1. Ch. 1 Essentials of Geometry
Review

2. Lesson 1-1 Point, Line, Plane
Coplanar Objects
**Remember: Any 3 non-collinear points determine a plane!
Coplanar objects (points, lines, etc.) are objects that lie on the same plane. The plane does not have to be visible.
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

3. Front Side – True or False
Naming points, lines, rays, and planes
Finding intersections
Lesson 1-1 Point, Line, Plane

4. Lesson 1-1 Point, Line, Plane
Example 3
Point S is between point R and point T. Use the given information to write an equation in terms of x. Solve the equation. Then find both RS and ST.
RS = 3x – 16
ST = 4x – 8
RT = 60
I I
3x-16
I x I
Lesson 1-1 Point, Line, Plane

5. Lesson 1-1 Point, Line, Plane
EXAMPLE 2
Use algebra with segment lengths
Point M is the midpoint of VW . Find the length of VM .
ALGEBRA
Lesson 1-1 Point, Line, Plane 5

6. Lesson 1-1 Point, Line, Plane
GUIDED PRACTICE
line l
Identify the segment bisector of
Then find PQ.
Lesson 1-1 Point, Line, Plane

7. Lesson 1-1 Point, Line, Plane
MIDPOINT FORMULA
The midpoint of two points P(x1, y1) and Q(x2, y2) is
M(X,Y) = M(x1 + x2, x2 +y2)
Think of it as taking the average of the x’s and the average of the y’s to make a new point.
Lesson 1-1 Point, Line, Plane 7

8. Lesson 1-1 Point, Line, Plane
EXAMPLE 3
Use the Midpoint Formula
a. FIND MIDPOINT The endpoints of RS are R(1,–3) and S(4, 2). Find the coordinates of the midpoint M.
Lesson 1-1 Point, Line, Plane 8

9. Lesson 1-1 Point, Line, Plane
EXAMPLE 3
Use the Midpoint Formula
SOLUTION
a. FIND MIDPOINT Use the Midpoint Formula. 2 5
1 + 4
– 3 + 2 = , M – 1
The coordinates of the midpoint M are 1 , – 5 2
ANSWER
Lesson 1-1 Point, Line, Plane 9

10. Lesson 1-1 Point, Line, Plane
EXAMPLE 3
Use the Midpoint Formula
b. FIND ENDPOINT The midpoint of JK is M(2, 1). One endpoint is J(1, 4). Find the coordinates of endpoint K.
FIND ENDPOINT Let (x, y) be the coordinates of endpoint K. Use the Midpoint Formula.
STEP 1
Find x.
STEP 2
Find y.
1+ x 2 =
4+ y 1 2 =
1 + x = 4
4 + y = 2
x = 3
y = – 2
The coordinates of endpoint K are (3, – 2).
ANSWER
Lesson 1-1 Point, Line, Plane 10

11. Lesson 1-1 Point, Line, Plane
Distance Formula
The distance between two points A and B is
Lesson 1-1 Point, Line, Plane 11

12. Lesson 1-1 Point, Line, Plane
EXAMPLE 4
Standardized Test Practice
SOLUTION
Use the Distance Formula. You may find it helpful to draw a diagram.
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13. Lesson 1-1 Point, Line, Plane
Naming Angles
Name the three angles in diagram.
Name this one angle in 3 different ways.
WXY, WXZ, and YXZ
The vertex of the angle
What always goes in the middle?
Lesson 1-1 Point, Line, Plane

14. Lesson 1-1 Point, Line, Plane
EXAMPLE 2
Find angle measures o
ALGEBRA Given that m LKN =145 , find m LKM and m MKN.
SOLUTION
STEP 1
Write and solve an equation to find the value of x.
m LKN = m LKM + m MKN
Angle Addition Postulate
145 = (2x + 10) + (4x – 3) o
Substitute angle measures.
145 = 6x + 7
Combine like terms.
138 = 6x
Subtract 7 from each side.
23 = x
Divide each side by 6.
Lesson 1-1 Point, Line, Plane

15. Lesson 1-1 Point, Line, Plane
EXAMPLE 2
Find angle measures
STEP 2
Evaluate the given expressions when x = 23.
m LKM = (2x + 10)° = ( )° = 56°
m MKN = (4x – 3)° = (4 23 – 3)° = 89°
So, m LKM = 56° and m MKN = 89°.
ANSWER
Lesson 1-1 Point, Line, Plane

16. Lesson 1-1 Point, Line, Plane
GUIDED PRACTICE
Find the indicated angle measures.
Given that KLM is straight angle, find m KLN and m NLM.
SOLUTION
STEP 1
Write and solve an equation to find the value of x.
m KLM + m NLM
= 180°
Straight angle
(10x – 5)° + (4x +3)°
= 180°
Substitute angle measures.
14x – 2
= 180
Combine like terms.
14x
= 182
Subtract 2 from each side. x
= 13
Divide each side by 14.
Lesson 1-1 Point, Line, Plane

17. Lesson 1-1 Point, Line, Plane
GUIDED PRACTICE
STEP 2
Evaluate the given expressions when x = 13.
m KLM = (10x – 5)° = ( – 5)° = 125°
m NLM = (4x + 3)° = ( )° = 55°
ANSWER
m KLM
= 125°
m NLM
= 55°
Lesson 1-1 Point, Line, Plane

18. Lesson 1-1 Point, Line, Plane
EXAMPLE 3
Double an angle measure
In the diagram at the right, YW bisects XYZ, and m XYW = 18. Find m XYZ. o
SOLUTION
By the Angle Addition Postulate, m XYZ = m XYW + m WYZ. Because YW bisects XYZ you know that XYW WYZ. ~
So, m XYW = m WYZ, and you can write
M XYZ = m XYW + m WYZ = 18° + 18° = 36°.
Lesson 1-1 Point, Line, Plane

19. Lesson 1-1 Point, Line, Plane
Example 4
Lesson 1-1 Point, Line, Plane

20. Lesson 1-1 Point, Line, Plane
EXAMPLE 2
Find measures of a complement and a supplement
a. Given that 1 is a complement of and m 1 = 68°,
find m 2.
SOLUTION
a. You can draw a diagram with complementary adjacent angles to illustrate the relationship.
m = 90° – m 1 = 90° – 68° = 22
Lesson 1-1 Point, Line, Plane

21. Lesson 1-1 Point, Line, Plane
EXAMPLE 3
Find angle measures
Sports
When viewed from the side, the frame of a ball-return net forms a pair of supplementary angles with the ground. Find m BCE and m ECD.
Remember in geometry we take are knowledge of the diagram and properties of the angle pairs to set up an equation.
Lesson 1-1 Point, Line, Plane

22. Lesson 1-1 Point, Line, Plane
EXAMPLE 3
Find angle measures
SOLUTION
STEP 1
Use the fact that the sum of the measures of supplementary angles is 180°.
m BCE + m ∠ ECD = 180°
Write equation.
(4x+ 8)° + (x + 2)° = 180°
Substitute.
5x + 10 = 180
Combine like terms.
5x = 170
Subtract 10 from each side.
x = 34
Divide each side by 5.
Lesson 1-1 Point, Line, Plane

23. Lesson 1-1 Point, Line, Plane
EXAMPLE 3
Find angle measures
STEP 2
Evaluate: the original expressions when x = 34.
m BCE = (4x + 8)° = ( )° = 144°
m ECD = (x + 2)° = ( )° = 36°
The angle measures are 144° and 36°.
ANSWER
Lesson 1-1 Point, Line, Plane

24. Angles Formed by the Intersection of 2 Lines
Click on picture!!
What angles always are equal, which ones add to equal 180 degree?
What are properties of linear pairs (adjacent angles) and vertical angles? How might this be useful? Time saving?
When would the vertical and adjacent angles all be equal?
 Click Me!
Lesson 1-1 Point, Line, Plane

25. Lesson 1-1 Point, Line, Plane
EXAMPLE 4
Identify angle pairs
Identify all of the linear pairs and all of the vertical angles in the figure at the right.
SOLUTION
To find linear pairs, look for adjacent angles whose noncommon sides are opposite rays.
1 and 4 are a linear pair and are also a linear pair.
ANSWER
To find vertical angles, look or angles formed by intersecting lines.
1 and are vertical angles.
ANSWER
Lesson 1-1 Point, Line, Plane

26. Lesson 1-1 Point, Line, Plane
Example 5
Two angles form a linear pair. The measure of one angle is 5 times the measure of the other. Find the measure of each angle.
Lesson 1-1 Point, Line, Plane

27. Lesson 1-1 Point, Line, Plane
Example 6
Given that m5 = 60 and m3 = 62, use your knowledge of linear pairs and vertical angles to find the missing angles.
Lesson 1-1 Point, Line, Plane

28. Lesson 1-1 Point, Line, Plane
EXAMPLE 1
Identify polygons
Tell whether the figure is a polygon and whether it is convex or concave. a. b. c. d.
SOLUTION
Some segments intersect more than two segments, so it is not a polygon. a. b.
The figure is a convex polygon.
Part of the figure is not a segment, so it is not a polygon. c. d.
The figure is a concave polygon.
Lesson 1-1 Point, Line, Plane

29. Lesson 1-1 Point, Line, Plane
# of sides
Type of Polygon
triangle
quadrilateral
pentagon
hexagon
heptagon
octagon
nonagon
decagon
dodecagon
n-gon 3 4 5 6 7 8 9 10 12 n
What is a polygon with 199 sides called?
199-gon
Lesson 1-1 Point, Line, Plane

30. Lesson 1-1 Point, Line, Plane
EXAMPLE 2
Classify polygons
Classify the polygon by the number of sides. Tell whether the polygon is equilateral, equiangular, or regular. Explain your reasoning. a. b.
SOLUTION
The polygon has 6 sides. It is equilateral and equiangular, so it is a regular hexagon. a.
The polygon has 4 sides, so it is a quadrilateral. It is not equilateral or equiangular, so it is not regular. b.
Lesson 1-1 Point, Line, Plane

31. Lesson 1-1 Point, Line, Plane
EXAMPLE 3
Find side lengths
A table is shaped like a regular hexagon.The expressions shown represent side lengths of the hexagonal table. Find the length of a side.
ALGEBRA
SOLUTION
First, write and solve an equation to find the value of x. Use the fact that the sides of a regular hexagon are congruent.
3x + 6
4x – 2 =
Write equation. 6 =
x – 2
Subtract 3x from each side. 8 = x
Add 2 to each side.
Lesson 1-1 Point, Line, Plane

32. Lesson 1-1 Point, Line, Plane
EXAMPLE 3
Find side lengths
Then find a side length. Evaluate
one of the expressions when x = 8. 30
3(8) + 6 =
3x + 6
The length of a side of the table is 30 inches.
ANSWER
Lesson 1-1 Point, Line, Plane

33. Lesson 1-1 Point, Line, Plane
Perimeter/Area
Rectangle
Square
Triangle
Circle
Lesson 1-1 Point, Line, Plane

34. Lesson 1-1 Point, Line, Plane
Area
The area of the triangle is 14 square inches and its height is 7 inches. Find the base of the triangle.
Lesson 1-1 Point, Line, Plane

35. Lesson 1-1 Point, Line, Plane
Perimeter
The perimeter of a rectangle 84.6 centimeters. The length of the rectangle is twice as long as its width. Find the length and width of the rectangle.
Lesson 1-1 Point, Line, Plane