Chapter 1 Sect.1 Inducve reasoning is reasoning that is based on paerns you observe. If you observe a paern in a sequence, you can use inducve reasoning to tell what the next terms in the sequence will be.

A conclusion you reach using inducve reasoning is called a conjecture.

Make a conjecture about the sum of the first 30 odd numbers. Find the first few sums.

Not all conjectures turn out to be true. You can prove that a conjecture is false by finding one counterexample. A counterexample to a conjecture is an example for which the conjecture is incorrect.

Notice that each sum is a perfect square. Using inductive reasoning, you can conclude that the sum of the first 30 odd numbers is 302, or 900.

Example 1 Identify the patterns, and determine the next to terms of the sequence.

A. 3, 6, 12, 24..._____, ______

B.

Example 2

Example 3Compare the points on the circle to the number of non- overlapping regions. State a conjecture for the pattern. Check to see if the conjecture is true.

Points Regions2345

Conjecture based on first four points:

Results:

A skateboard shop finds that over a period of five consecutive months, sales of small-wheeled skateboards decreased.

Use inductive reasoning. Make a conjecture about the number of small-wheeled skateboards the shop will sell in June.

Example 4

In geometry, some words such as point, line, and plane are undefined. In order to define these words you need to use words that need further defining. It is important however, to have general descriptions of their meanings.

Chapter 1 Sect.2

Think of a point as a location. A point has no size. It is represented by a small dot and is named by a capital letter. A geometric figure is a set of points. is defined as the set of all points

You can think of a line as a series of points that extends in two opposite directions without end. You can name a line by any two points on the line, such as line A B (read "line AB"). Another way to name a line is with a single lowercase letter, such as line t (see above). Points that lie on the same line are collinear points.

Are the following points collinear? If so, name the line on which they lie. :

E, F, and C

E, F, and D

A.

B.

Example 1

A plane is a flat surface that has no thickness. A plane contains many lines and extends without end in the directions of all its lines. You can name a plane by either a single capital letter or by at least three noncollinear points. Points and lines in the same plane are coplanar.

Each surface of the ice cube represents part of a plane. Name the plane represented by the:

front of the ice cube-

The top-

Example 2

A postulate or axiom is an accepted statement of fact.

What is the intersection of plane HGFE and plane BCGF?

Name two planes from that intersect in BF

Example 3

Shade the plane that contains points: A, B, and C E, H, and C

Example 4

Many geometric figures, such as squares and angles, are formed by parts of lines called segments or rays.

A segment is the part of a line consisting of two endpoints and all points between them.

A ray is the part of a line consisting of one endpoint and all the points of the line on one side of the endpoint.

Opposite rays are two collinear rays with the same endpoint. Opposite rays always form a line.

Name the segments and rays in the figure at the right.Example 1

Chapter 1 Sect. 3

Lines that do not intersect may or may not be coplanar.

Parallel lines are coplanar lines that do not intersect.

Skew lines are noncoplanar; therefore, they are not parallel and do not intersect.

Name all labeled segments that are:

parallel to DC-

skew to DC-

Example 2

Parallel planes are planes that do not intersect.

Name two pairs of parallel planes in the figure. Example 3

Chapter 1 Sect. 4

The distance between points C and D on the ruler is 3. You can use the Ruler Postulate to find the distance between points on a number line.

Two segments with the same length are congruent (=) segments. In other words, if AB = CD, then Segment AB = CD or CD = AB.

~~ ~

Find the lengths of AB and BC. Example 1

Examine the lengths of segment AB and segment BC in Example 1. Notice that

AB + BC = 6. Notice that AC = 6. This suggests the following postulate.

If DT = 60, find the value of x. Then find DS and ST.

Example 2

A midpoint of a segment is a point that divides a segment into two congruent segments. A midpoint, or any line, ray, or other segment through a midpoint, is said to bisect the segment.

C is the midpoint of segment AB. Example 3

Find:AC­ CB­ AB­

An angle ( ) is formed by two rays with the same endpoint. The rays are the sides of the angle. The endpoint is the vertex of the angle. The sides of the angle shown here are ray BT and ray BQ. The vertex is B. You could name this angle B, TBQ, QBT, or 1. Notice when naming angles using three letter, the vertex always is the middle letter.

1B

T

Q

Example 4Name 1 two other ways.

One way to measure an angle is in degrees. To indicate the size or degree measure of an angle, write a lowercase m in front of the angle symbol. The degree measure of angle A is 80. You show this by writing m A = 80.

You can classify angles according to their measures.

Find the measure of each angle. Classify each as acute, right, obtuse, or straight.

Example 5

The Angle Addition Postulate is similar to the Segment Addition Postulate.

Example 6What is m TSW if m RST = 50 and m RSW = 125?

Angles with the same measure are congruent angles. In other words, if m 1 = m 2, then 1 = 2.~

1 2

Chapter 1 Sect. 5 construction ­ When you use a straightedge and a compass to draw a geometric figure.

straightedge­ a ruler with no markings on it

compass­ a geometric tool used to draw circles and parts of circles called arcs.

congruent segments­ segments with the same length

perpendicular lines­ two lines that intersect to form right angles.The symbol means "is perpendicular to."

perpendicular bisector­ a line, segment, or ray that is perpendicular to the segment at its midpoint

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

Constructing Congruent Segments

Constructing Congruent AnglesGiven:

Given:

Constructing Perpendicular BisectorGiven:

Constructing Perpendicular BisectorGiven:

KN bisects JKL so that m JKN = 5x − 25 and m NKL = 3x + 5. Solve for x and find m JKN.

Example:

Chapter 1 Sect. 5

In a construction you use a straightedge and a compass to draw a geometric figure. A straightedge is a ruler with no markings on it. A compass is a geometric tool used to draw circles and parts of circles called arcs.

Four basic constructions involve constructing congruent segments, congruent angles, and bisectors of segments and angles.

Construct a segment congruent to a given segment. Construct: segment CD so that segment CD approximately equal to segment AB.

Step 1: Draw a ray with endpoint C.

Step 2: Open the compass to the length of segment AB.

Step 3: With the same compass setting, put the compass point on point C. Draw an arc that intersects the ray. Label the point of intersection D.

Example 1

Example 2Construct an angle congruent to a given angle.

Construct: angle S so that angle S approximately equal to angle A

Step 1: Draw a ray with endpoint S.

Step 2:With the compass point on point A, draw an arc that intersects the sides of angle A. Label the points of intersection B and C.

Step 3:With the same compass setting, put the compass point on point S. Draw an arc and label its point of intersection with the ray as R.

Step 4: Open the compass to the length BC. Keeping the same compass setting, put the compass point on R. Draw an arc to locate point T.

Step 5: Draw ray S T.

Perpendicular lines are two lines that intersect to form right angles. The symbol means "is perpendicular to."

A perpendicular bisector of a segment is a line, segment, or ray that is perpendicular to the segment at its midpoint, thereby bisecting the segment into two congruent segments.

Chapter 1 Sect. 6

Find the distance between T(5, 2) and R(−4, −1) to the nearest tenth.

Find the distance between A(1, -3) and R(−4, 4) to the nearest tenth.

Example 1

Example 2

To find the coordinate of the midpoint of a segment on a number line, find the average or mean of the coordinates of the endpoints.

Find the midpoint between A(3, 5) and R(7, -9) to the nearest tenth.

Find the midpoint between A(8,-5) and R(7, -9) to the nearest tenth.

The midpoint of segment AB is M(3, 4). One endpoint is A(−3, −2). Find the coordinates of the other endpoint B.

Example 3

Example 4

Example 5

Midpt (3, ­4) Endpt :(2, 3)

Example 1

Chapter 1 Sect. 7

The units of measurement for perimeter and circumference include inches, feet, yards, miles, centimeters, meters, and kilometers. When measuring area, use square units such as square inches (in.2), square centimeters (cm2), square meters (m2), and square miles (mi2).

Your pool is 15 feet wide and 20 feet long with a 3-foot wide deck surrounding it. You want to build a fence around the deck. How much fencing will you need?

Example 2Find the perimeter of the triangle.

Example 3Find the circumference of circle A in terms of π. Then find the circumference to the nearest tenth. (Treat π like a variable.)

Example 4You are designing a rectangular banner for the front of the museum. The banner will be 4 ft wide and 7 yd high. How much material do you need? (Always convert first!!!)

The following postulates are useful in finding areas of figures with irregular shapes.

Example 5Find the area of the figure at the right.