CPM Homework Help : GC Problem 3-31

Examine the diagram below. Name the geometric relationships of the angles below.

A transversal line cuts two parallel lines. About the point of intersection of the top parallel line and the transversal line are angles starting at top left going clockwise and labeled as follows: a, b, d, and c. About the point of intersection of the bottom parallel line and the transversal are angles starting at the top left going clockwise and labeled as follows: e, f, h, and g.

$d$ and $e$
Hint (a):
See More Angle Pair Relationships below.

$e$ and $h$
Hint (b):
See part (a).

$a$ and $e$
Answer (c):
Corresponding angles

$c$ and $d$
Answer (d):
Supplementary and/or adjacent angles

More Angle Pair Relationships

Vertical angles are the two opposite (that is, non-adjacent) angles formed by two intersecting lines, such as angles $∠c$ and $∠g$ in the diagram at right. $∠c$ by itself is not a vertical angle, nor is $∠g$ , although $∠c$ and $∠g$ together are a pair of vertical angles. Vertical angles always have equal measure.

Corresponding angles lie in the same position but at different points of intersection of the transversal. For example, in the diagram at right, $∠m$ and $∠d$ form a pair of corresponding angles, since both of them are to the right of the transversal and above the intersecting line. Corresponding angles are congruent when the lines intersected by the transversal are parallel.

$∠f$ and $∠m$ are alternate interior angles because one is to the left of the transversal, one is to the right, and both are between (inside) the pair of lines. Alternate interior angles are congruent when the lines intersected by the transversal are parallel.

A vertical transversal line that cuts two lines. About the point of intersection of the top line & the transversal line are these angles: exterior left, c, exterior right, d, interior right, g, & interior left, f, shaded. About the point of intersection of the bottom line & the transversal line are these angles: interior left, h, interior right, m, shaded, exterior right, n, & exterior left, k.

$∠g$ and $∠m$ are same-side interior angles because both are on the same side of the transversal and both are between the pair of lines. Same-side interior angles are supplementary when the lines intersected by the transversal are parallel.

A vertical transversal line that cuts two lines. About the point of intersection of the top line & the transversal line are these angles: exterior left, c, exterior right, d, interior right, g, shaded, & interior left, f. About the point of intersection of the bottom line & the transversal line are these angles: interior left, h, interior right, m, shaded, exterior right, n, & exterior left, k.