Question

Show how you got each answer:

What is the measurement of ∟1 if gg || h?
What is the measurement of ∟3 if gg || h?

Answer 1

Explanation:

angles 1 and 3 are supplementary angles (together they have 180°), because together they make sure that line f is a straight line, and there are no segments of the line angling "off". a straight line can be always seen as prolonged diameter of a circle. each side represents a half-circle with 180°.

a line intersecting 2 parallel lines has the same intersection angles with both lines, as parallel lines perfectly copy each other's behavior and attributes except for the y-intercept.

the angles of intersecting lines are the same in both sides of any of the 2 lines, they are just left-right mirrored.

therefore,

angle 1 = 82° = angle 2

angle 3 = 180 - angle 1 = 180 - 82 = 98° = angle 4

Answer 2

∠1 = 82°

∠3 = 98°

Explanation:

You want the measures of angles 1 and 3 where a transversal crosses parallel lines and one of the angles is marked as 82°.

Alternate interior angles

The "alternate interior angles theorem" tells you that the alternate interior angles created by a transversal crossing parallel lines are congruent. Hence angle 1 is congruent to the one marked 82°.

∠1 = 82°

Linear pair

The angles of a linear pair are supplementary. Angles 1 and 3 form a linear pair, so ...

∠3 = 180° -∠1 = 180° -82°

∠3 = 98°

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Additional comment

"Interior" angles are between the parallel lines. "Exterior" angles are outside the parallel lines. "Alternate" angles are on opposite sides of the transversal. "Consecutive" or "same-side" angles are on the same side of the transversal. (Angles 2 and 4 are "consecutive".)

"Corresponding" angles are in the same direction from the point of intersection of the transversal with the parallel lines. In this figure, angle 2 and the one marked 82° are corresponding. Corresponding angles are congruent. If you remember that, and that vertical angles are congruent, and linear pairs are supplementary, you can figure out all of the other relationships.

In the end, when the lines are parallel, all of the acute angles are congruent, and all of the obtuse angles are congruent. The acute and obtuse angles are supplementary. The angle relations themselves are pretty simple; the rest is a lot of vocabulary.