Question

1. given km || eg, ∠klj ≅ ∠fjh prove: km || ij options: a. alternate interior angles theorem b. corresponding angles theorem c. converse of the alternate interior angles theorem d. converse of the corresponding angles theorem statement reason km || eg given ∠klj ≅ ∠fjh given ∠klj ≅ ∠efa ∠efa ≅ ∠fjh km || ij transitive property

Explanation:

Step1: State given information

Given $KM\parallel EG$ and $\angle KLI\cong\angle FIH$.

Step2: Use parallel - line properties

Since $KM\parallel EG$, corresponding angles are equal. But we want to prove $KM\parallel IJ$. We know that $\angle KLI\cong\angle FIH$ and $\angle EFA\cong\angle FIH$ (vertically - opposite angles).

Step3: Apply transitive property

By the transitive property of congruence, since $\angle KLI\cong\angle FIH$ and $\angle EFA\cong\angle FIH$, we have $\angle KLI\cong\angle EFA$.

Step4: Use converse theorem

$\angle KLI$ and $\angle EFA$ are corresponding angles. By the Converse of the Corresponding Angles Theorem, if corresponding angles are congruent, then the lines are parallel. So $KM\parallel IJ$.

Answer:

1. Given: $KM\parallel EG$, $\angle KLI\cong\angle FIH$
2. $\angle EFA\cong\angle FIH$ (Vertically - opposite angles)
3. $\angle KLI\cong\angle EFA$ (Transitive Property)
4. $KM\parallel IJ$ (Converse of the Corresponding Angles Theorem)