Question

in the figure shown, cf intersects ad and eh at points b and f, respectively.

part a

• given: ∠cbd ≅ ∠bfe

prove: ∠abf ≅ ∠bfe
select from the drop-down menus to support each line of the proof.

statement: ∠cbd ≅ ∠bfe reason: given
statement: ∠cbd ≅ ∠abf reason: vertical angles
statement: ∠abf ≅ ∠bfe reason: transitive pro...

part b

• given: m∠cbd = m∠bfe

prove: m∠bfe + m∠dbf = 180°
select from the drop-down menus to support each line of the proof.

statement: m∠cbd = m∠bfe reason:
statement: m∠cbd + m∠dbf = 180° reason:
statement: m∠bfe + m∠dbf = 180° reason:

Explanation:

Part A

Step 1: First Statement and Reason

• Statement: $\angle CBD \cong \angle BFE$
• Reason: Given (as provided in the problem, this is the initial information given to start the proof).

Step 2: Second Statement and Reason

• Statement: $\angle CBD \cong \angle ABF$
• Reason: Vertical angles are congruent. $\angle CBD$ and $\angle ABF$ are vertical angles (formed by the intersection of lines $AD$ and $CF$), so they are congruent.

Step 3: Third Statement and Reason

• Statement: $\angle ABF \cong \angle BFE$
• Reason: Transitive property of congruence. If $\angle CBD \cong \angle BFE$ (from step 1) and $\angle CBD \cong \angle ABF$ (from step 2), then by the transitive property, $\angle ABF \cong \angle BFE$.

Part B

Step 1: First Statement and Reason

• Statement: $m\angle CBD = m\angle BFE$
• Reason: Given (this is the initial information provided for Part B).

Step 2: Second Statement and Reason

• Statement: $m\angle CBD + m\angle DBF = 180^\circ$
• Reason: Linear pair postulate. $\angle CBD$ and $\angle DBF$ form a linear pair (they are adjacent angles that form a straight line along $CF$), so their measures add up to $180^\circ$.

Step 3: Third Statement and Reason

• Statement: $m\angle BFE + m\angle DBF = 180^\circ$
• Reason: Substitution property. Since $m\angle CBD = m\angle BFE$ (from step 1), we can substitute $m\angle CBD$ with $m\angle BFE$ in the equation $m\angle CBD + m\angle DBF = 180^\circ$ (from step 2) to get $m\angle BFE + m\angle DBF = 180^\circ$.

Final Answers (for the drop - down selections)

Part A

• First reason: Given
• Second reason: Vertical angles are congruent
• Third reason: Transitive property of congruence

Part B

• First reason: Given
• Second reason: Linear pair postulate
• Third reason: Substitution property

Answer:

Part A

Step 1: First Statement and Reason

• Statement: $\angle CBD \cong \angle BFE$
• Reason: Given (as provided in the problem, this is the initial information given to start the proof).

Step 2: Second Statement and Reason

• Statement: $\angle CBD \cong \angle ABF$
• Reason: Vertical angles are congruent. $\angle CBD$ and $\angle ABF$ are vertical angles (formed by the intersection of lines $AD$ and $CF$), so they are congruent.

Step 3: Third Statement and Reason

• Statement: $\angle ABF \cong \angle BFE$
• Reason: Transitive property of congruence. If $\angle CBD \cong \angle BFE$ (from step 1) and $\angle CBD \cong \angle ABF$ (from step 2), then by the transitive property, $\angle ABF \cong \angle BFE$.

Part B

Step 1: First Statement and Reason

• Statement: $m\angle CBD = m\angle BFE$
• Reason: Given (this is the initial information provided for Part B).

Step 2: Second Statement and Reason

• Statement: $m\angle CBD + m\angle DBF = 180^\circ$
• Reason: Linear pair postulate. $\angle CBD$ and $\angle DBF$ form a linear pair (they are adjacent angles that form a straight line along $CF$), so their measures add up to $180^\circ$.

Step 3: Third Statement and Reason

• Statement: $m\angle BFE + m\angle DBF = 180^\circ$
• Reason: Substitution property. Since $m\angle CBD = m\angle BFE$ (from step 1), we can substitute $m\angle CBD$ with $m\angle BFE$ in the equation $m\angle CBD + m\angle DBF = 180^\circ$ (from step 2) to get $m\angle BFE + m\angle DBF = 180^\circ$.

Final Answers (for the drop - down selections)

Part A

• First reason: Given
• Second reason: Vertical angles are congruent
• Third reason: Transitive property of congruence

Part B

• First reason: Given
• Second reason: Linear pair postulate
• Third reason: Substitution property