1. Lesson 1-1 Point, Line, Plane

2. Lesson 1-1 Point, Line, Plane
Points
Points have no dimension.
How to Sketch:
Using dots
How to label:
Use capital letters
Never name two points with the same letter
(in the same sketch). A B C A
Lesson 1-1 Point, Line, Plane

3. Lesson 1-1 Point, Line, Plane
Lines
Lines have 1 dimension.
How to sketch : using arrows at both ends.
How to name: 2 ways
(1) small cursive letter – line n
(2) any two points on the line -
Never name a line using three points - n A B C
Lesson 1-1 Point, Line, Plane

4. Lesson 1-1 Point, Line, Plane
Collinear Points
Collinear points are points that lie on the same line. A point lies on the line if the coordinates of the point satisfy the equation of the line. A B C
Collinear C A B
Non collinear
Lesson 1-1 Point, Line, Plane

5. Lesson 1-1 Point, Line, Plane
Planes
A plane is a flat surface that extends indefinitely in all directions.
How to sketch: Use a parallelogram (four sided figure)-with opposite sides parallel.
How to name: 2 ways
(1) Capital cursive script letter – Plane M
(2) Any 3 or more non collinear points in the plane - Plane: ABC/ ACB / BAC / BCA / CAB / CBA A M B C
Horizontal Plane
Vertical Plane
Other
Lesson 1-1 Point, Line, Plane

6. Different planes in a figure:B
Plane ABCD
Plane EFGH
Plane BCGF
Plane ADHE
Plane ABFE
Plane CDHG
Etc. D C E F H G
Lesson 1-1 Point, Line, Plane

7. Other planes in the same figure:
Any three non collinear points determine a plane!
Plane AFGD
Plane ACGE
Plane ACH
Plane AGF
Plane BDG
Etc.
Lesson 1-1 Point, Line, Plane

8. Lesson 1-1 Point, Line, Plane
Coplanar Objects
Coplanar objects (points, lines, etc.) are objects that lie on the same plane.
Are the following points coplanar?
A, B, C ?
Yes
A, B, C, F ? No
H, G, F, E ?
Yes
E, H, C, B ?
Yes
A, G, F ?
Yes
C, B, F, H ? No
Lesson 1-1 Point, Line, Plane

9. Lesson 1-1 Point, Line, Plane
Intersection of Lines
The intersection of two lines is a point. m
Line m and line n intersect at point P. P n
Continued…….
Lesson 1-1 Point, Line, Plane

10. 3 Possibilities of Intersection of a Line and a Plane
(1) Line passes through plane – intersection is a point.
(2) Line lies on the plane - intersection is a line.
(3) Line is parallel to the plane - no common points.
Lesson 1-1 Point, Line, Plane

11. Intersection of Two Planes is a Line.B P A R
Plane P and Plane R intersect at the line
Lesson 1-1 Point, Line, Plane

12. Lesson 1-2: Segments and Rays

13. Lesson 1-2: Segments and Rays
Postulates
Definition: An assumption that needs no explanation.
Examples:
Through any two points there is
exactly one line.
A line contains at least two points.
Through any three nonlinear points, there is
exactly one plane.
A plane contains at least three nonlinear points.
Lesson 1-2: Segments and Rays

14. Lesson 1-2: Segments and Rays
Postulates
Examples:
If two planes intersect,
then the intersecting is a line.
If two points lie in a plane,
then the line containing the two
points lie in the same plane.
Lesson 1-2: Segments and Rays

15. Lesson 1-2: Segments and Rays
Between
Definition: X is between A and B if AX + XB = AB.
AX + XB = AB
AX + XB > AB
Lesson 1-2: Segments and Rays

16. Lesson 1-2: Segments and Rays
Definition:
A line with an endpoint at each end.
How to sketch:
How to name:
AB (without a symbol) means the length of the segment or the distance between points A and B.
Lesson 1-2: Segments and Rays

17. The Segment Addition Postulate
If C is between A and B, then AC + CB = AB.
Example:
If AC = x , CB = 2x and AB = 12, then, find x, AC and CB. 2x x 12
Step 1: Draw a figure
Step 2: Label fig. with given info.
AC + CB = AB
x x = 12
3x = 12
x = 4
Step 3: Write an equation
x = 4
AC = 4
CB = 8
Step 4: Solve and find all the answers
Lesson 1-2: Segments and Rays

18. Lesson 1-2: Segments and Rays
Congruent Segments
Definition:
Segments with equal lengths. (congruent symbol: )
Congruent segments can be marked with dashes.
If numbers are equal the objects are congruent.
AB: the segment AB ( an object )
AB: the distance from A to B ( a number )
Correct notation:
Incorrect notation:
Lesson 1-2: Segments and Rays

19. Lesson 1-2: Segments and Rays
Midpoint
Definition:
A point that divides a segment into
two congruent segments
Formulas:
On a number line, the coordinate of the midpoint of a segment whose endpoints have coordinates a and b is
In a coordinate plane, the coordinates of the midpoint of a segment whose endpoints have coordinates and is
Lesson 1-2: Segments and Rays

20. Midpoint on Number Line - Example
Find the coordinate of the midpoint of the segment PK.
Now find the midpoint on the number line.
Lesson 1-2: Segments and Rays

21. Lesson 1-2: Segments and Rays
Segment Bisector
Definition:
Any segment, line or plane that divides a segment into two congruent parts is called segment bisector.
Lesson 1-2: Segments and Rays

22. Lesson 1-2: Segments and Rays
Definition:
RA : A line with an endpoint on one end.
How to sketch:
How to name:
( the symbol RA is read as “ray RA” )-endpoint must come first.
Lesson 1-2: Segments and Rays

23. Lesson 1-2: Segments and Rays
Opposite Rays
Definition:
If A is between X and Y, AX and AY are opposite rays.
( Opposite rays must have the same “endpoint” and make a straight line )
opposite rays
not opposite rays
Lesson 1-2: Segments and Rays

24. Lesson 1-4
Angles
Lesson 1-4: Angles

25. Angle and Points ray vertex ray
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Angle and Points
An Angle is a figure formed by two rays with a common endpoint, called the vertex.
ray
vertex
ray
Angles can have points in the interior, in the exterior or on the angle. A E D B C
Points A, B and C are on the angle. D is in the interior and E is in the exterior.
B is the vertex.
Lesson 1-4: Angles

26. 4/28/2017
Naming an angle: (1) Using 3 points (2) Using 1 point (3) Using a number – next slide
Using 3 points:
vertex must be the middle letter
This angle can be named as
Using 1 point:
using only vertex letter
* Use this method is permitted when the vertex point is the vertex of one and only one angle.
Since B is the vertex of only this angle, this can also be called A C B
Lesson 1-4: Angles

27. Naming an Angle - continued
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Naming an Angle - continued
Using a number:
A number (without a degree symbol) may be used as the label or name of the angle. This number is placed in the interior of the angle near its vertex. The angle to the left can be named as A B 2 C
* The “1 letter” name is unacceptable when …
more than one angle has the same vertex point. In this case, use the three letter name or a number if it is present.
Lesson 1-4: Angles

28. Example Therefore, there is NO in this diagram.
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Example
K is the vertex of more than one angle.
Therefore, there is NO in this diagram.
There is
Lesson 1-4: Angles

29. 4 Types of Angles Acute Angle: Right Angle: Obtuse Angle:
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4 Types of Angles
Acute Angle:
an angle whose measure is less than 90.
Right Angle:
an angle whose measure is exactly 90 .
Obtuse Angle:
an angle whose measure is between
90 and 180.
Straight Angle:
an angle that is exactly 180 .
Lesson 1-4: Angles

30. 4/28/2017
Measuring Angles
Just as we can measure segments, we can also measure angles.
We use units called degrees to measure angles.
A circle measures _____
A (semi) half-circle measures _____
A quarter-circle measures _____
One degree is the angle measure of 1/360th of a circle.
360º ?
180º ? ?
90º
Lesson 1-4: Angles

31. 4/28/2017
Adding Angles
When you want to add angles, use the notation m1, meaning the measure of 1.
If you add m1 + m2, what is your result?
m1 + m2 = 58.
m1 + m2 = mADC also.
Therefore, mADC = 58.
Lesson 1-4: Angles

32. Angle Addition Postulate
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Angle Addition Postulate
Postulate:
The sum of the two smaller angles will always equal the measure of the larger angle.
Complete:
m  ____ + m  ____ = m  _____
MRK
KRW
MRW
Lesson 1-4: Angles

33. Example: Angle Addition
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Example: Angle Addition
K is interior to MRW, m  MRK = (3x), m KRW = (x + 6) and mMRW = 90º. Find mMRK.
First, draw it!
3x + x + 6 = 90
4x + 6 = 90
– 6 = –6
4x = 84
x = 21 3x
x+6
Are we done?
mMRK = 3x = 3•21 = 63º
Lesson 1-4: Angles

34. Congruent Angles Definition:
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Congruent Angles
Definition:
If two angles have the same measure, then they are congruent.
Congruent angles are marked with the same number of “arcs”.
The symbol for congruence is  3 5
Example:
3   5.
Lesson 1-4: Angles

35. 4/28/2017
Angle Bisector
An angle bisector is a ray in the interior of an angle that splits the angle into two congruent angles.
Example:
Since 4   6, is an angle bisector. 5 3
Lesson 1-4: Angles

36. Example Draw your own diagram and answer this question:
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Example
Draw your own diagram and answer this question:
If is the angle bisector of PMY and mPML = 87, then find:
mPMY = _______
mLMY = _______
Lesson 1-4: Angles

37. Lesson 1-4: Pairs of Angles

38. Lesson 1-4: Pairs of Angles
Adjacent Angles
Definition:
A pair of angles with a shared vertex and common side but do not have overlapping interiors.
Examples:
1 and 2 are adjacent. 3 and 4 are not.
1 and ADC are not adjacent. 4 3
Adjacent Angles( a common side )
Non-Adjacent Angles
Lesson 1-4: Pairs of Angles

39. Lesson 1-4: Pairs of Angles
Complementary Angles
Definition:
A pair of angles whose sum is 90˚
Examples:
Adjacent Angles
( a common side )
Non-Adjacent Angles
Lesson 1-4: Pairs of Angles

40. Lesson 1-4: Pairs of Angles
Supplementary Angles
Definition:
A pair of angles whose sum is 180˚
Examples:
Adjacent supplementary angles are also called “Linear Pair.”
Non-Adjacent Angles
Lesson 1-4: Pairs of Angles

41. Lesson 1-4: Pairs of Angles
Vertical Angles
Definition:
A pair of angles whose sides form opposite rays.
Examples:
Vertical angles are non-adjacent angles formed by intersecting lines.
Lesson 1-4: Pairs of Angles

42. Example: If m4 = 67º, find the measures of all other angles.
Step 1: Mark the figure with given info.
Step 2: Write an equation.
67º
Lesson 1-4: Pairs of Angles

43. Lesson 1-4: Pairs of Angles
Example: If m1 = 23 º and m2 = 32 º, find the measures of all other angles.
Answers:
Lesson 1-4: Pairs of Angles

44. Lesson 1-4: Pairs of Angles
Example: If m1 = 44º, m7 = 65º find the measures of all other angles.
Answers:
Lesson 1-4: Pairs of Angles

45. Lesson 1-4: Pairs of Angles
Algebra and Geometry
Common Algebraic Equations used in Geometry:
( ) = ( )
( ) + ( ) = ( )
( ) + ( ) = 90˚
( ) + ( ) = 180˚
If the problem you’re working on has a variable (x),
then consider using one of these equations.
Lesson 1-4: Pairs of Angles