

Calculation:

Using the theorem.

Since the arcs are unequal i.e. mCD<mAB and from the given figure.

The angle corresponding to arc mCD is mCQD and the angle corresponding to mAB is mAQB, we conclude that mCQD<mAQB.

Now considering the AQB, AQ¯=QB¯ as they are both radii.

Hence, AQB is an isosceles triangle.

Thus, the base angles are equal i.e. mQAB=mQBA.

The sum of all the angles of a triangle equals 180°. So mQAB+mAQB+mQBA=180°.

Substituting, mQAB=mQBA,

mQAB+mQAB=180°mAQB

Simplifying,

mAQB=180°2mQAB

Similarly, in QCD, CQ¯=QD¯ as they are both radii

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