section 9-1
DESCRIPTION
Section 9-1. Points, Lines, Planes, and Angles. Points, Lines, Planes, and Angles. The Geometry of Euclid Points, Lines, and Planes Angles. 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. - PowerPoint PPT PresentationTRANSCRIPT
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
AD
El
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 NameBetween 0° and 90° Acute Angle
90° Right AngleGreater 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)°
Solution3x + 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 23 4
5 67 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