Download Chapter 5 Triangles and the Pythagorean Theorem Notes

1.
2.
Go to the “Brain Pop” app and search
watch the “Angles” video. (If you don’t
have earbuds, watch with captions)
Take the quiz. It will record your results.
Symbols
Define it in your
own words
Draw it
Describe a realworld example of it
Parallel
Perpendicular
∥
⊥
Interior angles:
∠3, ∠4, ∠5, ∠6
Exterior angles:
∠1, ∠2, ∠7, ∠8
Alternate Interior angles:
𝑚∠4 = 𝑚∠6,
𝑚∠3 = 𝑚∠5
Alternate Exterior angles:
𝑚∠1 = 𝑚∠7
𝑚∠2 = 𝑚∠8
Corresponding angles:
𝑚∠1 = 𝑚∠5,
𝑚∠2 = 𝑚∠6
Classify each pair of angles in
the figure as alternate interior,
alternate exterior, or
corresponding.
a. ∠1 𝑎𝑛𝑑 ∠7
alternate exterior angles
b. ∠2 𝑎𝑛𝑑 ∠6
corresponding angles
Classify the relationship
between ∠4 and ∠6
If m∠1 = 50˚, find m∠2, m∠3, and m∠4.
m∠2 = 130˚ because ∠1 and
∠2 are supplementary.
m∠4 = 130˚
because ∠1 and
∠4 are
supplementary.
m∠3 = 50˚ because ∠1 and
∠3 are vertical angles.
A furniture designer
built the bookcase
shown. Line a is parallel
to line b. If m∠2 = 105˚,
find m∠6 and m∠3.
Justify your answer.
Since ∠2 and ∠6 are
supplementary, the
m∠6 = 75˚.
Since ∠6 and ∠3 are
interior angles, so the m∠3
is 75˚.
Find the measure of
angle 4.
In the figure, line m is
parallel to line n, and
line q is perpendicular
to line p. The measure
of ∠1 is 40˚. What is the
measure of ∠7.
Since ∠1 and ∠6 are alternate exterior angles, m∠6 = 40˚.
Since ∠6, ∠7, and ∠8 form a straight line, the sum is 180˚.
40 + 90 + m∠7 = 180
So m∠7 is 50˚.
Every time Bill watches his
favorite team on TV, the team
loses. So, he decides to not
watch the team play on TV.
In order to play sports, you need to
have a B average. Simon has a B
average, so he concludes that he
can play sports.
All triangles have 3 sides and
3 angles. Mariah has a figure
with 3 sides and 3 angles so it
must be a triangle.
After performing a science
experiment, LaDell concluded
that only 80% of tomato seeds
would grow into plants.
Deductive
Reasoning
Inductive
Reasoning
STEP 1: List the given information, or
what you know. Draw a diagram if
needed.
STEP 2: State what is to be proven.
STEP 3: Create a deductive argument by
forming a logical chain of statements
linking the given information.
STEP 4: Justify each statement with
definitions, properties, and theorems
STEP 5: State what it is you have proven.
A proof is a logical argument
where each statement is justified
by a reason.
A paragraph proof or informal proof involves
writing a paragraph.
A two-column proof or formal proof contains
statements and reason organized in two columns.
Once a statement has been proven, it is a theorem.
The diamondback rattlesnake has a diamond pattern on its
back. An enlargement of the skin is shown. If m∠1 = m∠4,
write a paragraph proof to show that m∠2 = m∠3.
Given: m∠1 = m∠4
Prove: m∠2 = m∠3
Proof: m∠1 = m∠2 because they are vertical angles. Since
m∠1 = m∠4, and m∠2 = m∠4. The measure of angle 3 and 4
are the same since they are vertical angles. Therefore, m∠2 =
m∠3.
Refer to the diagram shown.
AR = CR and DR = BR.
Write a paragraph proof to
show that AR + DR = CR + BR.
Given: AR = ___________ and DR = ____________.
Prove: _________________ = CR + BR.
Proof: You know that AR = CR and DR = BR.
AR + DR = CR + BR by the _____________ Property of
Equality. So, AR + DR = CR + BR by ___________________.
Write a two-column proof to show that if two
angles are vertical angles, then they have the same measure.
Given: lines m and n intersect; ∠1 and ∠3 are vertical.
Prove: m∠1 = m∠3
Statements
Reasons
a. Lines m and n intersect;
Given
∠1 and ∠3 are vertical.
b. ∠1 and ∠2 are a linear
Definition of linear pair
pair and ∠3 and ∠2 are a
linear pair.
c. m ∠1 and m∠2 = 180˚
Definition of supplemental
m∠3 and m∠2 = 180˚
angles
d. m ∠1 and m∠2 = m∠3 and m∠2 Substitution
e. m∠1 = m∠3
Subtraction Property of Equality
The statements for a two-column proof to
show that if m∠Y = m∠Z, then x =100 are
given below. Complete the proof by
providing the reasons.
Statements
a. m ∠Y = m∠Z,
m ∠Y = 2x – 90
m∠Z = x + 10
b. 2x – 90 = x + 10
c. x – 90 = 10
d. x = 100
Reasons
Given
1. What is true about the measures of
∠1 and ∠2? Explain.
2. What is true about the measures of ∠3 and ∠4?
3. What kind of angle is formed by ∠1, ∠5, and ∠3? Write an
equation representing the relationship between the 3 angles.
4. Draw a conclusion about ΔABC.
Words: The sum of the measures of the interior
angles of a triangle is 180˚.
Symbols:
Model:
x + y + z = 180˚.
Find the value of x in the
Antigua and Barbuda flag.
x + 55 + 90 = 180
x + 145 = 180
x = 35
The value of x is 35.
In ΔXYZ, if m∠X = 72˚ and m∠Y = 74˚, what is m∠Z?
The measures of the angles of ΔABC are in the ratio 1:4:5.
What are the measures of the angles?
Let x represent angle A, 4x angle B, and 5x angle C
x + 4x + 5x = 180
10x = 180
x = 18
Angle A = 18˚
Angle B = 18(4) = 72˚
Angle C = 18(5) = 90˚
The measures of the angles of ΔLMN are in the ratio 2:4:6.
What are the measures of the angles?
Words: The measure of an exterior angle is
equal to the sum of the measures of its two
remote interior angles.
Symbols:
Model:
m∠A + m∠B = m∠1
Each exterior angle of the triangle has two
remote interior angles that are not adjacent to
the exterior angle.
4
1
interior
exterior
3
5
2
6
∠4 is an exterior angle.
It’s two remote angles are
∠2 and ∠3.
m∠4 = m∠2 + m∠3
Suppose m∠4 = 135˚. Find the measure of ∠2.
First Way:
Angle 4 is the exterior
angle with angle 2 and
angle K as the remote
interior.
Second Way:
∠4 and ∠ 1 are
supplementary, so
they equal 180˚.
∠ 4 + ∠ 1 = 180
135 + ∠ 1 = 180
∠ 1 = 45
∠2 + ∠K = ∠ 4
∠ 2 + 90 = 135
∠ 2 = 45˚
∠ 1 + ∠ 2 + ∠ K = 180
45 + ∠ 2 + 90 = 180
∠ 2 = 45˚
Suppose m ∠ 5 = 147˚. Find m ∠ 1.
A polygon is a closed figure with three of more line
segments. List the states that are in a shape of a polygon.
Words: The sum of the measures of the interior
angles of a polygon is (n – 2)180, where n is the
number of sides.
Symbols: S = (n – 2)180
Regular Polygons – an equilateral (all sides
are the same) and a equiangular (all angles
are the same)
Find the sum of the measures of the interior angles
of a decagon.
S = (n -2)180
S = (10 – 2)180
S = (8)180
S = 1,440
The sum of the interior angles of a 10-sided
polygon is 1,440˚.
Find the sum of the measures of the interior angles
of each polygon.
a.
Hexagon
b.
Octagon
c.
15-gon
Each chamber of a bee honeycomb is a regular hexagon.
Find the measure of an interior angle of a regular hexagon.
STEP 1:
Find the sum of the measures of angle.
S = (n – 2)180
S = (6 – 2)180
S = (4)180
S = 720˚
STEP 2:
Divide 720 by 6, since there are six angles in a hexagon.
720˚÷ 6 = 120
Each angle in a hexagon is 120˚
Find the measure of one interior angle in each regular
polygon. Round to the nearest tenth if necessary.
a. octagon
b. heptagon
c. 20-gon
Words: The sum of the measures of the exterior angles,
one at each vertex, is 360˚.
Symbols: m∠1 + m ∠ 2 + m ∠ 3 + m ∠ 4 + m ∠ 5 = 360˚
Model:
Examples:
Find the measure of an exterior angle in a regular
hexagon.
A hexagon has a 6 exterior angles.
6x = 360
x = 60
Each exterior angle is 60˚.
Find the measure of an exterior angle in a regular
polygon.
a. triangle
b. quadrilateral
c. octagon
Words: In a right triangle, the sum of the squares
of the legs equal the square of the hypotenuse.
Symbols: a2 + b2 = c2
Model:
c
a
b
Find the missing length. Round to the nearest tenth.
c
12 in
9 in
a2 + b2 = c2
92 + 122 = c2
81 + 144 = c2
225 = c2
± 𝟐𝟐𝟓 = c
c = 15 and -15
The equation has two solutions,
-15 and 15. However, the length of
the side must be positive.
The hypotenuse is 15 inches long.
Find the missing length. Round to the nearest tenth.
b
24 cm
8 cm
a2 + b2 = c2
82 + b2 = 242
64 + b2 = 576
64 – 64 + b2 = 576 - 64
b2 = 512
b = ± 𝟓𝟏𝟐
b ≈ 22.6 or -22.6
The length of leg b is 22.6 cm long.
Find the missing length. Round to the nearest tenth
if necessary.
a.
b.
STATEMENT:
If a triangle is a right triangle, then a2 + b2 = c2.
CONVERSE:
If a2 + b2 = c2, then a triangle is a right triangle.
The converse of the Pythagorean Theorem is also true.
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
The triangle is a right triangle.
Determine if these side lengths makes a right triangle.
a. 36 in, 48 in, 60 in
b. 4 ft, 7ft, 5ft
Write an equation that can be used to find the length of the
ladder. Then solve. Round to the nearest tenth.
a2 + b2 = c2
8.752 + 182 = x2
76.5625 + 324 = x2
400.5625 = x2
± 𝟒𝟎𝟎. 𝟓𝟔𝟐𝟓 = x
20.0 ≈ x
The ladder is about 20 feet.
Write an equation that can be used to find the length of the
ladder. Then solve. Round to the nearest tenth.
a2 + b2 = c2
102 + b2 = 122
100 + b2 = 144
b2 = 44
b = ± 𝟒𝟒
b ≈ 6.6
The height of the plane is
about 6.6 miles.
Mr. Parsons wants to build a new banister for the staircase
shown. If the rise of the stairs of a building is 5 feet and the
run is 12 feet, what will be the length of the new banister?
A 12-foot flagpole is placed in the center of a square area. To
stabilize the pole, a wire will stretch from the top of the pole
to each corner of the square. The flagpole is 7 feet from each
corner of the square. what is the length of each wire. Round
to the nearest tenth.
a2 + b2 = c2
72 + 122 = c2
49 + 144 = c2
193 = c2
± 𝟏𝟗𝟑 = c2
13.9 ≈ c
The length of the wire is about
13.9 feet.
The top part of a circus tent is in the shape of a cone. The tent
has a radius of 50 feet. The distance from the top of the tent
to the edge is 61 feet. How tall is the top part of the tent?
Round to the nearest whole number.
Graph the ordered pairs (3, 0) and (7, 5). Then find the
distance c between the two points. Round to the nearest
tenth.
a2 + b 2 = c2
5 2 + 4 2 = c2
25 + 16 = c2
41 = c2
± 𝟒𝟏 = c2
6.4 ≈ c
The points are about 6.4
units apart.
Graph the ordered pairs (1, 3) and (-2, 4). Then find the
distance c between the two points. Round to the nearest
tenth.
Symbols: The distance d between two points
with coordinates (x1, y1) and (x2, y2) is given by
the formula
d=
Model:
𝑥2 − 𝑥1
2
+ 𝑦2 − 𝑦1
2
On the map, each unit represents 45 miles. West Point, New
York is located at (1.5, 2) and Annapolis, Maryland, is located
at (-1.5, -1.5). What is the approximate distance between
West Point and Annapolis?
METHOD 1:
Use the Pythagorean Theorem
a2 + b2 = c2
32 + 3.52 = c2
21.25 = c2
± 𝟐𝟏. 𝟐𝟓 = c
± 4.6 ≈ c
Since the map units equals 45 miles, the distance
between the cities is 4.6(45) or about 207 miles.
On the map, each unit represents 45 miles. West Point, New
York is located at (1.5, 2) and Annapolis, Maryland, is located
at (-1.5, -1.5). What is the approximate distance between
West Point and Annapolis?
METHOD 2:
Use the Distance Formula
c = 𝑥2 − 𝑥1 2 + 𝑦2 − 𝑦1 2
c=
−1.5 − 1.5
c=
−3
2
2
+ −1.5 − 2
+ −3.5
c = 9 + 12.25
c = 21.25 ≈ ± 4.6
2
2
Since the map units equals
45 miles, the distance
between the cities is 4.6(45)
or about 207 miles.
Cromwell Field is located at (2.5, 3.5) and Deadwoods Field
is at (1.5, 4.5) on a map. If each map unit is 0.1 mile, about
how far apart are the fields?
Use the Distance Formula to find the distance between
X(5, -4) and Y(-3, -2). Round to the nearest tenth if necessary.
d=
𝟓 − (−𝟑
𝟐
+ −𝟒 − (−𝟐
𝟐
d = 𝟖𝟐 + 𝟐𝟐
d = 𝟔𝟒 + 𝟒
d = 𝟔𝟖
d ≈ ± 8.2
This distance between the points is about 8.2 units.