Answer:
- b - corresponding
- a - alternate interior
- d - vertical
- c - alternate exterior
- e - supplementary
- b - corresponding
Explanation:
This is basically a vocabulary question. Angle pairs around a transversal crossing a pair of parallel lines all have names. Those names are used to refer to different theorems about the relationships of the angles to each other. Once you understand the words being used, the vocabulary is easier to remember.
Linear Pairs and Vertical Angles
Angles formed where one line crosses another form a linear pair if they share a side and the non-shared sides are opposite rays. If they share a vertex, but not a side, they are vertical angles. The angles of a linear pair are supplementary: they total 180°. Vertical angles are supplementary to the same angle, so vertical angles are congruent. These relations hold anywhere lines cross.
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Angle Names at a Transversal
Where a single line crosses two (parallel) lines, that line is called a "transversal". Four angles will be formed at each intersection. Some of those angles are between the (parallel) lines, so are referred to as interior angles. The ones that are not between the (parallel) lines are referred to as exterior angles. The angles are alternate if they are on opposite sides of the transversal, and they are consecutive, or same-side if they are on the same side of the transversal.
Angles that lie in the same direction from the crossing point (to the northeast, for example) are corresponding angles.
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In your figure, we can identify the following pairs:
- corresponding: (a, f), (b, g), (c, h), (e, i)
- alternate exterior: (a, i), (c, g)
- alternate interior: (b, h), (e, f)
- consecutive exterior: (a, g), (c, i)
- consecutive interior: (b, f), (e, h)
- vertical: (a, e), (b, c), (f, i), (g, h)
- linear pairs: (a, b), (a, c), (b, e), (c, e), (f, g), (f, h), (g, i), (h, i)
The theorems referenced above will tell you if and only if the lines are parallel ...
- corresponding, alternate interior, alternate exterior angles are congruent
- consecutive interior, consecutive exterior angles are supplementary (total 180°)
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Using the above, we find the answers to your questions to be ...
1. (a, f) - corresponding angles (b)
2. (b, h) - alternate interior angles (a)
3. (h, g) - vertical angles (d)
4. (c, g) - alternate exterior angles (c)
5. (a, c) - supplementary angles (e)
6. (e, i) - corresponding angles (b)
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Additional comment
In the above, we put (parallel) in parentheses, because these same definitions apply where the crossed lines may not be parallel. The relevant angle relationships can be used to determine if the lines are parallel or not. For example, if "corresponding" angles are congruent, the lines will be parallel. If "consecutive interior" angles do not total 180°, the lines cannot be parallel.
These same relationships hold in geometric figures, such as rectangles, trapezoids, parallelograms. For example, each side of a parallelogram is a transversal crossing a pair of parallel lines (the adjacent sides). Sometimes, you're asked to prove particular angle relationships in these figures. You can often do so by making use of the angle theorems related to parallel lines.