Question

practice problems

1. complete the two - column proof.

given: ∠hkm and ∠fdk are supplementary.
prove: gk∥fc
statement\treason
∠hkm and ∠fdk are supplementary.\tgiven
m∠hkm + m∠fdk = 180°\tgiven
∠gkd≅∠hkm\tvertical angles are congruent.
\tdefinition of congruence
m∠gkd + m∠fdk = 180°
∠gkd and ∠fdk are supplementary.\tdefinition of supplementary angles
gk∥fc

Explanation:

Step1: State given fact

$\angle HKM$ and $\angle FDK$ are supplementary (Given)

Step2: Write supplementary - angle equation

$m\angle HKM + m\angle FDK=180^{\circ}$ (Given)

Step3: Use vertical - angle property

$\angle GKD\cong\angle HKM$ (Vertical angles are congruent)

Step4: Apply congruence definition

$m\angle GKD = m\angle HKM$ (Definition of congruence)

Step5: Substitute angles

$m\angle GKD + m\angle FDK = 180^{\circ}$ (Substitution property of equality)

Step6: Define supplementary angles

$\angle GKD$ and $\angle FDK$ are supplementary (Definition of supplementary angles)

Step7: Use parallel - line theorem

$GH\parallel FC$ (If two interior angles on the same side of a transversal are supplementary, then the two lines are parallel)

Answer:

$GH\parallel FC$