chapter 9. lesson 9-1 introduction to geometry: points, lines, and planes

Chapter 9

Lesson 9-1

Introduction to Geometry: Points, Lines, and Planes

Sample Name Symbol Description

∙A Point Point A A location in space. It has no size.

LineABBA

or n

A series of points that extends in opposite directions without end. A lowercase letter can name a line.

PlaneABCD or

M

A flat surface with no thickness. It contains many lines and extends without end in the directions of all its lines.

Line Segment

or segment

PQQP

A part of a line. It has two endpoints. PQ represents the length of PQ.

Ray CRA part of a line. It has exactly one endpoint. Name its endpoint first.

M

A B

CD

P

Q

C

R

Intersecting, Parallel, and Skew Lines

• Two lines that lie in the same plane and do not intersect are parallel. Use the symbol ║to indicate “is parallel to”.

• Two lines intersect if they have exactly one point in common.

• Skew lines are lines that do not lie in the same plane.

AB ║ PQ

EF intersects BF

AB and DE are skew

Parts of an Angle

• An angle is formed by two rays with a common endpoint.

• The rays are the sides of the angle.

• The common endpoint is the vertex.

B

A

C

●

●

Angle

Endpoint or Vertex

Ray

Classifying Angles

• An acute angle is less than 90○

.

• A right angle is 90○.

• An obtuse angle is greater than 90○

and less than 180○.

• A straight angle is equal to 180○

. Straight Angle

Obtuse Angle

Right Angle

Acute Angle

Lesson 9-2

Angle Relationships

and Parallel Lines

Adjacent and Vertical Angles

• Adjacent angles share a vertex and a side but no points in their interiors.

• Vertical angles are formed by two intersecting lines and are opposite each other.

Common Side

1

2

34

Angles 1 & 2 are vertical angles.

Angles 3 & 4 are vertical angles.

Angle Relationships

• If the sum of the measures of two angles is 90○, the angles are complementary.

• If the sum of the measures of two angles is 180○, the angles are supplementary.

Complementary Angles

Supplementary Angles

Relating Angles and Parallel Lines

A line that intersects two other lines in different points is a transversal.

When a transversal intersects two parellel lines, corresponding and alternate interior angles are congruent.

Alternate interior angles are in the interior of a pair of lines and on opposite sides of the transversal. d and e are alternate interior angles.

Corresponding angles lie on the same side of the transversal and in corresponding positions. d and h are corresponding angles.

Lesson 9-3

Classifying Polygons

Classifying Triangles• A triangle is a polygon with three

sides.

Acute triangle three acute sides

Right triangle one right angle

Obtuse triangle one obtuse angle

Equilateral triangle three congruent sides

Isosceles triangle at least two congruent sides

Scalene triangle no congruent sides

Classifying Quadrilaterals

Trapezoid exactly one pair of

parallel sides

Quadrilateral four

sides

Parallelogram both pairs of

opposite sides parallel

Rhombus four congruent

sides

Square four 90○ angles and

four congruent sides

Rectangle four 90○ angles

Classifying Quadrilaterals Cont.

• All parallelograms have opposite sides parallel.

• Parallelograms include rectangles, rhombuses, and squares.

• Quadrilaterals that have four right angles include the rectangles and squares.

Regular Polygons• A regular polygon has all sides

congruent and all angles congruent.• The formula for the perimeter of a

regular polygon is P = number of sides the length of the sides.

Triangle

Square

Pentagon

Hexagon

Quadrilaterals and Their Properties

Quadrilateral Quest: Do You Know Their Properties?

Lesson 9-5

Congruence

Congruent Triangles

• Congruent figures have the same size and shape, and their corresponding parts have equal measures.

• Triangles are congruent when all corresponding sides and interior angles are congruent.

• You use corresponding parts of triangles to identify congruent triangles.

Congruent Triangles

Side-Side-Side (SSS) If three sides of one triangle are congruent to three sides of a second

triangle, the two triangles are congruent.

Angle-Side-Angle (ASA) If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, the

triangles are congruent.

Side-Angle-Side (SAS) If two sides and the included angle are

congruent to two sides and the included angle of a second triangle, the two

triangles are congruent.

Congruent figures have the same size and shape, and their corresponding

parts have equal measures.

Lesson 9-6

Circles

Circle

Radius is a segment that has one endpoint at the center and the other point on the circle.

Circumference is the distance around the circle.

Diameter is a chord that passes through the center of a circle.

Chord is a segment whose endpoints are on the circle.

Circumference of a Circle

• The circumference of a circle is π times the diameter.

C = π d C = 2 π r

6 ft

C = π d Write the formula

C ≈ (3.14)6 Replace π with 3.14

and d with 6

= 18.84 Simplify

Making a Circle Graph• To make a circle graph, you find the measure of

each central angle.

• A central angle is an angle whose vertex is the center of a circle.

• There are 360○ in a circle.

• Use proportions to find the measures of the central angles.

20 = _r 25 = _r_

100 360 100 360

r = 72 ○

r = 90 ○

• Use a compass to draw a circle.

•Draw the central angles with a protractor.

•Label each section.

•Add a title and necessary information.

Lesson 9-7

Constructions

Construction Vocabulary

Perpendicular lines, segments, or rays intersect to form right angles.

A segment bisector is a line, segment, or ray that divides a segment into two congruent segments.

A perpendicular bisector is a line, segment, or ray that is perpendicular to the segment it bisects.

Construction Vocabulary Cont.

An angle bisector is a ray that divides an angle into two congruent angles.

Steps for Constructing a(n) . . .

Congruent SegmentPearson Prentice Hall Mathematics VideoCongruent AnglePearson Prentice Hall Mathematics VideoPerpendicular BisectorPearson Prentice Hall Mathematics VideoAngle BisectorPearson Prentice Hall Mathematics Video

Lesson 9-8

Translations

Translation Vocabulary• You perform a translation by sliding, flipping, or turning an

object.

• A transformation is a change of position or size of a figure.

• A translation is a transformation that moves points the same distance and in the same direction.

• The figure you get after a transformation is called the image.

• Use prime notation (A1) to name the image of a point.

Example of a SlideExample of a Flip

Example of a Turn

Examples of Translations

Pearson Prentice Hall Mathematics Video

Translating a Figure

Writing a Rule to Describe a Translation

Pearson Prentice Hall Mathematics Video

Symmetry and Reflections

Lesson 9-9

Symmetry• A figure has reflectional symmetry when

one half is a mirror image of the other half.• A line of symmetry divides a figure with

reflectional symmetry into two congruent halves.

Reflections• A reflection is a

transformation that flips a figure over a line of reflection.

• The reflected figure, or image, is congruent to the original figure.

• Together, an image and its reflection have line symmetry, the line of reflection being the line of symmetry.

Graphing Reflections of a Shape

Pearson Prentice Hall Mathematics Video

Rotations

Lesson 9-10

Rotations

• A rotation is a transformation that turns a figure about a fixed point called the center of rotation.

• The angle measure of the rotation is the angle of rotation.

Rotational Symmetry

• A figure has rotational symmetry if you can rotate it 360

○, or less, so

that its image matches the original figure.

• The angle (or its measure) through which the figure rotates is the angle of rotation.