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Elementary Geometry for College St...

6th Edition
Daniel C. Alexander + 1 other
ISBN: 9781285195698

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BuyFindarrow_forward

Elementary Geometry for College St...

6th Edition
Daniel C. Alexander + 1 other
ISBN: 9781285195698
Textbook Problem
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In Exercises 30 and 31, complete each proof.

Given: Diameters A B ¯ and C D ¯ in E
Prove: A C D B
Chapter 6.1, Problem 30E, In Exercises 30 and 31, complete each proof. Given: Diameters AB and CD in E Prove: ACDB PROOF
PROOF
Statements Reasons
1. ? 1. Given
2. A E C D E B 2. ?
3. m A E C = m D E B 3. ?
4. m A E C = m A C and m D E B = m D B 4. ?
5. m A C = m D B 5. ?
6. ? 6. If two arcs of a circle have the same measure, they are

To determine

To prove: The ACDB by using the provided figure.

Explanation

Given: The diameters AB¯ and CD¯ in E

Postulate used:

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:

We need prove that if AB¯ and CD¯ are diameters of circle E, then ACDB.

PROOF
Statements Reasons
1. AB¯ and CD¯ are diameters of circle E 1. Given

Step 2:

The diameters AB¯ and CD¯ are form the vertical angles, AEC and DEB.

Since, vertical angles are congruent.

Therefore, the angles are congruent, then AEC and DEB have equal measures.

PROOF
Statements Reasons
1. AB¯ and CD¯ are diameters of circle E 1. Given
2. AECDEB 2. vertical angles are congruent.
3. mAEC=mDEB 3. congruent angles have equal measures.

Step 3:

Since, AEC and DEB are central angles.

Therefore, the measures degree of a central angle is equal to the degree measure of its intercepted arc.

Thus, mAC is the intercept arc of AEC and mDB is the intercept arc of DEB

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