1. Unit 1: Angle Pair Relationships
Geometry
Unit 1: Angle Pair Relationships

2. Notebook Check Sheet Notes 9-9 Angle Relationships Assignment 9-10
Due 9-11
Angle Pair Relationships WKST

3. ≅ Definitions You read this symbol by saying “is congruent to”.
Side AB is congruent to side AC
𝑨𝑩 ≅ 𝑨𝑪
3.1 cm
3.1 cm B C

4. Definitions Angle: Formed when two rays share a common endpoint. A B C
You can name this angle ABC or CBA. You can use the angle symbol ∠ to say ∠ 𝑨𝑩𝑪 or ∠𝐂𝐁𝐀
If there is only one angle with vertex B, you can name this angle ∠𝑩

5. Naming Angles
If a vertex has more than one angle, you must use the endpoints and vertex to name the angle.
What is the name of the red angle? A
∠𝑨𝑩𝑫 or ∠𝑫𝑩𝑨 D
What is the name of the green angle?
∠𝑨𝑩𝑪 or ∠𝑪𝑩𝑨 B C
What is the name of the purple angle?
∠𝑫𝑩𝑪 or ∠𝑪𝑩𝑫

6. Definitions
Vertex: Common endpoint of the two rays that make an angle. A
Point B is the vertex of ∠𝑨𝑩𝑪 B C
Sides: The two rays that make up the angle. A
side
𝑩𝑨 and 𝑩𝑪 are sometimes called sides. B C
side

7. Definitions
Measure of an angle: Smallest amount of rotation about the vertex from one ray to the other. Use a curved line to indicate the angle. A
𝟐𝟒° B C
Degrees: The unit of measure for angles. Indicates the amount of rotation.
∠𝑨𝑩𝑪=𝟐𝟒°

8. Definitions
Congruent Angles: Angles that have the same degree measure. Used a curved line above an equal sign to indicate congruence. A X ≅ B Y C Z
∠𝑨𝑩𝑪≅∠𝑿𝒀𝒁
Said: “Angle ABC is congruent to angle XYZ.”

9. Practice Name all the angles in this figure. P ∠𝑷𝑿𝑮 ∠𝑮𝑿𝑴 X ∠𝑷𝑿𝑴 ∠𝑿𝑴𝑮
∠𝑿𝑮𝑨
∠𝑿𝑮𝑴
∠𝑴𝑮𝑨 M G A

10. Definitions
Adjacent Angles: Angles that have a common ray or side and a common vertex, but points inside either one of the angles are not inside the other. A B C Z
∠𝑨𝑩𝑪
is adjacent to
∠𝑪𝑩𝒁

11. Angle Addition PostulateR
Angle Addition Postulate: ∠𝑹𝑪𝑴+
∠𝑴𝑪𝑷
= ∠𝑹𝑪𝑷. C M P

12. Definitions
Angle Bisector: Contains the vertex and divides an angle into two equal halves. Splits the angle in half. X B Y Z
𝒀𝑩 bisects ∠𝑿𝒀𝒁

13. Example Angle Bisector X B Y Z 𝒀𝑩 bisects ∠𝑿𝒀𝒁 ∠𝑿𝒀𝒁=𝟗𝟎°
What is the degree measure of ∠𝑿𝒀𝑩 ?
=𝟒𝟓°
∠𝑩𝒀𝒁 ?
=𝟒𝟓°

14. Types of Angles Right Angle: Angle that measures exactly 𝟗𝟎°. A B C
Acute Angle: Angle that measures less than 𝟗𝟎°. A 50 B C

15. Types of Angles
Obtuse: Angle that measures more than 𝟗𝟎° A
𝟏𝟐𝟎° B C

16. Types of Angles
Straight: The angle measure for a straight line is 𝟏𝟖𝟎°
𝟏𝟖𝟎° A B C

17. Types of Angles
Vertical Angles: congruent angles formed by intersecting lines or line segments.

18. Types of Angles What are the vertical angles in this figure? ∠𝑨𝑬𝑩≅∠𝑪𝑬𝑫B A
∠𝑨𝑬𝑩≅∠𝑪𝑬𝑫 E
∠𝑨𝑬𝑪≅∠𝑩𝑬𝑫 C D

19. Example Find the value of the variable. 𝟐𝟓𝒙−𝟖=𝟗𝟐 B +𝟖 +𝟖 A 𝟐𝟓𝒙=𝟏𝟎𝟎 𝟐𝟓
𝟗𝟐° E
𝒙=𝟒 C D

20. Example Find the value of the variable. 𝟑𝟓𝒙−𝟐𝟓=𝟑𝟐𝒙−𝟏𝟎 −𝟑𝟐𝒙 −𝟑𝟐𝒙 B
𝟑𝒙−𝟐𝟓=−𝟏𝟎 A
+𝟐𝟓
+𝟐𝟓
𝟑𝒙=𝟏𝟓 𝟑 𝟑
𝟑𝟓𝒙−𝟐𝟓 E
𝟑𝟐𝒙−𝟏𝟎
𝒙=𝟓 C D

21. Example Find the value of the variable. B A 𝟑𝒙+𝟔 +(𝟏𝟎𝒙−𝟖)=𝟏𝟖𝟎
𝟏𝟑𝒙−𝟐=𝟏𝟖𝟎
(𝟑𝒙+𝟔)° +𝟐 +𝟐
𝟏𝟑𝒙=𝟏𝟖𝟐 𝟏𝟑 𝟏𝟑
𝒙=𝟏𝟒
(𝟏𝟎𝒙−𝟖)° E D C

22. Example Find the value of the variables. 𝟐𝟎𝒙+𝟏𝟐 +(𝟏𝟓𝒙−𝟕)=𝟏𝟖𝟎 A
𝟑𝟓𝒙+𝟓=𝟏𝟖𝟎 −𝟓 −𝟓 D
𝟑𝟓𝒙=𝟏𝟕𝟓
(𝟐𝟎𝒙+𝟏𝟐)°
𝟐𝟎 𝟓 +𝟏𝟐° 𝟑𝟓 𝟑𝟓 𝒚
𝒙=𝟓 E
(𝟏𝟓𝒙−𝟕)°
𝟐𝟎 𝟓 +𝟏𝟐° B
𝟏𝟏𝟐°
𝟏𝟖𝟎°−𝟏𝟏𝟐°=𝒚 C
𝒚=𝟔𝟖

23. Types of Angles A D B E S P Q T R U
Complementary Angles: two or more angles that add up to 𝟗𝟎°.
∠𝑨𝑩𝑫 and ∠𝑫𝑩𝑬 are adjacent complementary angles A D
∠𝑨𝑩𝑫 +∠𝑫𝑩𝑬=𝟗𝟎° B E S
∠𝑷𝑸𝑹 𝒂𝒏𝒅 ∠𝑺𝑻𝑼 P
are nonadjacent complementary angles
𝟑𝟓°
𝟓𝟓° Q T R U
∠𝑷𝑸𝑹+∠𝑺𝑻𝑼=𝟗𝟎°

24. Supplementary Angles: two angles that add up to 𝟏𝟖𝟎°. D
∠𝑫𝑩𝑬 + ∠𝑫𝑩𝑪=𝟏𝟖𝟎° E B C
∠𝑫𝑩𝑬 𝒂𝒏𝒅 ∠𝑫𝑩𝑪 are adjacent supplementary angles
They would also be called a linear pair because they form a straight line
∠𝑨𝑹𝑻 𝒂𝒏𝒅 ∠𝑩𝑳𝑴 are nonadjacent
supplementary angles
∠𝑨𝑹𝑻+∠𝑩𝑳𝑴=𝟏𝟖𝟎° B A
𝟏𝟐𝟓°
𝟓𝟓° L R T M

25. Example D A B C 𝑩𝑫 bisects ∠𝑨𝑩𝑪. Find the value of x. (𝟒𝒙+𝟏𝟎)°
(𝟏𝟎𝒙−𝟔𝟖)° B C
𝟒𝒙+𝟏𝟎=𝟏𝟎𝒙−𝟔𝟖
What if you were asked the degree measure of ∠𝑨𝑩𝑫?
−𝟒𝒙
−𝟒𝒙
𝟏𝟎=𝟔𝒙−𝟔𝟖
𝟒 𝟏𝟑 +𝟏𝟎
+𝟔𝟖
+𝟔𝟖
𝟓𝟐+𝟏𝟎
𝟕𝟖=𝟔𝒙
𝟔𝟐° 𝟔 𝟔
𝟏𝟑=𝒙

26. Example D A B C 𝑩𝑫 bisects ∠𝑨𝑩𝑪. Find the value of x. Find ∠𝑨𝑩𝑫
(𝟔𝒙−𝟏𝟗)°
(𝟐𝒙 +𝟏𝟕)° B C
𝟔𝒙−𝟏𝟗=𝟐𝒙+𝟏𝟕
−𝟐𝒙
−𝟐𝒙
∠𝑨𝑩𝑫=𝟔𝒙−𝟏𝟗
𝟒𝒙−𝟏𝟗=𝟏𝟕
𝟔 𝟗 −𝟏𝟗
+𝟏𝟗
+𝟏𝟗
𝟓𝟒−𝟏𝟗
𝟒𝒙=𝟑𝟔
𝟑𝟓° 𝟒 𝟒
𝒙=𝟗

27. Example
Use the marks on the diagram to name the congruent segments and congruent angles. A
𝑨𝑩 ≅ 𝑩𝑪
∠𝑫𝑨𝑩≅ ∠𝑫𝑪𝑩
𝑨𝑫 ≅ 𝑪𝑫
∠𝑨𝑫𝑩≅ ∠𝑪𝑫𝑩 B D C

28. Definitions
Parallel Lines: Lying on the same plane, but they never intersect. Are like railroad tracks.
Perpendicular Lines: Lines intersecting at right angles.
Polygon: Literally means “many sides”. Closed figure with three or more straight sides.

29. Algebra Review

30. Number Line Think of arithmetic as movement on the number line.
You have a starting point, you move positive or negative, and then arrive at your destination.
Here we start at negative nine.
Then we move positive seventeen units.
To arrive at positive eight.

31. Example

32. Example

33. Example

34. Example

35. Example

36. Example

37. Example

38. Example

39. Example

40. Sign Confusion
A negative number times a positive number is a negative:
A negative number times a negative number is a positive:
A negative number divided by a positive number is a negative:
A negative number divided by a negative number is a positive:

41. Solving for x
Equation: statement that two quantities are equivalent. Solving an Equation: Totally isolate the variable you are attempting to solve for.

42. Solving for x 9𝑥−7=11 +7 +7 9𝑥=18 9 9 𝑥=2
Rule 1: You can do anything to an equation as long as you do it to both sides. Rule 2: To move a term from one side of the equation to another perform the opposite mathematical operation. Rule 3: The variable you solve for must be made positive.
9𝑥−7=11 +7 +7
9𝑥=18 9 9
𝑥=2

43. Solving for x −11+4𝑥=−33 +11 +11 4𝑥=−22 4 4 𝑥=−5.5
This is an equation.
It has three terms.
−11+4𝑥=−33
To solve for x we need to isolate the variable.
+11
+11
Our first step is to add eleven to both sides.
4𝑥=−22 4 4
We know we have to add eleven because of rule two.
𝑥=−5.5
Our next step is to divide both sides by four.
We know to divide by four because of rule two.

44. Solving for y 12−4𝑦=−16 −12 −12 −4𝑦=−28 −4 −4 𝑦=7 This is an equation.
It has three terms.
12−4𝑦=−16
To solve for y we need to isolate the variable.
−12
−12
Our first step is to subtract twelve from both sides.
−4𝑦=−28 −4 −4
We know we have to subtract twelve because of rule two.
𝑦=7
Our next step is to divide both sides by negative four.
Remember a negative divided by a negative is a positive.

45. Solving for x
5𝑥−11=19
+11
+11
5𝑥=30 5 5
𝑥=6

46. Solving for x
6𝑥+18=54
−18
−18
6𝑥=36 6 6
𝑥=6

47. Solving for x
−18+5𝑥=22
+18
+18
5𝑥=40 5 5
𝑥=8

48. Solving for x
−14−9𝑥=−59
+14
+14
−9𝑥=−45 −9 −9
𝑥=5

49. Solving for x
𝑥+9 3 =21
3∙ 𝑥+9 3 =3∙21
x+9=63 −9 −9
x=54

50. Solving for x
9𝑥−7=38 +7 +7
9𝑥=45 9 9
𝑥=5

51. P. E. M. D. A. S. Order of Operations 𝑥+5 𝑥 3 𝑥∙8 𝑥÷4 𝑥+2 𝑥−11
parenthesis E.
𝑥+5 M.
exponents
𝑥 3 D.
multiplication
𝑥∙8 A.
division S.
𝑥÷4
addition
𝑥+2
subtraction
𝑥−11

52. This is an equation.
5𝑥+12 8 =4
It has two terms.
The term on the left is a fraction.
8 5𝑥 =4∙8
5x + 12 is being divided by 8.
To get rid of 8 perform the opposite operation: multiply.
5𝑥+12=32
−12
−12
Rule 2 tells us to do the opposite operation.
5𝑥=20 5 5
𝑥=4

53. Variables on both sides
Isolate the variable.
−9𝑥+7=−4𝑥−13
Be careful to only combine like terms.
+13
+13
−9𝑥+20=−4𝑥
+9𝑥
+9𝑥
20=5𝑥 5 5
4=𝑥

54. Lots of the same variable
What are the like terms?
3𝑥+4 + 4𝑥−3 + 3𝑥−6 =
180
3𝑥+4𝑥+3𝑥
4−3−6
10𝑥 −5
=180 +5 +5
10𝑥=185 10 10
𝑥=18.5

55. Evaluating an Expression
Step 1: Substitute a specific value for each variable (plug a number in for the variable). Put this number in parenthesis.
Evaluate the following expression for 𝒙=𝟕.
3𝑥−8
3 7 −8
21−8 13
Step 2: Perform all operations using the correct order of operations.

56. Evaluating an Expression
Evaluate the following expressions for 𝒙=𝟗.
6𝑥−11
4𝑥+3
11𝑥−14
6 9 −11
4 9 +3
11 9 −14
54−11
36+3
99−14 43 39 85

57. Variables on both sides
Isolate the variable.
7𝑥+9=−4𝑥−24
Be careful to only combine like terms.
+4𝑥
+4𝑥
11𝑥+9=−24 −9 −9
11𝑥=−33 11 11
𝑥=−3

58. Lots of the same variable
What are the like terms?
𝑥−7 + 𝑥−2 + 2𝑥+1 =
180
𝑥+𝑥+2𝑥
−7−2+1 4𝑥 −8
=180 +8 +8
4𝑥=188 4 4
𝑥=47

59. Common Errors #1
4𝑥+8𝑥+𝑥=27
12𝑥=27
𝑥𝑥 𝑥𝑥 + 𝑥𝑥𝑥𝑥 𝑥𝑥𝑥𝑥 + 𝑥 =27
13𝑥=27

60. Common Errors #1 Directi0ns say to round to the nearest tenth.
4𝑥+8𝑥+𝑥=27
13𝑥=27 13 13
𝑥=2.07
𝑥=2. 1

61. Common Errors #3 3𝑥+7 + 2𝑥−4 + 8𝑥−9 = 7 3𝑥+2𝑥+8𝑥 7−4−9 18𝑥 −6 18𝑥=−6
Don’t do it this way
3𝑥+7 + 2𝑥−4 + 8𝑥−9 = 7
3𝑥+2𝑥+8𝑥
7−4−9
18𝑥 −6
18𝑥=−6

62. Common Errors #3 3𝑥+7 + 2𝑥−4 + 8𝑥−9 = 7 3𝑥+2𝑥+8𝑥 7−4−9 13𝑥 −6 =7 +6 +6
Do it this way
3𝑥+7 + 2𝑥−4 + 8𝑥−9 = 7
3𝑥+2𝑥+8𝑥
7−4−9
13𝑥 −6 =7 +6 +6
13𝑥=13 13 13
𝑥=1