prove: ( overline{gh} cong overline{eh} ) ( overline{dh} cong overline{fh} ) what is the reason for step 5 in this proof? proof

| statement | reason || --- | --- || 1. quadrilateral defg is a parallelogram. | given || 2. ( overline{de} parallel overline{gf} ) ( overline{dg} parallel overline{ef} ) | definition of a parallelogram || 3. draw ( overline{df} ) and ( overline{ge} ). these line segments are transversals cutting two pairs of parallel lines: ( overleftrightarrow{de} ) and ( overleftrightarrow{gf} ) and ( overleftrightarrow{dg} ) and ( overleftrightarrow{ef} ). | drawing line segments || 4. place point h where ( overline{df} ) and ( overline{ge} ) intersect. | defining a point || 5. ( angle hgd cong angle hef ) ( angle hdg cong angle hfe ) | || 6. ( overline{dg} cong overline{ef} ) | opposite sides of a parallelogram are congruent. || 7. | asa criterion for congruence || 8. ( overline{gh} cong overline{eh} ) ( overline{dh} cong overline{fh} ) | corresponding sides of congruent triangles are congruent. |

a. vertical angles theorem
b. opposite interior angles of a parallelogram are congruent.
c. all angles of a rectangle have the same measure.
d. alternate interior angles theorem

Question

prove: ( overline{gh} cong overline{eh} ) ( overline{dh} cong overline{fh} ) what is the reason for step 5 in this proof? proof

| statement | reason || --- | --- || 1. quadrilateral defg is a parallelogram. | given || 2. ( overline{de} parallel overline{gf} ) ( overline{dg} parallel overline{ef} ) | definition of a parallelogram || 3. draw ( overline{df} ) and ( overline{ge} ). these line segments are transversals cutting two pairs of parallel lines: ( overleftrightarrow{de} ) and ( overleftrightarrow{gf} ) and ( overleftrightarrow{dg} ) and ( overleftrightarrow{ef} ). | drawing line segments || 4. place point h where ( overline{df} ) and ( overline{ge} ) intersect. | defining a point || 5. ( angle hgd cong angle hef ) ( angle hdg cong angle hfe ) |  || 6. ( overline{dg} cong overline{ef} ) | opposite sides of a parallelogram are congruent. || 7.  | asa criterion for congruence || 8. ( overline{gh} cong overline{eh} ) ( overline{dh} cong overline{fh} ) | corresponding sides of congruent triangles are congruent. |

a. vertical angles theorem
b. opposite interior angles of a parallelogram are congruent.
c. all angles of a rectangle have the same measure.
d. alternate interior angles theorem

Accepted Answer

# Brief Explanations:
- **Vertical Angles Theorem**: States that vertical angles (opposite angles formed by two intersecting lines) are congruent. But here we are dealing with angles formed by parallel lines and transversals.
- **Opposite interior angles of a parallelogram are congruent**: This is not a standard theorem. The standard ones are about opposite sides and opposite angles (not "interior" in this context).
- **All angles of a rectangle have the same measure**: A rectangle is a special case of a parallelogram, but we are not told this is a rectangle. The problem starts with a general parallelogram.
- **Alternate Interior Angles Theorem**: When a transversal cuts two parallel lines, alternate - interior angles are congruent. In the proof, since $DE\parallel GF$ (from the definition of a parallelogram) and $GE$ is a transversal, $\angle HGD\cong\angle HEF$ (alternate - interior angles). Also, since $DG\parallel EF$ (definition of a parallelogram) and $DF$ is a transversal, $\angle HDG\cong\angle HFE$ (alternate - interior angles).

# Answer:
D. Alternate Interior Angles Theorem

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