Question

Fill in the missing statement and reason in the proof of the corresponding angles theorem. Segment AB is parallel to segment CD, and transversal EF intersects segment AB at G and segment CD at H. It is given that segment AB is parallel to segment CD and points E, G, H, and F are collinear. The measure of ∠EGF is 180°, by the definition of a straight angle. ∠AGE and ∠AGF are adjacent, so the measure of ∠AGE plus the measure of ∠AGF equals the measure of ∠EGF, by the Angle Addition Postulate. Then, substituting for the measure of ∠EGF it can be said that the measure of ∠AGE plus the measure of ∠AGF equals 180°. ________, so the measure of ∠CHE plus the measure of ∠AGF equals 180°. Substituting once again means that the measure of ∠AGE plus the measure of ∠AGF equals the measure of ∠CHE plus the measure of ∠AGF. The measure of ∠AGE is equal to the measure of ∠CHE ________. Finally, by the definition of congruence, ∠AGE is congruent to ∠CHE.

Answer 1

The corresponding angles theorem has been proved by using the Angle Addition Postulate and the fact that the angles ∠AGE and ∠CHE are congruent.

Step-by-step explanation:

To complete the proof, we can use the fact that segment AB is parallel to segment CD and points E, G, H, and F are collinear. Since segment AB is parallel to segment CD, the measure of ∠AGE plus the measure of ∠AGF must be equal to 180°, by the Angle Addition Postulate. We can substitute the measure of ∠EGF (which is 180°) in place of the measure of ∠AGE plus the measure of ∠AGF, resulting in the equation: the measure of ∠AGE plus the measure of ∠AGF equals 180°.

Using the same logic, we can also conclude that the measure of ∠CHE plus the measure of ∠AGF equals 180°. By substituting the measure of ∠EGF (which is 180°) for the measure of ∠CHE plus the measure of ∠AGF, we can say that the measure of ∠AGE plus the measure of ∠AGF equals the measure of ∠CHE plus the measure of ∠AGF.

Since both sides of the equation have the measure of ∠AGF, we can subtract it from both sides to get the measure of ∠AGE equals the measure of ∠CHE. Finally, since the angles have equal measures, we can use the definition of congruence to conclude that ∠AGE is congruent to ∠CHE.

Answer 2

Corresponding Angles Theorem; ∠EGB and ∠EHD are congruent.

Step-by-step explanation:

got around 87 on the test but this answer was correct.