Question

select the correct answer from each drop - down menu. in the given image, line ab is parallel to line dc and $overline{df}congoverline{bf}$. complete the table.

statements	reasons
$angle fdccongangle fba$
$angle afbcongangle$
$overline{df}congoverline{bf}$
$\triangle dfccong\triangle bfa$

options for reasons: alternate interior angles theorem, same - side interior angles, vertical angles theorem, corresponding angles theorem

Explanation:

Step1: Identify angle - angle relationship for ∠FDC and ∠FBA

Since line AB || line DC, by the alternate - interior angles theorem, when a transversal (the line passing through F) intersects two parallel lines AB and DC, ∠FDC and ∠FBA are alternate - interior angles. So the reason for ∠FDC≅∠FBA is the alternate interior angles theorem.

Step2: Identify the vertical - angle pair

∠AFB and ∠DFC are vertical angles. By the vertical angles theorem, vertical angles are congruent. So the angle that ∠AFB is congruent to is ∠DFC and the reason is the vertical angles theorem.

Step3: Prove triangle congruence

We have ∠FDC≅∠FBA (alternate interior angles), ∠AFB≅∠DFC (vertical angles), and DF≅BF (given). By the Angle - Angle - Side (AAS) congruence criterion, △DFC≅△BFA.

Answer:

Statements	Reasons
∠FDC≅∠FBA	alternate interior angles theorem
∠AFB≅∠DFC	vertical angles theorem
DF≅BF	given
△DFC≅△BFA	AAS congruence criterion