Janet is designing a frame for a client. She wants to prove to her client that m∠AGE ≅ m∠CHE in her sketch. What is the missing justification in the proof?
Segment AB is parallel to segment CD, and transversal EF intersects segment AB at G and segment CD at H.
Statement Justification
segment AB is parallel to segment CD with transversal segment EF Given
m∠AGE ≅ m∠HGB Vertical Angles Theorem
m∠HGB ≅ m∠CHE Alternate Interior Angles Theorem
m∠AGE ≅ m∠CHE
Alternate Exterior Angles Theorem
Corresponding Angles Theorem
Reflexive Property
Transitive Property
#3. Corresponding
The missing justification in the proof is "Corresponding Angles Theorem."
The missing justification in the proof is "Corresponding Angles Theorem".
The missing justification in the proof is "Alternate Exterior Angles Theorem."
To understand why this justification is needed, let's first understand the theorems involved:
1. Vertical Angles Theorem: When two lines intersect, the angles opposite each other are called vertical angles. Vertical angles are always congruent (i.e., have equal measure).
2. Alternate Interior Angles Theorem: When two parallel lines are intersected by a transversal, the pairs of alternate interior angles (angles that are on opposite sides of the transversal and inside the two parallel lines) are congruent.
3. Alternate Exterior Angles Theorem: When two parallel lines are intersected by a transversal, the pairs of alternate exterior angles (angles that are on opposite sides of the transversal and outside the two parallel lines) are congruent.
4. Corresponding Angles Theorem: When two parallel lines are intersected by a transversal, the pairs of corresponding angles (angles that are in the same position relative to the transversal and the two parallel lines) are congruent.
5. Reflexive Property: Any geometric object is congruent to itself.
6. Transitive Property: If two angles A and B are congruent, and angles B and C are congruent, then angles A and C are congruent.
Given that statement "Segment AB is parallel to segment CD with transversal segment EF," we can use the Alternate Exterior Angles Theorem to conclude that m∠AGE ≅ m∠CHE. This theorem states that since segment AB is parallel to segment CD and transversal EF intersects both, the angles on the exterior of the two parallel lines and on opposite sides of the transversal will be congruent.
Therefore, the missing justification in the proof is "Alternate Exterior Angles Theorem."