Question

example 3 prove the alternate interior angles converse
prove that if two lines are cut by a transversal so the alternate interior
angles are congruent, then the lines are parallel.
solution
given ∠4 ≅ ∠5
prove g ∥ h
statement | reason

1. ∠4 ≅ ∠5 | 1. given
2. ∠4 ≅ ∠1 | 2. vert. ∠s ≅
3. ∠5 ≅ ∠1 | 3. trans. property of ≅
4. g ∥ h | 4. corresponding angles converse postulate

Explanation:

To properly prove the Alternate Interior Angles Converse, we can use a more precise and complete two - column proof:

Step 1: List the given information

We are given that $\angle4\cong\angle5$. In the two - column proof, we write:

Statement	Reason

Step 2: Use the Vertical Angles Theorem

Vertical angles are congruent. $\angle4$ and $\angle1$ are vertical angles. So we have:

Statement	Reason

Step 3: Use the Transitive Property of Congruence

We know that $\angle4\cong\angle5$ (from step 1) and $\angle4\cong\angle1$ (from step 2). By the transitive property of congruence, if $\angle A\cong\angle B$ and $\angle B\cong\angle C$, then $\angle A\cong\angle C$. So, $\angle5\cong\angle1$.

Statement	Reason

Step 4: Apply the Corresponding Angles Converse

The Corresponding Angles Converse states that if two lines are cut by a transversal and the corresponding angles are congruent, then the two lines are parallel. Here, $\angle1$ and $\angle5$ are corresponding angles (when we consider lines $g$ and $h$ cut by the transversal). Since $\angle5\cong\angle1$, by the Corresponding Angles Converse, we can conclude that $g\parallel h$.

Statement	Reason

So, through the use of the Vertical Angles Theorem, the Transitive Property of Congruence, and the Corresponding Angles Converse, we have proven that if two lines are cut by a transversal such that the alternate interior angles are congruent, then the lines are parallel.

Answer:

To properly prove the Alternate Interior Angles Converse, we can use a more precise and complete two - column proof:

Step 1: List the given information

We are given that $\angle4\cong\angle5$. In the two - column proof, we write:

Statement	Reason

Step 2: Use the Vertical Angles Theorem

Vertical angles are congruent. $\angle4$ and $\angle1$ are vertical angles. So we have:

Statement	Reason

Step 3: Use the Transitive Property of Congruence

We know that $\angle4\cong\angle5$ (from step 1) and $\angle4\cong\angle1$ (from step 2). By the transitive property of congruence, if $\angle A\cong\angle B$ and $\angle B\cong\angle C$, then $\angle A\cong\angle C$. So, $\angle5\cong\angle1$.

Statement	Reason

Step 4: Apply the Corresponding Angles Converse

The Corresponding Angles Converse states that if two lines are cut by a transversal and the corresponding angles are congruent, then the two lines are parallel. Here, $\angle1$ and $\angle5$ are corresponding angles (when we consider lines $g$ and $h$ cut by the transversal). Since $\angle5\cong\angle1$, by the Corresponding Angles Converse, we can conclude that $g\parallel h$.

Statement	Reason

So, through the use of the Vertical Angles Theorem, the Transitive Property of Congruence, and the Corresponding Angles Converse, we have proven that if two lines are cut by a transversal such that the alternate interior angles are congruent, then the lines are parallel.