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

• 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.

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.

Complementary & Supplementary

• Complementary – the sum of the measures of two angles is 90º

• Supplementary – the sum of the measures of two angles is 180º

Vertical Angles

• Angles opposite one another at the intersection of two lines.

• Vertical angles are congruent.

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

c

a

b

1 2

3 4

5 6

7 8

•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.

• 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

• 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)

Example

• Find x and y.

x

5y

75º

Solution:

X + 5y= 180º Supplementary angles

75º + 5y = 180º Substituted

5y = 105º sub. 75 from both sides

y=21º divided both sides by 5

Objectives

• TSW define a triangle.

• TSW categorize the different types of triangles.

• TSW identify relationships among triangles.

Triangles

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

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.

Classifying Triangles by Their Sides

• EQUILATERAL – 3 congruent sides

• ISOSCELES – at least two sides

congruent

• SCALENE – no sides congruent

EQUILATERAL

ISOSCELES

SCALENE

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

Congruent Triangles

• Congruent triangles are triangles whose corresponding angles and sides are congruent. 

• They are exactly the same size and shape.

The Hypotenuse

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

Question 1

• If a triangle has three unequal sides it is a ______ triangle.• A) Scalene

• B) Obtuse

• C) Isosceles

Question 2

• A triangle which has three equal angles is a ______ triangle.• A) Acute

• B) Equilateral

• C) Equiangular

Question 3

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

• B) Isosceles

• C) Congruent

Question 4

• Two triangles that are equal are called ______ triangles• A) Acute

• B) Obtuse

• C) Congruent

Question 5

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

• B) Right

• C) Equiangular

Question 6

• A triangle with an angle of 90° is a ______ triangle.• A) Acute

• B) Right

• C) Congruent

Question 7

• A triangle in which all three angles are less than 90° is a ______ triangle.• A) Right

• B) Acute

• C) Scalene

Question 8

• A triangle in which one of the angles is greater than 90° is a ______ triangle.• A) Obtuse

• B) Isosceles

• C) Acute

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.

Answers

1. A) Equilateral

2. C) Equiangular

3. B) Isosceles

4. C) Congruent

5. A) Equilateral

6. B) Right 7. B) Acute

8. A) Obtuse

9. B) The side opposite the right angle.

• 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.

Objectives

Pythagorean Theorem

• Pythagoras was a Greek philosopher that was known as “the father of numbers.”

• He is best known for the Pythagorean Theorem.

leg

leg

hypotenuse

Right angle

The Right Triangle

hypotenuseleg a

leg b

leg a

leg bhyp

ote

nuse

hypotenuse

leg b

leg a

leg a

leg

b

hypotenuse

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.

Pythagorean Theorem

• For a right triangle with legs a and b and hypotenuse c, a2+b2=c2.

Practicea² + b² = c²9² + 12² = c²81 + 144 = c²225 = c²√225 = c15 = c

Practicea² + b² = c²a² + 8² = 24²a² + 64 = 576a² + 64 - 64 = 576 - 64a² = √512 a² ≈ 22.6

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.

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 = c²5² + 12² = 13²25 + 144 = 169169 = 169

Yes! This is a right triangle

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

Is the following a right triangle?

10 cm9 cm

7 cm

Find the value of b

b

20 feet

12 feet

Find x

x

8 inches

6 inches

What is the measure of the diagonal of the following rectangle?

9 feet

12 feet

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 didn’t 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?

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?

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

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)