1. JRLeon Discovering Geometry Chapter 1.1-1.2 HGSH
1.1 Building Blocks of Geometry
Three building blocks of geometry are points, lines, and planes.
POINT
A point is the most basic building block of geometry.
It has no size (no dimension).
It has only location (a place).
You represent a point with a dot, and you name it with a capital letter. The point shown below is called P.
It is important to understand that a point is not a thing, but a location (place).
A tiny seed is a physical model of a point.
A point, however, is smaller than any seed that ever existed.
. P
JRLeon Discovering Geometry Chapter HGSH

2. JRLeon Discovering Geometry Chapter 1.1-1.2 HGSH
1.1 Building Blocks of Geometry
Three building blocks of geometry are points, lines, and planes.
LINE
A line is a straight, continuous arrangement of infinitely many points.
It has infinite length, but no thickness. It has zero width.
It extends forever in two directions. A line is one-dimensional.
You name a line by giving the letter names of any two points on the line and by placing the line symbol above the letters, for example, AB or BA.
A piece of spaghetti is a physical model of a line. A line, however, is longer,
straighter, and thinner than any piece of spaghetti ever made. . A B
If you draw a line with a pencil, examination with a microscope would show that the pencil mark has a measurable width. The pencil line is just a way to illustrate the idea on paper. In geometry however, a line has no width.
A straight line is the shortest distance between any two points
JRLeon Discovering Geometry Chapter HGSH

3. JRLeon Discovering Geometry Chapter 1.1-1.2 HGSH
1.1 Building Blocks of Geometry
Three building blocks of geometry are points, lines, and planes.
PLANE
A plane has
length and width,
but no thickness.
It is like a flat surface that extends infinitely along its length and width. You represent a plane with a four-sided figure, like a tilted piece of paper, drawn in perspective. Of course, this actually illustrates only part of a plane. You name a plane with a script capital letter, such as P. P
If you draw a line with a pencil, examination with a microscope would show that the pencil mark has a measurable width. The pencil line is just a way to illustrate the idea on paper. In geometry however, a line has no width.
A straight line is the shortest distance between any two points
A flat piece of rolled out dough is a physical model of a plane. A plane, however, is broader, wider, and thinner than any piece of dough you could ever roll.
JRLeon Discovering Geometry Chapter HGSH

4. JRLeon Discovering Geometry Chapter 1.1-1.2 HGSH
1.1 Building Blocks of Geometry
A definition is a statement that clarifies or explains the meaning of a word or a phrase.
However, it is impossible to define point, line, and plane without using words or phrases that themselves need definition. So these terms remain undefined. Yet, they are the basis for all of geometry.
Using the undefined terms point, line, and plane, you can define all other geometry terms and geometric figures.
Here are thefirst definitions. Begin a list and draw sketches for all definitions.
Collinear means on the same line Coplanar means on the same plane. . A B C
DEFINITIONS
Ask students to study and discuss the definitions given in the student book. Demonstrate how students should organize the definition list in their notebooks, including sketches, and help them get started with these new terms. You might use the Example of a Definition List transparency.
Points A, B, and C are collinear. P . E F D
Points D, E, and F are coplanar.
JRLeon Discovering Geometry Chapter HGSH

5. JRLeon Discovering Geometry Chapter 1.1-1.2 HGSH
1.1 Building Blocks of Geometry
A line segment consists of two points called the endpoints of the segment and all the points between them that are collinear with the two points. A B
Endpoints
Line segment
You can write line segment AB, using a segment symbol, as AB or BA. There are two ways to write the length of a segment.
Write AB  2 in., meaning the distance from A to B is 2 inches.
You can also use an m for “measure” in front of the segment name, and write the distance as mAB  2 in.
JRLeon Discovering Geometry Chapter HGSH

6. JRLeon Discovering Geometry Chapter 1.1-1.2 HGSH
1.1 Building Blocks of Geometry
If no measurement units are used for the length of a segment, it is understood that the choice of units is not important, or, is based on the length of the smallest square in the grid.
JRLeon Discovering Geometry Chapter HGSH

7. JRLeon Discovering Geometry Chapter 1.1-1.2 HGSH
1.1 Building Blocks of Geometry
Two segments are congruent if and only if they have equal measures, or lengths.
JRLeon Discovering Geometry Chapter HGSH

8. JRLeon Discovering Geometry Chapter 1.1-1.2 HGSH
1.1 Building Blocks of Geometry
When drawing figures, you show congruent segments by making identical markings.
JRLeon Discovering Geometry Chapter HGSH

9. JRLeon Discovering Geometry Chapter 1.1-1.2 HGSH
1.1 Building Blocks of Geometry
The midpoint of a segment is the point on the segment that is the same distance from both endpoints.
The midpoint bisects the segment, or divides the segment into two congruent segments.
Example:
Study the diagrams below.
a. Name each midpoint and the segment it bisects.
b. Name all the congruent segments. Use the congruence symbol to write your answers.
a. CF  FD, so F is the midpoint of CD; JK  KL, so K is the midpoint of JL.
b. CF  FD, HJ  HL, and JK  KL.
Even though EF and FG appear to have the same length, you cannot assume they are congruent without the markings.
The same is true for MN and NP.
JRLeon Discovering Geometry Chapter HGSH

10. JRLeon Discovering Geometry Chapter 1.1-1.2 HGSH
1.1 Building Blocks of Geometry
Ray AB is the part of AB that contains point A and all the points on AB that are on the
same side of point A as point B. Imagine cutting off all the points to the left of point A.
A Y B
A Y B
You can write ray AB, using the ray symbol, as AB
In the figure above, AY and AB are two ways to name the same ray. Note that AB IS NOT THE SAME AS BA!
A ray begins at a point and extends infinitely in one direction. You need two letters to name a ray. The first letter is the endpoint of the ray, and the second letter is any other point that the ray passes through.
A B AB
A B BA
JRLeon Discovering Geometry Chapter HGSH

11. JRLeon Discovering Geometry Chapter 1.1-1.2 HGSH
Midpoint : Using Your Algebra Skills
A midpoint is the point on a line segment that is the same distance from both
endpoints.
You can think of a midpoint as being halfway between two locations.
You know how to mark a midpoint. But when the position and location matter, such as in navigation and geography, you can use a coordinate grid and some algebra to find the exact location of the midpoint. You can calculate the coordinates of the
midpoint of a segment on a coordinate grid using a formula.
JRLeon Discovering Geometry Chapter HGSH

12. JRLeon Discovering Geometry Chapter 1.1-1.2 HGSH
Midpoint : Using Your Algebra Skills
JRLeon Discovering Geometry Chapter HGSH

13. JRLeon Discovering Geometry Chapter 1.1-1.2 HGSH
VOCABULARY:
A POINT is the most basic building block of geometry. It has no size. It has only location. You represent a point with a dot, and you name it with a capital letter.
A LINE is a straight, continuous arrangement of infinitely many points.
A PLANE has length and width, but no thickness. It is like a flat surface that extends infinitely along its length and width.
A definition is a statement that clarifies or explains the meaning of a word or a phrase.
COLINEAR means on the same line.
COPLANAR means on the same plane.
A LINE SEGMENT consists of two points called the endpoints of the segment and all the points between them that are collinear with the two points.
Two segments are CONGRUENT if and only if they have equal measures, or lengths.
The MIDPOINT of a segment is the point on the segment that is the same distance from both endpoints.
The midpoint BISECTS the segment, or divides the segment into two congruent segments.
JRLeon Discovering Geometry Chapter HGSH

14. JRLeon Discovering Geometry Chapter 1.1-1.2 HGSH
1.2 – Poolroom Math
People use angles every day.
Plumbers measure the angle between connecting pipes to make a good fitting.
Woodworkers adjust their saw blades to cut wood at just the correct angle.
Air traffic controllers use angles to direct planes.
Good pool players must know their angles to plan their shots.
JRLeon Discovering Geometry Chapter HGSH

15. JRLeon Discovering Geometry Chapter 1.1-1.2 HGSH
1.2 – Poolroom Math
ANGLE:
We can use the terms that defined in Lesson 1.1 to write a precise definition of angle. An angle is formed by two rays that share a common endpoint, provided that the two rays are noncollinear. In other words, the rays cannot lie on the same line. The common endpoint of the two rays is the vertex of the angle. The two rays are the sides of the angle.
You can name the angle in the figure below angle TAP or angle PAT, or use the angle symbol and write TAP or PAT. Notice that the vertex must be the middle letter, and the first and last letters each name a point on a different ray. Since there are no other angles with vertex A, you can also simply call this A.
JRLeon Discovering Geometry Chapter HGSH

16. JRLeon Discovering Geometry Chapter 1.1-1.2 HGSH
1.2 – Poolroom Math
EXAMPLE A: Name all the angles in these drawings.
Which angles in Example A seem big to you? Which seem small?
The measure of an angle is the smallest amount of rotation about the vertex from one ray to the other, measured in degrees.
According to this definition, the measure of an angle can be any value between 0° and 180°. The largest amount of rotation less than 360° between the two rays is called the reflex measure of an angle.
JRLeon Discovering Geometry Chapter HGSH

17. JRLeon Discovering Geometry Chapter 1.1-1.2 HGSH
1.2 – Poolroom Math
The geometry tool you use to measure an angle is a protractor.
How to use: Animation
JRLeon Discovering Geometry Chapter HGSH

18. 1.2 – Poolroom Math

19. 1.2 – Poolroom Math

20. JRLeon Discovering Geometry Chapter 1.1-1.2 HGSH
1.2 – Poolroom Math
JRLeon Discovering Geometry Chapter HGSH

21. JRLeon Discovering Geometry Chapter 1.1-1.2 HGSH
1.2 – Poolroom Math
JRLeon Discovering Geometry Chapter HGSH

22. JRLeon Discovering Geometry Chapter 1.1-1.2 HGSH
1.2 – Poolroom Math
Pocket billiards, or pool, is a game of angles. When a ball bounces off the pool table’s cushion, its path forms two angles with the edge of the cushion.
The incoming angle is formed by the cushion and the path of the ball approaching the cushion.
The outgoing angle is formed by the cushion and the path of the ball leaving the cushion. As it turns out, the measure of the outgoing angle equals the measure of the incoming angle.
JRLeon Discovering Geometry Chapter HGSH

23. JRLeon Discovering Geometry Chapter 1.1-1.2 HGSH
1.2 – Poolroom Math
JRLeon Discovering Geometry Chapter HGSH

24. JRLeon Discovering Geometry Chapter 1.1-1.2 HGSH
Homework (Home Learning)
1.1: P # 2-14, 18, 22, 24
P37 # 1-4
1.2: P42 # 1, 2, 5, 7-14, 16-20, 22-24, 32, 34, 36, 39
JRLeon Discovering Geometry Chapter HGSH