1. PROPERTIES OF PLANE FIGURES
ANGLES & TRIANGLES

2. Angles
Angles are formed by the intersection of 2 lines, 2 rays, or 2 line segments. The point at which the lines, rays, or line segments intersect is called the vertex.
Vertex
Vertex
Vertex

3. Measuring Angles
The measure of an angle is the amount of rotation in degrees about the vertex from one side to the other. The measure of any angle is between 0º and 180º. The wider the “mouth,” the greater the measure of the angle.

4. Classifying Angles
There are 4 types of angles: acute angle – angle that is greater than 0° and less than 90° right angle – angle that is exactly 90° obtuse angle – angle that is greater than 90° and less than 180° straight angle – angle that is exactly 180°

5. acute
right
obtuse
straight

6. Naming Angles
Angles are named according to their vertex and the points through which each side passes. Angles can be labeled by only the letter or number that represents their vertex. If more than 1 angle share a vertex, then label the angle with the points on each side of the angle and the vertex. Make sure that the vertex is always the middle letter.

7. Homework
Complete all of the problems on “PPF Practice Problems 1.” Also, answer the following questions: What do acute people and acute angles have in common? What do obtuse people and obtuse angles have in common?

8. Angle Addition Postulate
When more than 1 angle share a vertex, the sum of the measure of the smaller angles equal the measure of the largest angle.
Example 1

9. Another Example of Applying the Angle Addition Postulate

10. You Try

11. Angle Relationships
Adjacent Angles – two angles that are next to each other Complementary Angles – two adjacent angles that form a right angle Supplementary Angles – two adjacent angles that form a straight angle Vertical Angles – two angles that are opposite of each other when two lines cross

12. More Angle Relationships
The following relationships are formed when a transversal line intersects two parallel lines: Corresponding Angles – two angles that are on the same side of the transversal line, but one is inside and the other is outside of the parallel lines Alternate Exterior Angles – two angles that are on opposite sides of the transversal line, but both are outside of the parallel lines Alternate Interior Angles – two angles that are on opposite sides of the transversal line, but both are inside of the parallel lines Consecutive (Co-interior) Angles – two angles that are on the same side of the transversal line and are both inside of the parallel lines

13. Adjacent Angles
CAB and BAD are adjacent angles

14. Complementary Angles

15. Supplementary Angles

16. Vertical Angles

17. Corresponding Angles

18. Alternate Exterior Angles

19. Alternate Interior Angles

20. Consecutive (Co-Interior) Angles

21. You Try
Identify the relationship
Identify the relationship

22. Homework
Complete all of the problems on PPF Practice Problems 2.

23. More Info On Angle Relationships
Complementary Angles – add up to 90° Supplementary Angles – add up to 180° Vertical Angles – have the same measure Alternate Exterior Angles – have the same measure Alternate Interior Angles – have the same measure Corresponding Angles – have the same measure Consecutive Angles – add up to 180°

24. You Try
Find the measure of angle b
Find the measure of angle b

25. You Try More
Find the measure of angle b
Solve for x

26. Homework
Complete all of the problems on PPF Practice Problems 3.

27. Geometric Notations

28. More Geometric Notations

29. Congruent Angles and Segments

30. Parallel Lines

31. Perpendicular Lines

32. Homework
Complete all of the problems on PPF Practice Problems 4

33. Basic Facts About Triangles
Three-sided polygon
Each “corner” is a vertex (vertices for plural)
The area of a triangle is ½ · base · height
The angles in a triangle add up to 180°
Triangles are named by their vertices
Triangles are classified by their sides and angles

34. Classifying Triangles by Angles
Acute Triangle – all 3 angles are acute
Right Triangle – has 1 right angle
Obtuse Triangle – has 1 obtuse angle

35. Classifying Triangles by Sides
Equilateral Triangle – all sides and all angles are congruent
Isosceles Triangle – 2 sides are congruent and base angles are congruent
Scalene Triangle – no sides are congruent

36. Classifying Triangles by Angles and Sides
Right Scalene
Acute Scalene
Right Isosceles
Obtuse Scalene
Acute Isosceles
Obtuse Isosceles

37. You Try: Classify Triangles by Their Angles and Sides4 1 2 5 6 3

38. Triangle Angle Sum The measure of the unknown angle is
180° – (81° + 58°) = 41°
The measure of the unknown angle is
180° – (90° + 40°) = 50°

39. Triangle Angle Sum Extended
This angle is also 102° due to the property of vertical angles
This angle is
180° – (102° + 52°) = 26°
The measure of the unknown angle is 180° – 26° = 154° due to the property of supplementary angles.

40. You Try: Find the Measure of the Unknown Angles1 2 3

41. Homework
Complete all of the problems on PPF Practice Problems 5 and 6. For extra credit you may complete the “Additional Triangle Angle Sum Practice Problems” sheet.

42. Triangle Congruence
Congruent triangles are exactly the same size (same side lengths and angle measures)
Corresponding sides are the congruent sides of congruent triangles
Corresponding angles are the congruent angles of congruent triangles

43. Examples of Congruent Triangles
Note: We must name the congruent triangles correctly according to the corresponding angles!

44. You Try: Write a Statement Indicating the Pairs of Triangle Are Congruent1 2

45. Proving Triangle Congruence
SSS (side-side-side) – if all 3 pairs of corresponding sides are congruent, then the triangles are congruent SAS (side-angle-side) – if 2 pairs of corresponding sides and the pair of corresponding angles between them are congruent, then the triangles are congruent ASA (angle-side-angle) – if 2 pairs of corresponding angles and the pair of corresponding sides between them are congruent, then the triangles are congruent AAS (angle-angle-side) – if 2 pairs of corresponding angles and the pair of corresponding sides not between them are congruent, then the triangles are congruent

46. Examples of Proving Triangle Congruence
SSS
ASA
SSS
ASA
SAS
AAS
SAS

47. You Try: State Why the Triangles Are Congruent1 2

48. Proving Right Triangle Congruence
LA (leg-angle) – if a pair of corresponding legs and a pair of corresponding angles other than the right angles are congruent, then the right triangles are congruent LL (leg-leg) – if 2 pairs of corresponding legs are congruent, then the right triangles are congruent HA (hypotenuse-angle) – if the pair of hypotenuses and a pair of corresponding angles other than the right angles are congruent, then the right triangles are congruent HL (hypotenuse-leg) – if the pair of hypotenuses and a pair of corresponding legs are congruent, then the right triangles are congruent

49. Examples of Proving Right Triangle CongruenceLA HA HL HL LL

50. You Try: State Why the Right Triangles Are Congruent1 2 3

51. Homework
Complete all of the problems on PPF Practice Problems 7 For extra credit do Triangle Congruence Additional Practice Problems

52. Properties of Isosceles Triangles
Two sides are congruent
Base angles are congruent

53. Using Properties of Isosceles Triangles to Find Angle Measures
This angle is 124° (triangle angle sum)
This angle is 28° (base angles of isosceles)
This angle is also 28° (vertical)
This angle is 124° (supplementary)
This angle is 28° because it is congruent to the other base angle and together they must add up to 56°
So, by the properties of supplementary angles, angle x is 56°.

54. You Try: Find the Measure of the Unknown Angle

55. Homework
Complete all of the problems on PPF Practice Problems 8. For extra credit do Properties of Isosceles and Equilateral Triangles Additional Practice Problems.

56. Triangle Inequality Theorem
Any side of a triangle is always shorter than the sum of the other two sides but greater than the difference between the other two sides. The shortest side of a triangle is always greater than the difference between the other two sides, and the longest side of a triangle is always less than the sum of the other two sides.

57. Example of Using The Triangle Inequality Theorem
Example 1 – State whether or not the following three numbers can be lengths of the sides of a triangle: 8, 8, 14 Solution – 1. Check to see if the longest side is less than the sum of the other two sides. 14 < ? Yes 2. Check to see if the shortest side is greater than the difference between the other two sides. 8 > 14 – 8? Yes Therefore, these three numbers can be lengths of the sides of a triangle.

58. You Try: State If The Three Numbers Can Be Lengths of the Sides of a Triangle
13, 16, 20
2, 12, 22
4, 11, 15

59. Finding The Range of Possible Lengths of The Third Side of a Triangle
Example 1 – What is the range of possible values for x, the length of the third side of the triangle, when given the lengths of the other two sides? 10, 13, x Solution – 1. x must be less than = x must be greater than 13 – 10 = 3 Therefore, 3 < x < 23.

60. You Try
Find the range of x, the possible lengths of the third side of the triangle, when given the lengths of the other two sides: 9, 13, x
Find the range of x, the possible lengths of the third side of the triangle, when given the lengths of the other two sides: 5, 5, x

61. Homework
Complete all of the problems on PPF Practice Problems 9. For extra credit do Triangle Inequality Theorem Additional Practice Problems.

62. Triangle Sides and Angles
The longer the side, the larger the angle that is across from it.
The larger the angle, the longer the side that is across from it.

63. Example of Ordering Sides According to Angle Measures

64. Example of Ordering Angles According Side Lengths

65. You Try
In ΔSTU,
TU = 8 ¼
SU = 8 4/5
ST = 9
Order the angles from smallest to largest.
1. Order the sides of the triangle from shortest to longest.
2. Order the angles of the triangle from smallest to largest.

66. Homework
Complete all of the problems on PPF Practice Problems 10.

67. The Pythagorean Theorem
Only applies to right triangles
Used to find the unknown side length of any right triangle
c² = a² + b²
a and b are the legs, and c is the hypotenuse (the longest side)
The hypotenuse is always across from the right angle. a c b
16² = x² + 10²
16² - 10² = x²
256 – 100 = x²
156 = x²
√156 = x
2√39 = x ≈ 12.5

68. You Try
Use the Pythagorean Theorem to find the unknown side length of the following triangles: 1 2

69. Real-world Application of the Pythagorean Theorem
Ex 1: A ladder that is 13 feet long is placed against a wall such that the base of the ladder is 5 feet from the wall. How many feet above the ground is the top of the ladder?

70. Homework
Complete all of the problems on PPF Practice Problems 11.

71. The Pythagorean Theorem and The Area of Triangle
The area of a triangle is
½ · base · height
The base and the height of a right triangle are its legs
Ex 1: Find the area of the triangle
Solution:
Step 1 – use the Pythagorean to find the length of the other leg, which is also the height of the triangle
Step 2 – plug the values for the base and the height into the formula of the area of triangles.

72. You Try: Find the Area of the Triangles1 2

73. Extension of the Pythagorean Theorem and Area of Triangles
Find the area of the triangle.

74. You Try
Find the area of the triangle.

75. Homework
Complete all of the problems on PPF Practice Problems 12. Study for quiz. For extra credit do The Pythagorean Theorem and the Area of a Triangle Additional Practice Problems.

76. 45°-45°-90° Right Triangles Also known as right isosceles triangles
Legs are congruent
Hypotenuse is √2 times longer than the legs

77. Finding Missing Side Lengths of 45°-45°-90° Right Triangles
Example 1 – Find the missing side lengths. Leave your answer as a radical in simplest form if possible.
Solution -
Hint – Since we know the length of the hypotenuse, we must find the length of the legs, which are congruent to one another, by dividing the hypotenuse by √2.

78. Finding Missing Side Lengths of 45°-45°-90° Right Triangles
Solution -
Example 2 – Find the missing side lengths. Leave your answers as radicals in simplest form if possible.
Hint – Since we know the length of one leg, we know the length of the other leg as well because they are congruent. To find the length of the hypotenuse, we must multiply the length of the leg by √2.

79. Rules To Finding Missing Side Lengths of 45°-45°-90° Right Triangles
Rule 1 – If you know the length of the hypotenuse, then divide it by √2 to find the length of the legs. Rule 2 – If you know the length of one of the legs, then multiply it by √2 to find the length of the hypotenuse.

80. You Try
1. Find the missing side lengths. Leave your answer as a radical in simplest form if possible.
2. Find the missing side lengths. Leave your answer as a radical in simplest form if possible.

81. Homework
Complete all of the problems on PPF Practice Problems 13.

82. 30°-60°-90° Right Triangles The short leg is across from the 30° angle
The long leg is across from the 60° angle and is √3 times longer than the short leg
The hypotenuse is 2 times longer than the short leg

83. Finding the Unknown Side Length of 30°-60°-90° Right Triangles
The length of the short leg is 5.
y is the long leg, so its length is 5√3
x is the hypotenuse, so its length is 5 ∙ 2 = 10
The length of the long leg is √15
y is the short leg, so its length is
√15 / √3 = √5
x is the hypotenuse, so its length is 2√5

84. You Try

85. Solving Multi-step 30°-60°-90° Right Triangle Problems
Example 1 – Find the unknown side length, x.
Solution – First find the length of the hypotenuse of the triangle on the right, which is also the long leg of the triangle on the left.
9 ∙ 2 = 18
Next, find the length of x, which is the short leg of the triangle on the left.
= ∙ = = 𝟔 𝟑 18

86. Solving Multi-step 30°-60°-90° Right Triangle Problems
Example 2 – Find the unknown side length, x.
Solution – First, find the length of the long leg of the triangle on the left, which is also the hypotenuse of the triangle on the right.
Next, find the short leg of the triangle on the right, which is also x.
÷ 2 = ∙ =

87. You Try

88. Solving Multi-step Special Right Triangle Problems

89. Solving Multi-step Special Right Triangle Problems