Download Angles

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.
• 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.
Measuring Angles
• Complementary – the sum of the measures of two angles
is 90º
• Supplementary – the sum of the measures of two angles
is 180º
Complementary & Supplementary
• Angles opposite one
another at the
intersection of two
lines.
• Vertical angles are
congruent.
Vertical 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
Angles
1
2
a
3
4
5
6
7
b
8
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.
• 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)
Solution:
X + 5y= 180º
Supplementary angles
75º + 5y = 180º Substituted
5y = 105º
sides
sub. 75 from both
• Find x and y.
x
75º
5y
Example
y=21º divided both sides by 5
• TSW define a triangle.
• TSW categorize the different types of triangles.
• TSW identify relationships among triangles.
Objectives
Triangles
• A triangle is a 3-sided
polygon. Every triangle
has three sides and three
angles. When added
together, the three angles
equal 180°.
• 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.
Different Types of
Triangles
• EQUILATERAL – 3 congruent
sides
• ISOSCELES – at least two sides
congruent
• SCALENE – no sides congruent
EQUILATERAL
ISOSCELES
Classifying Triangles by
Their Sides
SCALENE
• EQUIANGULAR – all angles are congruent
• ACUTE – all angles are acute
• RIGHT – one right angle
EQUIANGULAR
• OBTUSE – one obtuse angle
Classifying Triangles by
Their Angles
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.
• If a triangle has three
unequal sides it is a
______ triangle.
• A) Scalene
• B) Obtuse
• C) Isosceles
Question 1
• A triangle which has
three equal angles is a
______ triangle.
• A) Acute
• B) Equilateral
• C) Equiangular
Question 2
• A triangle with (at
least) two equal sides
is a ______ triangle.
• A) Scalene
• B) Isosceles
• C) Congruent
Question 3
• Two triangles that are
equal are called
______ triangles
• A) Acute
• B) Obtuse
• C) Congruent
Question 4
• A triangle with all
three sides of equal
length is a ______
triangle.
• A) Equilateral
• B) Right
• C) Equiangular
Question 5
• A triangle with an
angle of 90° is a
______ triangle.
• A) Acute
• B) Right
• C) Congruent
Question 6
• A triangle in which all
three angles are less
than 90° is a ______
triangle.
• A) Right
• B) Acute
• C) Scalene
Question 7
• A triangle in which
one of the angles is
greater than 90° is a
______ triangle.
• A) Obtuse
• B) Isosceles
• C) Acute
Question 8
• What is the
Hypotenuse of a right
triangle?
• A) The smallest angle.
• B) The side opposite
the right angle.
• C) The right angle.
Question 9

1. A) Equilateral

6. B) Right

2. C) Equiangular

7. B) Acute

3. B) Isosceles

8. A) Obtuse

4. C) Congruent

9. B) The side opposite the
right angle.

5. A) Equilateral
Answers
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.
Pythagorean Theorem
• Pythagoras was a
Greek philosopher that
was known as “the
father of numbers.”
• He is best known for
the Pythagorean
Theorem.
The Right Triangle
hypotenuse
leg
leg
Right angle
leg a
leg b
leg b
leg a
leg a
leg b
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.
Practice
a² + b² = c²
9² + 12² = c²
81 + 144 = c²
225 = c²
√225 = c
15 = c
Practice
a² + b² = c²
a² + 8² = 24²
a² + 64 = 576
a² + 64 - 64 = 576 - 64
a² = √512
a² ≈ 22.6
• 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.
Converse of the
Pythagorean Theorem
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
• 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
Practice
Is the following a right
triangle?
10 cm
9 cm
7 cm
Find the value of b
20 feet
b
12 feet
Find x
6 inches
8 inches
x
What is the measure of
the diagonal of the
following rectangle?
9 feet
12 feet
• 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?
Applications of the
Pythagorean Theorem
• 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?
Example
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/coor
ddist.html
• 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)
Distance on a Coordinate
Plane