Question

given m∠1 = m∠3
prove m∠eba = m∠cbd

3.
statements reasons

1. ______________ 1. given
2. m∠eba = m∠2 + m∠3 2.
3. m∠eba = m∠2 + m∠1 3.
4. m∠cbd = __________ 4. angle addition postulate
5. ______________ 5.

substitution property of equality transitive property of equality m∠1 = m∠3
m∠2 + m∠1 angle addition postulate m∠eba = m∠cbd

4.
given ac = ab + ab
prove ab = bc

statements reasons

1. ac = ab + ab 1. given
2. ab + bc = ac 2. ______________
3. ab + ab = ab + bc 3. ______________
4. ab = bc 4. ______________

addition property of equality subtraction property of equality substitution
transitive property of equality symmetric property of equality segment addition postulate

Explanation:

Problem 3 (Angle Proof)

Step1: State Given

\( m\angle 1 = m\angle 3 \) (Given)

Step2: Angle Addition for \( \angle EBA \)

\( m\angle EBA = m\angle 2 + m\angle 3 \) (Angle Addition Postulate)

Step3: Substitute \( \angle 1 \) for \( \angle 3 \)

\( m\angle EBA = m\angle 2 + m\angle 1 \) (Substitution Property of Equality, since \( m\angle 1 = m\angle 3 \))

Step4: Angle Addition for \( \angle CBD \)

\( m\angle CBD = m\angle 2 + m\angle 1 \) (Angle Addition Postulate)

Step5: Conclude Equality

\( m\angle EBA = m\angle CBD \) (Transitive Property of Equality, as both equal \( m\angle 2 + m\angle 1 \))

Filling the Table:

1. STATEMENT: \( m\angle 1 = m\angle 3 \)

REASON: Given

1. STATEMENT: \( m\angle EBA = m\angle 2 + m\angle 3 \)

REASON: Angle Addition Postulate

1. STATEMENT: \( m\angle EBA = m\angle 2 + m\angle 1 \)

REASON: Substitution Property of Equality (replace \( m\angle 3 \) with \( m\angle 1 \))

1. STATEMENT: \( m\angle CBD = m\angle 2 + m\angle 1 \)

REASON: Angle Addition Postulate

1. STATEMENT: \( m\angle EBA = m\angle CBD \)

REASON: Transitive Property of Equality

Problem 4 (Segment Proof)

Step1: State Given

\( AC = AB + AB \) (Given)

Step2: Segment Addition for \( AC \)

\( AB + BC = AC \) (Segment Addition Postulate: If \( B \) is between \( A \) and \( C \), then \( AB + BC = AC \))

Step3: Substitute \( AC \)

\( AB + AB = AB + BC \) (Substitution: Replace \( AC \) in Step 2 with \( AB + AB \) from Step 1)

Step4: Subtract \( AB \)

\( AB = BC \) (Subtraction Property of Equality: Subtract \( AB \) from both sides of \( AB + AB = AB + BC \))

Filling the Table:

1. STATEMENT: \( AC = AB + AB \)

REASON: Given

1. STATEMENT: \( AB + BC = AC \)

REASON: Segment Addition Postulate

1. STATEMENT: \( AB + AB = AB + BC \)

REASON: Substitution (replace \( AC \) with \( AB + AB \))

1. STATEMENT: \( AB = BC \)

REASON: Subtraction Property of Equality (subtract \( AB \) from both sides)

Answer:

Step1: State Given

\( AC = AB + AB \) (Given)

Step2: Segment Addition for \( AC \)

\( AB + BC = AC \) (Segment Addition Postulate: If \( B \) is between \( A \) and \( C \), then \( AB + BC = AC \))

Step3: Substitute \( AC \)

\( AB + AB = AB + BC \) (Substitution: Replace \( AC \) in Step 2 with \( AB + AB \) from Step 1)

Step4: Subtract \( AB \)

\( AB = BC \) (Subtraction Property of Equality: Subtract \( AB \) from both sides of \( AB + AB = AB + BC \))

Filling the Table:

1. STATEMENT: \( AC = AB + AB \)

REASON: Given

1. STATEMENT: \( AB + BC = AC \)

REASON: Segment Addition Postulate

1. STATEMENT: \( AB + AB = AB + BC \)

REASON: Substitution (replace \( AC \) with \( AB + AB \))

1. STATEMENT: \( AB = BC \)

REASON: Subtraction Property of Equality (subtract \( AB \) from both sides)