Idiot\'s Guides Basic Math and Pre-Algebra

Chapter 12: Basics of Geometry 161

Look at only the angles that are between the parallel lines, ’3, ’4, ’5, and ’6. Choose one
from the top cluster, say ’3, and the one from the bottom cluster on the other side of the
transversal, ’ 6, and you have a pair of alternate interior angles. Take one angle from each side of
the transversal so that they are not between the parallels but outside them, like ’1 and ’8 or ’ 2
and ’7, and you have a pair of alternate exterior angles. Alternate exterior angles are congruent if
the lines are parallel.

DEFINITION

Alternate interior angles are two angles formed when a transversal intersects parallel

lines that are on opposite sides of the transversal and between the parallel lines.

Alternate exterior angles are two angles formed when a transversal intersects parallel

lines that are on opposite sides of the transversal and outside the parallel lines.

When parallel lines are cut by a transversal, alternate interior angles are congruent. They have
the same measurements. ’ 3 # ’6 and ’ 4 # ’5.

Using these facts about parallel lines and corresponding angles and alternate interior angles,
and the fact that vertical angles are congruent, you can figure out that ’ 1 # ’ 4 # ’ 5 # ’ 8
and ’ 2  # ’ 3 # ’ 6 # ’7. Add the fact that ’1 and ’2 are supplementary, and it becomes
possible to assign each of the angles one of two measurements. ’1, ’4, ’5, and ’8 have one
measurement, and ’2, ’3, ’6, and ’7 are supplementary to those. If you know the measurement
of one angle created when a transversal cuts parallel lines, you can find the measurements of
all eight angles. If m’1 is 40r, then ’4, ’5, and ’8 are also 40r, and the other four angles are
180 r – 40r = 140r.

CHECK POINT

Lines PQ and RT are parallel. Transversal AB intersects PQ at X and RT at Y.

16. ’PXY and ’XYT are a pair of ____________ angles.

17. ’AXQ and ’XYT are a pair of ____ angles.


18. If m’XYT = 68r, then m’PX A = __.


19. If m’PXY = 107r, then m’RYB = __.


20. If AB PQ, then m’XYR = __.