Question

practice problems

1. complete the two - column proof.

given: ∠hkm and ∠fdk are supplementary.
prove: (overline{gh}paralleloverline{fc})
statement: ∠hkm and ∠bdc are supplementary.
reason: given
statement: (mangle hkm + mangle fdk=180^{circ})
reason: given
statement: ∠ckd≅∠hkm
reason: vertical angles are congruent.
statement:
reason: definition of congruence
statement: (mangle ckd + mangle fdk = 180^{circ})
statement: ∠ckd and ∠fdk are supplementary.
reason: definition of supplementary angles

Explanation:

Step1: Recall supplementary - angle property

Given $\angle HKM$ and $\angle FDK$ are supplementary, so $m\angle HKM + m\angle FDK=180^{\circ}$ (by the definition of supplementary angles).

Step2: Use vertical - angle congruence

Since $\angle CKD$ and $\angle HKM$ are vertical angles, $\angle CKD\cong\angle HKM$ (vertical angles are congruent).

Step3: Apply congruence definition

By the definition of congruence, if $\angle CKD\cong\angle HKM$, then $m\angle CKD = m\angle HKM$.

Step4: Substitute angles

Substitute $m\angle CKD$ for $m\angle HKM$ in $m\angle HKM + m\angle FDK = 180^{\circ}$, we get $m\angle CKD+m\angle FDK = 180^{\circ}$.

Step5: Conclude parallel lines

Since $m\angle CKD+m\angle FDK = 180^{\circ}$, $\overline{GH}\parallel\overline{FC}$ (if two same - side interior angles are supplementary, then the two lines are parallel).

Answer:

The completed two - column proof is:

Statement	Reason
$m\angle HKM + m\angle FDK=180^{\circ}$	Given
$\angle CKD\cong\angle HKM$	Vertical angles are congruent
$m\angle CKD = m\angle HKM$	Definition of congruence
$m\angle CKD+m\angle FDK = 180^{\circ}$	Substitution
$\angle CKD$ and $\angle FDK$ are supplementary	Definition of supplementary angles
$\overline{GH}\parallel\overline{FC}$	If two same - side interior angles are supplementary, then the two lines are parallel