Title: Angles
1Angles 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.
3Measuring 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.
4Complementary Supplementary
- Complementary the sum of the measures of two
angles is 90º - Supplementary the sum of the measures of two
angles is 180º
5Vertical Angles
- Angles opposite one another at the intersection
of two lines. - Vertical angles are congruent.
6Angles
- 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
71
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)
10Example
Solution X 5y 180º Supplementary angles 75º
5y 180º Substituted 5y 105º sub. 75 from both
sides
x
75º
5y
y21º divided both sides by 5
11Objectives
- TSW define a triangle.
- TSW categorize the different types of triangles.
- TSW identify relationships among triangles.
12Triangles
- A triangle is a 3-sided polygon. Every triangle
has three sides and three angles. When added
together, the three angles equal 180.
13Different 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.
14Classifying Triangles by Their Sides
- EQUILATERAL 3 congruent
sides - ISOSCELES at least two sides
- congruent
- SCALENE no sides congruent
EQUILATERAL
ISOSCELES
SCALENE
15Classifying 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
16Congruent Triangles
- Congruent triangles are triangles whose
corresponding angles and sides are congruent. - They are exactly the same size and shape.
17The Hypotenuse
- The hypotenuse of a right triangle is the
triangle's longest side, i.e., the side opposite
the right angle.
18Question 1
- If a triangle has three unequal sides it is a
______ triangle. - A) Scalene
- B) Obtuse
- C) Isosceles
19Question 2
- A triangle which has three equal angles is a
______ triangle. - A) Acute
- B) Equilateral
- C) Equiangular
20Question 3
- A triangle with (at least) two equal sides is a
______ triangle. - A) Scalene
- B) Isosceles
- C) Congruent
21Question 4
- Two triangles that are equal are called ______
triangles - A) Acute
- B) Obtuse
- C) Congruent
22Question 5
- A triangle with all three sides of equal length
is a ______ triangle. - A) Equilateral
- B) Right
- C) Equiangular
23Question 6
- A triangle with an angle of 90 is a ______
triangle. - A) Acute
- B) Right
- C) Congruent
24Question 7
- A triangle in which all three angles are less
than 90 is a ______ triangle. - A) Right
- B) Acute
- C) Scalene
25Question 8
- A triangle in which one of the angles is greater
than 90 is a ______ triangle. - A) Obtuse
- B) Isosceles
- C) Acute
26Question 9
- What is the Hypotenuse of a right triangle?
- A) The smallest angle.
- B) The side opposite the right angle.
- C) The right angle.
27Answers
- 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
28Objectives
- 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.
29Pythagorean Theorem
- Pythagoras was a Greek philosopher that was known
as the father of numbers. - He is best known for the Pythagorean Theorem.
30The Right Triangle
hypotenuse
leg
leg
Right angle
31hypotenuse
leg a
leg b
32hypotenuse
leg b
leg a
33leg a
leg b
hypotenuse
34leg a
leg b
hypotenuse
35Pythagorean 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.
36Pythagorean Theorem
- For a right triangle with legs a and b and
hypotenuse c, a2b2c2.
37Practice
a² b² c² 9² 12² c² 81 144 c² 225
c² v225 c 15 c
38Practice
a² b² c² a² 8² 24² a² 64 576 a² 64
- 64 576 - 64 a² v512 a² 22.6
39Converse 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.
40Identify 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
41Practice
- Determine whether each set of side lengths form a
right triangle. Write yes or no. Then justify
your answer.
- 12, 16, 24
- 8, 15, 17
42Is the following a right triangle?
10 cm
9 cm
7 cm
43Find the value of b
20 feet
b
12 feet
44Find x
6 inches
8 inches
x
45What is the measure of the diagonal of the
following rectangle?
9 feet
12 feet
46Applications 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?
47Example
- 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?
48Distance 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
49Distance on a Coordinate Plane
- Graph each pair of ordered pairs. Then find the
distance between the points. Round to the
nearest tenth.
- (2, 0), (5, -4)
- (1, 3), (-2, 4)
- (-3,-4), (2, -1)