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 property

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

Explanation:

Part A

Step1: Identify Given

The first statement $\angle CBD \cong \angle BFE$ is given, so the reason is "Given".

Step2: Vertical Angles

$\angle CBD$ and $\angle ABF$ are vertical angles (formed by intersecting lines), so vertical angles are congruent. Thus, $\angle CBD \cong \angle ABF$ with reason "Vertical angles are congruent".

Step3: Transitive Property

If $\angle CBD \cong \angle BFE$ (from step 1) and $\angle CBD \cong \angle ABF$ (from step 2), by the transitive property of congruence, $\angle ABF \cong \angle BFE$. The reason is "Transitive property of congruence".

Step1: Identify Given

The first statement $m\angle CBD = m\angle BFE$ is given, so the reason is "Given".

Step2: Linear Pair

$\angle CBD$ and $\angle DBF$ form a linear pair (they are adjacent and their non - common sides form a straight line). The sum of the measures of angles in a linear pair is $180^{\circ}$, so $m\angle CBD + m\angle DBF=180^{\circ}$ with reason "Linear pair postulate" (angles in a linear pair are supplementary).

Step3: 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 $m\angle CBD + m\angle DBF = 180^{\circ}$. So $m\angle BFE+m\angle DBF = 180^{\circ}$ with reason "Substitution property".

Answer:

(for the drop - downs in Part A):

• For $\angle CBD \cong \angle BFE$: Given
• For $\angle CBD \cong \angle ABF$: Vertical angles are congruent
• For $\angle ABF \cong \angle BFE$: Transitive property of congruence

Part B