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Math
GeometryElementary Geometry for College Students6th Edition

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Chapter 1.5, Problem 29E
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Elementary Geometry for College St...

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

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Problem 23E
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Problem 29E
Problem 30E
Problem 31E
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Problem 33E
Problem 34E
Problem 35E
Problem 36E
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Elementary Geometry for College St...

6th Edition
Daniel C. Alexander + 1 other
ISBN: 9781285195698
Textbook Problem
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In Exercises 27 to 30, fill in the missing reasons for each geometric proof.

Given: B D → bisects ∠ A B C
Prove: m ∠ A B D = 1 2 ( m ∠ A B C )

Chapter 1.5, Problem 29E, In Exercises 27 to 30, fill in the missing reasons for each geometric proof. Given: BD bisects ABC

Exercises 29, 30

PROOF
Statements Reasons
1. B D → bisects ∠ A B C 1. ?
2. m ∠ A B D = m ∠ D B C 2. ?
3. m ∠ A B D + m ∠ D B C = m ∠ A B C 3. ?
4. m ∠ A B D + m ∠ A B D = m ∠ A B C 4. ?
5. 2 ( m ∠ A B D ) = m ∠ A B C 5. ?
6. m ∠ A B D = 1 2 ( m ∠ A B C ) 6. ?

To determine

To find:

The missing reasons for the geometric proof.

Explanation

Given:

The given table is shown below.

PROOF
Statements Reasons
1. BD→ bisects ∠ABC 1. ?
2. m∠ABD=m∠DBC 2. ?
3. m∠ABD+m∠DBC=m∠ABC 3. ?
4. m∠ABD+m∠ABD=m∠ABC 4. ?
5. 2(m∠ABD)=m∠ABC 5. ?
6. m∠ABD=12(m∠ABC) 6. ?

Property:

1. The angle bisector is defined as a line or line segment that divides an angle into two equal parts.

2. According to Angle-Addition Postulate, if point B lies in the interior of ∠AOC then, m∠AOB+m∠BOC=m∠AOC.

3. According to Substitution Property of Equality, if A=B, then A can be substituted for B and B can be substituted for A in any equation.

4. The Multiplication Property of Equality states that if both the sides of an equation is multiplied by a same number then the sides of the equation remains equal.

Calculation:

The first statement of the proof is given below.

BD→ bisects ∠ABC

Therefore, the first reason of the proof is Given.

Now, go from first statement to the second statement.

BD→ bisects ∠ABC

Use definition of angle bisector as BD→ bisects ∠ABC.

m∠ABD=m∠DBC

Therefore, the second reason is definition of angle bisector.

Now, go from second statement to the third statement.

m∠ABD=m∠DBC

According to Angle-Addition Postulate, the sum of the measures of ∠ABD and ∠DBC is equal to the measure of ∠ABC as D is the interior of ∠ABC.

m∠ABD+m∠DBC=m∠ABC

Therefore, the third reason for the proof is Angle-Addition Postulate.

Now, go from third statement to fourth statement.

m∠ABD+m∠DBC=m∠ABC

Use Substitution Property of Equality and substitute m∠ABD for m∠DBC as m∠ABD=m∠DBC.

m∠ABD+m∠ABD=m∠ABC

Therefore, the fourth reason for the proof is Substitution Property of Equality.

Now, go from fourth statement to fifth statement.

m∠ABD+m∠ABD=m∠ABC

Use Substitution Property of Equality and substitute 2m∠ABD for m∠ABD+m∠ABD

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