   Chapter 6.1, Problem 31E 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 30 and 31, complete each proof. Given: M N ¯ ∥ O P ¯  in  ⊙ O Prove: m M Q ⌢ = 2 ( m N P ⌢ ) PROOF Statements Reasons 1. ? 1. Given 2. ∠ 1 ≅ ∠ 2 2. ? 3. m ∠ 1 = m ∠ 2 3. ? 4. m ∠ 1 = 1 2 ( m M Q ⌢ ) 4. ? 5. m ∠ 2 = m N P ⌢ 5. ? 6. 1 2 ( m M Q ⌢ ) = m N P ⌢ 6. ? 7. m M Q ⌢ = 2 ( m N P ⌢ ) 7. Multiplication Property of Equality

To determine

To prove: The mMQ2(mNP) by using the provided figure.

Explanation

Given: The MN¯OP¯ in O

Postulate used:

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

Central angle postulate:

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

Definition:

In a circle or congruent circles, congruent arcs are arcs with equal measures.

Proof:

Step 1:

Since, MN¯OP¯ in O.

Now, 1 and 2 are alternate interior angles.

Thus, alternate interior angles are congruent.

Since, congruent angles have equal measures.

Thus, 1 and 2 have equal measures.

 PROOF Statements Reasons 1. MN¯∥OP¯ in ⊙O 1. Given 2. ∠1≅∠2 2. Alternate interior angle 3. m∠1=m∠2 3.congurent angles have equal measure

Step 2:

Since, 1 is an inscribed angle and MQ is an intercepted arc.

Therefore, the measure of an inscribed angle is equal to the one-half the measure of an inscribed arc.

Thus, m2=12(mMQ)

Also, the measure of a central angle is equal to the measure of its intercepted arc.

Thus, m2=mNP

Now, by substitution, using statements 3 and 5 we obtain 12(mMQ)=mNP

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