Points, lines, and planes are the primary elements of geometry. This chapter will explore the relationship between the lines in a plane. If two lines are in a plane, either they intersect or they are parallel.
Experiment
Discover the properties of lines in a plane.
Materials
Two pencils
A table
Procedure
1.Place two pencils on a table. The table represents a plane.
2.Let the two pencils represent two straight lines, but remember that straight lines extend infinitely. Make an X with the pencils. The pencils now intersect at exactly one point.
3.Try to make the pencils intersect at more than one point. It’s impossible! The only way to make two pencils intersect at more than one point is to put them on top of each other, but then they represent the same line.
Something to think about . . .
How many different ways can two lines intersect?
EXAMPLE:
These lines intersect at exactly one point, A.
However, two lines can intersect an infinite number of different ways. These lines could also intersect at point B, point C, or any other point along either line.
If two lines intersect they form four angles.
When two lines intersect, the sum of any two adjacent angles is 180 degrees.
EXAMPLE:
Angle 1 and angle 2 are adjacent angles and their sum is 180 degrees.
Angle 1 and angle 4 are adjacent angles and their sum is 180 degrees.
Angle 2 and angle 3 are adjacent angles and their sum is 180 degrees.
Angle 3 and angle 4 are adjacent angles and their sum is 180 degrees.
The sum of all four angles formed by two intersecting lines is 360 degrees.
The sum of angles 1, 2, 3, and 4 is 360 degrees.
If two lines intersect, they form four angles. The opposite angles formed are called vertical angles.
EXAMPLE:
Angles a and c are vertical angles. Angles b and d are vertical angles.
Theorem: Vertical angles are always congruent. Vertical angles always have the same measure.
EXAMPLE:
Angles a and c are vertical angles; therefore . . .
angles a and c are congruent.
angles a and c have the same measure.
Angles b and d are vertical angles; therefore . . .
angles b and d are congruent.
angles b and d have the same measure.
Use this diagram to solve the problems that follow.
1.If m∠a = 15, what is the measure of angle b?
2.If m∠a = 15, what is the measure of angle c?
3.If m∠a = 15, what is the m∠d?
4.What is the sum of the m∠a and the m∠b?
5.What is the sum of the m∠b and the m∠c?
6.What is the sum of the m∠c and the m∠d?
7.What is the sum of the m∠d and the m∠a?
8.What is the sum of the measures of angles a, b, c, and d?
(Answers)
Sometimes two lines intersect to form right angles.
Theorem: Perpendicular lines always intersect to form four right angles.
These lines are perpendicular to each other.
All four of these angles are right angles. The sum of all four of these angles is 90 + 90 + 90 + 90 = 360. The opposite angles are vertical angles and are congruent. All four angles are congruent to each other.
Experiment
Discover the number of lines that can be drawn from a point not on the line to a line.
Materials
Red pencil
Black pencil
Ruler
Paper
Procedure
1.Draw a red line on a piece of paper.
2.Use the black pencil to place a black point above the red line.
3.How many lines can you draw through the black point that also intersect the red line?
Something to think about . . .
How many lines can be drawn perpendicular to the red line that go through a single point?
•
Given a line and a point not on that line, there is exactly one line that passes through the point perpendicular to the line.
Experiment
Explore the relationship between a line and a point on the line.
Materials
Red pencil
Black pencil
Paper
Ruler
Procedure
1.Draw a red line on a piece of paper. Make a black dot on the red line.
2.How many lines can you draw through the black dot?
3.Draw a second red line on a piece of paper. Make a black dot on this line.
4.How many perpendicular lines can you draw through the black dot?
Something to think about . . .
How many perpendicular lines can you draw through two parallel lines?
Postulate
Through a point on a line, there is exactly one line perpendicular to the given line.
Theorem: If two lines intersect and the adjacent angles are congruent, then the lines are perpendicular.
MINI-PROOF
How can you prove the theorem: If two lines intersect and the adjacent angles are congruent, then the lines are perpendicular?
What do you know?
1.Angle 1 and angle 2 are adjacent angles.
2.Angle 1 and angle 2 are congruent.
3.The measure of angle 1 plus the measure of angle 2 is equal to a straight angle, which is 180.
What does this mean?
4.Since angles 1 and 2 are congruent, the measures of angles 1 and 2 are equal.
5.Since angle 1 is equal to angle 2 and they form a straight angle, they must each be equal to 90 degrees.
6.Since angle 1 and angle 2 are each 90 degrees, they are each right angles.
What can you conclude?
7.Since both angle 1 and angle 2 are right angles, the lines must be perpendicular.
Determine whether each of these statements is true or false.
1.Two intersecting lines are always perpendicular.
2.Two different lines can intersect at more than one point.
3.Perpendicular lines form acute angles.
4.The sum of the measures of the angles of two perpendicular lines is 360 degrees.
5.All right angles are congruent to each other.
(Answers)
Two lines are parallel if and only if they are in the same plane and they never intersect. Parallel lines are the same distance from each other over their entire lengths.
If two lines are cut by a transversal, eight angles are formed.
These angles have specific names. Memorize them!
•Exterior angles are angles that lie outside the space between the two lines. Angles 1, 2, 7, and 8 are exterior angles.
•Interior angles lie in the space between the two lines. Angles 3, 4, 5, and 6 are interior angles.
•Alternate interior angles are angles that lie between the two lines but on opposite sides of the transversal. Angles 4 and 5 are alternate interior angles. Angles 3 and 6 are alternate interior angles.
•Alternate exterior angles are angles that lie outside the two lines and on opposite sides of the transversal. Angles 1 and 8 are alternate exterior angles. Angles 2 and 7 are alternate exterior angles.
•Corresponding angles are non-adjacent angles on the same side of the transversal. One corresponding angle must be an interior angle and one must be an exterior angle. Angles 1 and 5 are corresponding angles. Angles 2 and 6 are corresponding angles. Angles 3 and 7 are corresponding angles. Angles 4 and 8 are corresponding angles.
•Consecutive interior angles are non-adjacent interior angles that lie on the same side of the transversal. Angles 3 and 5 are consecutive interior angles. Angles 4 and 6 are consecutive interior angles.
Determine the relationship between each of the following pairs of angles. Circle the correct letter(s) for each pair.
A = Adjacent
AI = Alternate Interior
AE = Alternate Exterior
C = Corresponding Angles
V = Vertical Angles
S = Supplementary Angles
1.∠1 and ∠2: A AI AE C V S
2.∠1 and ∠3: A AI AE C V S
3.∠1 and ∠4: A AI AE C V S
4.∠1 and ∠5: A AI AE C V S
5.∠1 and ∠6: A AI AE C V S
6.∠1 and ∠7: A AI AE C V S
7.∠1 and ∠8: A AI AE C V S
(Answers)
Parallel Postulate
Given a line and a point not on the line, there is only one line parallel to the given line.
Draw a red line. Draw a black dot, not on the line. Label it P.
How many lines can you draw through the black dot parallel to the red line?
Postulate
If two lines are cut by a transversal and the corresponding angles are equal, then the lines are parallel.
Theorem: If two parallel lines are cut by a transversal, then their alternate interior angles are congruent.
What does the term transversal mean?
A transversal is a line that intersects two lines at different points.
What does the term alternate interior mean?
Alternate means on different sides of the transversal.
Interior means between the two parallel lines.
Alternate interior means between the two parallel lines and on opposite sides of the transversal.
Look at these two parallel lines.
Angles 1 and 4 are alternate interior angles.
Angles 2 and 3 are alternate interior angles.
The measure of angle 1 is equal to the measure of angle 4.
The measure of angle 2 is equal to the measure of angle 3.
Remember, when two parallel lines are cut by a transversal, eight different angles are formed. If you know the measure of one of the eight angles, you can find the measure of all eight of the angles.
EXAMPLE:
If the measure of angle 3 is 60 degrees, you can find the measure of all the rest of the angles.
The measure of angle 2 is also 60 degrees, since it is vertical to angle 3.
The measure of angle 6 is also 60 degrees, since it is an alternate interior angle to angle 3.
The measure of angle 7 is also 60 degrees since it is a vertical angle to angle 6.
The measure of angle 1 is 120 degrees since it is supplemental to angle 3.
The measure of angle 4 is 120 degrees, since it is vertical to angle 1.
The measure of angle 5 is 120 degrees, since it is an alternate interior angle to angle 4.
The measure of angle 8 is 120 degrees, since it is a vertical angle to angle 5.
Assume that the measure of angle 4 is 110 degrees.
1.What is the m∠1?
2.What is the m∠2?
3.What is the m∠3?
4.What is the m∠5?
5.What is the m∠6?
6.What is the m∠7?
7.What is the m∠8?
(Answers)
Theorem: If two parallel lines are cut by a transversal, then their corresponding angles are congruent.
What are corresponding angles? Corresponding angles are angles that lie on the same side of the transversal. One corresponding angle lies on the interior of the parallel lines while the other corresponding angle lies on the exterior of the parallel lines.
EXAMPLE:
Angles 1 and 5 are corresponding angles, so they are congruent.
Angles 2 and 6 are corresponding angles, so they are congruent.
Angles 3 and 7 are corresponding angles, so they are congruent.
Angles 4 and 8 are corresponding angles, so they are congruent.
Theorem: If two parallel lines are cut by a transversal, their alternate exterior angles are congruent.
Exterior angles are found above or below the pair of parallel lines.
Alternate angles lie on opposite sides of the transversal. Alternate exterior angles lie outside the parallel lines, and on the opposite sides of the transversal.
Angles 1 and 8 are alternate exterior angles; therefore, they are congruent.
Angles 2 and 7 are alternate exterior angles; therefore, they are congruent.
Theorem: If two parallel lines are cut by a transversal, the consecutive interior angles are supplementary.
The consecutive interior angles are angles of the same side of the transversal and inside both parallel lines.
EXAMPLE:
Angles 3 and 5 are consecutive interior angles.
Angles 4 and 6 are consecutive interior angles.
Angles 3 and 5 are supplementary angles.
Angles 4 and 6 are supplementary angles.
Explore the relationship between angles formed by parallel lines and a transversal.
Materials
Pencil
Paper
Ruler
Protractor
Procedure
1.Draw two parallel lines.
2.Draw a transversal across the lines.
3.Label the eight angles formed angles 1, 2, 3, 4, 5, 6, 7, and 8.
4.Measure each of the angles and write its measure on the diagram.
5.Describe the relationship between each pair of angles as complementary, supplementary, equal, and unknown. Enter the results in the chart. Place a C for Complementary, S for Supplementary, E for Equal, and U for Unknown.
Something to think about . . .
What did you notice about the relationship of the angles?
When two parallel lines are cut by a transversal, the following pairs of angles are congruent:
•alternate interior angles
•corresponding angles
•alternate exterior angles
When two parallel lines are cut by a transversal, the following pairs of angles are supplementary:
•consecutive interior angles
Theorem: If two parallel lines are cut by a transversal, then any two angles are either congruent or supplementary.
Theorem: If a line is perpendicular to one of two parallel lines, then it is perpendicular to the other.
Proving lines parallel
There are four ways to prove two lines parallel. First, cut the two lines by a transversal. If any of the following are true, then the lines are parallel.
1.The alternate interior angles are congruent.
2.Their corresponding angles are congruent.
3.Their alternate exterior angles are congruent.
4.The interior angles on the same side of the transversal are supplementary.
Way 1—Alternate Interior Angles
If ∠4 and ∠5 are congruent, then line m is parallel to line n.
If ∠3 and ∠6 are congruent, then line m is parallel to line n.
Way 2—Corresponding Angles
If ∠1 and ∠5 are congruent, then line m is parallel to line n.
If ∠2 and ∠6 are congruent, then line m is parallel to line n.
If ∠3 and ∠7 are congruent, then line m is parallel to line n.
If ∠4 and ∠8 are congruent, then line m is parallel to line n.
Way 3—Alternate Exterior Angles
If ∠1 and ∠8 are congruent, then line m is parallel to line n.
If ∠2 and ∠7 are congruent, then line m is parallel to line n.
Are these two lines parallel?
If each of the following equations are true, decide if line a and line b are parallel. Answer Yes, No, or Don’t know.
1.m∠1 = m∠5
2.m∠1 = m∠4
3.m∠1 = m∠3
4.m∠4 = m∠5
5.m∠1 = m∠8
(Answers)
Fill in the blanks with the correct terms to find out how well you understand perpendicular and parallel lines.
1.If two lines intersect they form _______ angles.
2.Vertical angles are always _______.
3.Parallel lines never ______.
4.Perpendicular lines intersect to form four ______ angles.
5.If two parallel lines are cut by a transversal, then three types of angles are equal. What are they?
a._____________________________
b._____________________________
c._____________________________
6.If two parallel lines are cut by a transversal, then any two angles are either congruent or _____________.
7.Through a point not on a line there is (are) __________ perpendicular line(s) to the given line.
(Answers)
