Question

How can you use properties of corresponding angles and vertical angles to justify that alternate interior angles ∠BHE and ∠CGF are congruent? Answer in complete sentences and use specific angle names as you work your way through your justification.

Answer 1

As the line EF intersects the line AB, angles ∠BHE and ∠AHF are vertically opposite angles. Therefore, according to the Vertical Angles Theorem, these two angles are congruent.

Assuming line AB and line CD are parallel, the line EF intersects parallel lines. Therefore, angles ∠AHF and ∠CGF are in the same relative position. Therefore, according to the Corresponding Angles Theorem, these two angles are congruent.

According to the Substitution Property of Equality, as ∠BHE ≅ ∠AHF and ∠AHF ≅ ∠CGF, then ∠BHE ≅ ∠CGF.

Explanation:

Vertical Angles Theorem

When two straight lines intersect, the opposite vertical angles are congruent.

Corresponding Angles Theorem

When a straight line intersects two parallel straight lines, the angles in the same relative position are congruent.

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As the line EF intersects the line AB, angles ∠BHE and ∠AHF are vertically opposite angles. Therefore, according to the Vertical Angles Theorem, these two angles are congruent.

• ∠BHE ≅ ∠AHF

Assuming line AB and line CD are parallel, the line EF intersects parallel lines. Therefore, angles ∠AHF and ∠CGF are in the same relative position. Therefore, according to the Corresponding Angles Theorem, these two angles are congruent.

• ∠AHF ≅ ∠CGF

According to the Substitution Property of Equality, as ∠BHE ≅ ∠AHF and ∠AHF ≅ ∠CGF, then ∠BHE ≅ ∠CGF.