1. Section 9-1
Points, Lines, Planes, and Angles

2. Points, Lines, Planes, and Angles
The Geometry of Euclid
Points, Lines, and Planes
Angles

3. The Geometry of Euclid A point has no magnitude and no size.
A line has no thickness and no width and it extends indefinitely in two directions.
A plane is a flat surface that extends infinitely.

4. Points, Lines, and Planes
A capital letter usually represents a point. A line may named by two capital letters representing points that lie on the line or by a single letter such as l. A plane may be named by three capital letters representing points that lie in the plane or by a letter of the Greek alphabet such as l A E D

5. Half-Line, Ray, and Line Segment
A point divides a line into two half-lines, one on each side of the point.
A ray is a half-line including an initial point.
A line segment includes two endpoints.

6. Half-Line, Ray, and Line Segment
Name
Figure
Symbol
Line AB or BA
AB or BA
Half-line AB AB
Half-line BA BA
Ray AB
Ray BA
Segment AB or segment BA
A B
A B
A B
A B
A B
A B

7. Parallel and Intersecting Lines
Parallel lines lie in the same plane and never meet.
Two distinct intersecting lines meet at a point.
Skew lines do not lie in the same plane and do not meet.
Intersecting
Skew
Parallel

8. Parallel and Intersecting Planes
Parallel planes never meet.
Two distinct intersecting planes meet and form a straight line.
Parallel
Intersecting

9. Angles
An angle is the union of two rays that have a common endpoint. An angle can be named with the letter marking its vertex, and also with three letters: the first letter names a point on the side; the second names the vertex; the third names a point on the other side. A
Side
Vertex B
Side C

10. Angles
Angles are measured by the amount of rotation. 360° is the amount of rotation of a ray back onto itself.
90°
45°
10°
150°
360°

11. Greater than 90° but less than 180°
Angles
Angles are classified and named with reference to their degree measure.
Measure
Name
Between 0° and 90°
Acute Angle
90°
Right Angle
Greater than 90° but less than 180°
Obtuse Angle
180°
Straight Angle

12. Protractor
A tool called a protractor can be used to measure angles.

13. Intersecting Lines
When two lines intersect to form right angles they are called perpendicular.

14. Vertical Angles In the figure below the pair
are called vertical angles.
are also vertical angles. A D B E C
Vertical angles have equal measures.

15. Example: Finding Angle Measure
Find the measure of each marked angle below.
(3x + 10)°
(5x – 10)°
Solution
3x + 10 = 5x – 10
Vertical angles are equal.
2x = 20
x = 10
So each angle is 3(10) + 10 = 40°.

16. Complementary and Supplementary Angles
If the sum of the measures of two acute angles is 90°, the angles are said to be complementary, and each is called the complement of the other. For example, 50° and 40° are complementary angles
If the sum of the measures of two angles is 180°, the angles are said to be supplementary, and each is called the supplement of the other. For example, 50° and 130° are supplementary angles

17. Example: Finding Angle Measure
Find the measure of each marked angle below.
(2x + 45)°
(x – 15)°
Solution
2x x – 15 = 180
3x + 30 = 180
Supplementary angles.
3x = 150
x = 50
Evaluating each expression we find that the angles are 35° and 145°.

18. Angles Formed When Parallel Lines are Crossed by a Transversal
The 8 angles formed will be discussed on the next few slides.
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19. Angles Formed When Parallel Lines are Crossed by a Transversal
Name
Angle measures are equal.
Alternate interior angles
(also 3 and 6) 1
Angle measures are equal.
Alternate exterior angles 8
(also 2 and 7)

20. Angles Formed When Parallel Lines are Crossed by a Transversal
Name
Angle measures add to 180°.
Interior angles on same side of transversal 4 6
(also 3 and 5) 2
Angle measures are equal.
Corresponding angles 6
(also 1 and 5, 3 and 7, 4 and 8)

21. Example: Finding Angle Measure
Find the measure of each marked angle below.
(3x – 80)°
(x + 70)°
Solution
x = 3x – 80
Alternating interior angles.
2x = 150
x = 75
Evaluating we find that the angles are 145°.