When parallel lines are cut by another line, called a transversal, eight angles are formed. Different pairs from this group of eight are classified in different ways.
As the transversal crosses the first line, it creates a cluster of four angles, labeled 1, 2, 3, and
4 in this picture. As it crosses the second line, it creates another cluster of four angles, labeled
5, 6, 7, and 8. In each cluster, there is an angle in the upper left position (1 from the top cluster or 5 from the bottom). There are also angles in the upper right, lower left, and lower right positions. The angle from the upper cluster and the angle from the lower cluster that are in the same position are called corresponding angles.
DEFINITION
A transversal is a line that intersects two or more other lines.
Corresponding angles are a pair of angles created when a transversal intersects two parallel lines that are on the same side of the transversal and are both above or both below the parallel lines.
When parallel lines are cut by a transversal, corresponding angles are congruent. They have the same measurements. 1 # 5, 2 # 6, 3 # 7, and 4 # 8.
1 2
3 4
5 6
7 8
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 m1 is 40r, then 4, 5, and 8 are also 40r, and the other four angles are 180r – 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 mXYT = 68r, then mPXA = ______.
19. If mPXY = 107r, then mRYB = ______.
20. If AB PQ, then mXYR = ______.
Slopes
Geometry and algebra may feel like different worlds at times, but now and then, they come together, and that often happens on the coordinate plane. When we looked at graphing linear equations, you saw how the slope of a line controlled its tilt or angle. When you talk about parallel and perpendicular lines, the angles the lines make are important.
The graphs of two lines in the coordinate plane will be parallel lines if they have the same slope.
The matching slopes mean they run at the same angle and don’t tilt toward each other, so they never cross. The line y = 2x – 3 and the line y = 2x + 5 both have a slope of 2, and so they will be parallel.
In order for the graphs of two linear equations to be perpendicular lines, one must rise and one must fall, so the slopes must have opposite signs. That alone won’t get that exact right angle, however. To actually be perpendicular, the lines must have slopes that are negative reciprocals. If one line has a slope of 2, a line perpendicular to it will have a slope of -12. The graphs of y 3x
5 4
- 1
5 and y 5x
3 ⴚ1
ⴝ are perpendicular lines because their slopes, 3 5 - and 5
3, multiply to -1, so they are negative reciprocals.
0 6
0 6
Label each pair of lines as parallel, perpendicular, or neither.
21. Line a has a y-intercept of 2 and a slope of -3. Line b has a y-intercept of -1
The Least You Need to Know
•
Midpoints and bisectors divide segments into two congruent segments; angle bisectors divide angles into two congruent angles.•
Angles less than 90r are acute, angles greater than 90r are obtuse. Right angles are 90r and straight angles are 180r.•
Complementary angles are two angles that add to 90r, and supplementary angles are two angles that add to 180r.•
Vertical angles are congruent; linear pairs are supplementary.•
Parallel lines never intersect; perpendicular lines intersect at right angles.•
If parallel lines are cut by a transversal, pairs of corresponding angles, alternate interior angles, or alternate exterior angles are congruent.13
Triangles
In This Chapter
•
Classifying triangles by sides and angles•
How to find missing angles and measure exterior angles•
The Pythagorean theorem and how to use it•
How to find area and perimeter of triangles Geometry is a branch of mathematics that looks at shapes.That’s true, but “shapes” covers so much ground that the statement really doesn’t tell you much. To understand what geometry really looks at, what kinds of questions it tries to answer, you need to break that down a bit.
The first big distinction you can make is that geometry looks at shapes with straight sides, called polygons, and shapes that curve, or circles. There are some things you can say about all polygons, but the conversation gets more interesting when you start to break that group down even more according to the number of sides the polygon has.
In this chapter, we’ll explore triangles, polygons with three sides. We’ll look at some things that are true about all triangles, and then organize triangles into groups, based on facts about their sides or about their angles, or both. There will be lots to learn about right triangles, the most interesting group. We’ll also have a look at the very practical matters of area and perimeter.