Angles

1
Angles Triangles
Objectives TSW define an angle. TSW identify
relationships among angles. TSW identify the
angles created in a transversal. TSW determine
the measures of the angles formed by in a
transversal
2

• The point about which an angle is measured is
  called the angle's vertex
• If two angles have the same vertex and share the
  same side between, then the angles are called
  adjacent angles.

3
Measuring Angles

• Align the baseline of the protractor with one
  side of the angle and move the protractor left or
  right as necessary to place the angle vertex
  under the origin at the center of the baseline.

4
Complementary Supplementary

• Complementary the sum of the measures of two
  angles is 90º
• Supplementary the sum of the measures of two
  angles is 180º

5
Vertical Angles

• Angles opposite one another at the intersection
  of two lines.
• Vertical angles are congruent.

6
Angles

• Parallel lines are lines that do not intersect
• A transversal line intersects two or more lines
  at different points.
• Angles are formed when a transversal intersects a
  pair of parallel lines

7
1
2
a
3
4
5
6
b
8
7
c

• Line A and B are parallel Line C is a
  transversal.
• Angles 3,4,5, and 6 are interior angles because
  they are between the parallel lines.
• Angles 3 and 6 are alternate interior angles, as
  well as angles 4 and 5.

8

• Corresponding angles are angles that are in the
  same relative position compared to the two
  different parallel lines. Ex 2 and 6, 4 and 8, 1
  and 5, 3 and 7.
• The measure of corresponding angles and alternate
  interior angles are equal.
• Adjacent angles combine to form a straight line
  measuring 180º.
• Adjacent angles are called supplementary angles.
  Ex 1 and 2, 1 and 7

9

• Exterior angles are angles on the outer sides of
  two lines cut by a transversal. (Ex. Angles 1, 2,
  7, and 8)
• An alternate exterior angle is a pair of angles
  on the outer sides of two lines cut by a
  transversal, but on opposite sides of a
  transversal (Ex. Angles 1 and 8, angles 2 and 7)

10
Example
Solution X 5y 180º Supplementary angles 75º
5y 180º Substituted 5y 105º sub. 75 from both
sides

• Find x and y.

x
75º
5y
y21º divided both sides by 5
11
Objectives

• TSW define a triangle.
• TSW categorize the different types of triangles.
• TSW identify relationships among triangles.

12
Triangles

• A triangle is a 3-sided polygon. Every triangle
  has three sides and three angles. When added
  together, the three angles equal 180.

13
Different Types of Triangles

• There are several different types of triangles.
• You can classify a triangle by its sides and its
  angles.
• There are THREE different classifications for
  triangles based on their sides.
• There are FOUR different classifications for
  triangles based on their angles.

14
Classifying Triangles by Their Sides

• EQUILATERAL 3 congruent
  sides
• ISOSCELES at least two sides
• congruent
• SCALENE no sides congruent

EQUILATERAL
ISOSCELES
SCALENE
15
Classifying Triangles by Their Angles

• EQUIANGULAR all angles are congruent
• ACUTE all angles are acute
• RIGHT one right angle
• OBTUSE one obtuse angle

EQUIANGULAR
ACUTE
RIGHT
OBTUSE
16
Congruent Triangles

• Congruent triangles are triangles whose
  corresponding angles and sides are congruent. 
• They are exactly the same size and shape.

17
The Hypotenuse

• The hypotenuse of a right triangle is the
  triangle's longest side, i.e., the side opposite
  the right angle.

18
Question 1

• If a triangle has three unequal sides it is a
  ______ triangle.
• A) Scalene
• B) Obtuse
• C) Isosceles

19
Question 2

• A triangle which has three equal angles is a
  ______ triangle.
• A) Acute
• B) Equilateral
• C) Equiangular

20
Question 3

• A triangle with (at least) two equal sides is a
  ______ triangle.
• A) Scalene
• B) Isosceles
• C) Congruent

21
Question 4

• Two triangles that are equal are called ______
  triangles
• A) Acute
• B) Obtuse
• C) Congruent

22
Question 5

• A triangle with all three sides of equal length
  is a ______ triangle.
• A) Equilateral
• B) Right
• C) Equiangular

23
Question 6

• A triangle with an angle of 90 is a ______
  triangle.
• A) Acute
• B) Right
• C) Congruent

24
Question 7

• A triangle in which all three angles are less
  than 90 is a ______ triangle.
• A) Right
• B) Acute
• C) Scalene

25
Question 8

• A triangle in which one of the angles is greater
  than 90 is a ______ triangle.
• A) Obtuse
• B) Isosceles
• C) Acute

26
Question 9

• What is the Hypotenuse of a right triangle?
• A) The smallest angle.
• B) The side opposite the right angle.
• C) The right angle.

27
Answers

• 6. B) Right
• 7. B) Acute
• 8. A) Obtuse
• 9. B) The side opposite the right angle.

• 1. A) Equilateral
• 2. C) Equiangular
• 3. B) Isosceles
• 4. C) Congruent
• 5. A) Equilateral

28
Objectives

• TSW identify Pythagoras and his contributions to
  math.
• TSW identify the parts of a right triangle.
• TSW apply the Pythagorean Theorem in order to
  determine the missing side of a right triangle.
• TSW use the Pythagorean Theorem to solve
  contextual problems
• TSW apply the Pythagorean Theorem to determine
  the distance between two points on a coordinate
  plane.

29
Pythagorean Theorem

• Pythagoras was a Greek philosopher that was known
  as the father of numbers.
• He is best known for the Pythagorean Theorem.

30
The Right Triangle
hypotenuse
leg
leg
Right angle
31
hypotenuse
leg a
leg b
32
hypotenuse
leg b
leg a
33
leg a
leg b
hypotenuse
34
leg a
leg b
hypotenuse
35
Pythagorean Theorem

• The legs are the sides that form the right angle.
• The hypotenuse is the side opposite the right
  angle. It is the longest side of the triangle.
• The Pythagorean Theorem describes the
  relationship between the lengths of the legs and
  the hypotenuse for any right triangle.

36
Pythagorean Theorem

• For a right triangle with legs a and b and
  hypotenuse c, a2b2c2.

37
Practice
a² b² c² 9² 12² c² 81 144 c² 225
c² v225 c 15 c
38
Practice
a² b² c² a² 8² 24² a² 64 576 a² 64
- 64 576 - 64 a² v512 a² 22.6
39
Converse of the Pythagorean Theorem

• If you reverse the parts of the Pythagorean
  Theorem, you have formed its converse. The
  converse of the Pythagorean Theorem is also true.
• If the sides of a triangle have lengths a, b, and
  c units such that a² b² c², then the triangle
  is a right triangle.

40
Identify a Right Triangle

• The measures of three sides of a triangle are 5
  inches, 12 inches, and 13 inches. Determine
  whether the triangle is a right triangle.

a² b² c² 5² 12² 13² 25 144 169 169
169
Yes! This is a right triangle
41
Practice

• Determine whether each set of side lengths form a
  right triangle. Write yes or no. Then justify
  your answer.

1. 12, 16, 24
2. 8, 15, 17

42
Is the following a right triangle?
10 cm
9 cm
7 cm
43
Find the value of b
20 feet
b
12 feet
44
Find x
6 inches
8 inches
x
45
What is the measure of the diagonal of the
following rectangle?
9 feet
12 feet
46
Applications of the Pythagorean Theorem

• Omar drove a remote control boat across a creek
  that has a width of 9 feet. When the boat got to
  the other side, he realized that the boat didnt
  cross the creek directly because the current
  carried the boat downriver. He wants to
  determine how far down the creek the boat ended
  up. If the boat actually traveled 15 feet, how
  far downriver did the boat end up?

47
Example

• A lighthouse statue that is 12 feet tall casts a
  16-ft shadow on the surface of the water. What
  is the distance from the top of the lighthouse to
  the top of its shadow?

48
Distance on the Coordinate Plane

• Recall that a coordinate plane is formed by the
  intersection of a vertical and horizontal number
  line at their zero points. The coordinate plane
  is separated into 4 quadrants.
• You can use the Pythagorean Theorem to find the
  distance between
• two points on the
• coordinate plane.

http//www.mathopenref.com/coorddist.html
49
Distance on a Coordinate Plane

• Graph each pair of ordered pairs. Then find the
  distance between the points. Round to the
  nearest tenth.

1. (2, 0), (5, -4)
2. (1, 3), (-2, 4)
3. (-3,-4), (2, -1)