SECTION 9-1 Points, Lines, Planes, and Angles Slide 9-1-1.

SECTION 9-1

• Points, Lines, Planes, and Angles

Slide 9-1-1

POINTS, LINES, PLANES, AND ANGLES

• The Geometry of Euclid• Points, Lines, and Planes • Angles

Slide 9-1-2

THE GEOMETRY OF EUCLID

Slide 9-1-3

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.

POINTS, LINES, AND PLANES

Slide 9-1-4

A

DE

l

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 , , or .

HALF-LINE, RAY, AND LINE SEGMENT

Slide 9-1-5

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.

HALF-LINE, RAY, AND LINE SEGMENT

Slide 9-1-6

Name Figure Symbol

Line AB or BA AB or BA

Half-line AB AB

Half-line BA BA

Ray AB AB

Ray BA BA

Segment AB or segment BA

AB or BA

A B

A B

A B

A B

A B

A B

PARALLEL AND INTERSECTING LINES

Slide 9-1-7

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.

Parallel Intersecting Skew

PARALLEL AND INTERSECTING PLANES

Slide 9-1-8

Parallel planes never meet.

Two distinct intersecting planes meet and form a straight line.

Parallel Intersecting

ANGLES

Slide 9-1-9

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.

Vertex B

A

C

ABC,B

Side

Side

ANGLES

Slide 9-1-10

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

45°90°

10°

150°360°

ANGLES

Slide 9-1-11

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

PROTRACTOR

Slide 9-1-12

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

INTERSECTING LINES

Slide 9-1-13

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

VERTICAL ANGLES

Slide 9-1-14

In the figure below the pairare called vertical angles.are also vertical angles.

A

CB

D

E

and ABC DBE and DBA EBC

Vertical angles have equal measures.

EXAMPLE: FINDING ANGLE MEASURE

Slide 9-1-15

Find the measure of each marked angle below.

(3x + 10)° (5x – 10)°

Solution

3x + 10 = 5x – 10

So each angle is 3(10) + 10 = 40°.

Vertical angles are equal.2x = 20x = 10

COMPLEMENTARY AND SUPPLEMENTARY ANGLES

Slide 9-1-16

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

EXAMPLE: FINDING ANGLE MEASURE

Slide 9-1-17

Find the measure of each marked angle below.(2x + 45)° (x – 15)°

Solution2x + 45 + x – 15 = 180 3x + 30 = 180

Evaluating each expression we find that the angles are 35° and 145°.

Supplementary angles.

3x = 150x = 50

ANGLES FORMED WHEN PARALLEL LINES ARE CROSSED BY A TRANSVERSAL

Slide 9-1-18

1 2

3 4

5 6

7 8

The 8 angles formed will be discussed on the next few slides.

>

>

ANGLES FORMED WHEN PARALLEL LINES ARE CROSSED BY A TRANSVERSAL

Slide 9-1-19

Alternate interior angles

Alternate exterior angles

Angle measures are equal.

Angle measures are equal.

1

5 4

8

(also 3 and 6)

(also 2 and 7)

Name

ANGLES FORMED WHEN PARALLEL LINES ARE CROSSED BY A TRANSVERSAL

Slide 9-1-20

Interior angles on same side of transversal

Corresponding angles

Angle measures are equal.

Angle measures add to 180°.

46

2

6

(also 3 and 5)

(also 1 and 5, 3 and 7, 4 and 8)

Name

EXAMPLE: FINDING ANGLE MEASURE

Slide 9-1-21

Find the measure of each marked angle below.

(x + 70)°(3x – 80)°

Solution

Evaluating we find that the angles are 145°.

Alternating interior angles.x + 70 = 3x – 80 2x = 150

x = 75