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Elementary Geometry For College St...

7th Edition
Alexander + 2 others
ISBN: 9781337614085

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Section
BuyFindarrow_forward

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. 12 2. Alternate interior angle
3. m1=m2 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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