Transcript Unit 3 Triangles

Unit 3
Triangles
Chapter Objectives
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Classification of Triangles by Sides
Classification of Triangles by Angles
Exterior Angle Theorem
Triangle Sum Theorem
Adjacent Sides and Angles
Parts of Specific Triangles
5 Congruence Theorems for Triangles
Lesson 3.1
Classifying Triangles
Lesson 3.1 Objectives
• Classify triangles according to their side
lengths. (G1.2.1)
• Classify triangles according to their angle
measures. (G1.2.1)
• Find a missing angle using the Triangle Sum
Theorem. (G1.2.2)
• Find a missing angle using the Exterior Angle
Theorem. (G1.2.2)
Classification of Triangles by Sides
Name
Equilateral
Isosceles
Scalene
3 congruent
sides
At least 2
congruent
sides
No Congruent
Sides
Looks Like
Characteristics
Classification of Triangles by Angles
Name
Acute
Equiangular
Right
Obtuse
3 acute
angles
3 congruent
angles
1 right
angles
1 obtuse
angle
Looks Like
Characteristics
Example 3.1
Classify the following triangles by their
sides and their angles.
Scalene
Obtuse
Scalene
Right
Equilateral
Equiangular
Isosceles
Acute
Vertex
• The vertex of a triangle is any point at
which two sides are joined.
– It is a corner of a triangle.
• There are 3 in every triangle
Adjacent Sides and Adjacent Angles
• Adjacent sides are
those sides that
intersect at a common
vertex of a polygon.
– These are said to be
adjacent to an angle.
• Adjacent angles are
those angles that are
right next to each other
as you move inside a
polygon.
– These are said to be
adjacent to a specific
side.
More Parts of Triangles
• If you were to extend the sides you will
see that more angles would be formed.
• So we need to keep them separate
– The three angles are called interior angles
because they are inside the triangle.
– The three new angles are called exterior
angles because they lie outside the triangle.
Theorem 4.1: Triangle Sum Theorem
• The sum of the measures of the interior
angles of a triangle is 180o.
B
mA + mB + mC = 180o
C
A
Example 3.2
Solve for x and then classify the triangle
based on its angles.
Acute
75o
50o
3x + 2x + 55 = 180
Triangle Sum Theorem
5x + 55 = 180
Simplify
5x = 125
SPOE
x = 25
DPOE
Example 3.3
Solve for x and classify each triangle by angle measure.
1. mA  ( x  30) o
( x  30)  x  ( x  60)  180
mB  x o
mC  ( x  60) o
mA  60o
mB  30o
mC  90
2. mA  (6 x  11)o
mB  (3 x  2)o
mC  (5 x  1)o
o
3x  90  180
3x  90
x  30
Right
(6 x  11)  (3x  2)  (5 x  1)  180
14x  12  180
14x  168
o
mA  83
x  12
o
mB  34
Acute
mC  59o
Example 3.4
Draw a sketch of the triangle described.
1.
Mark the triangle with symbols to indicate the
necessary information.
Acute Isosceles
2.
Equilateral
3.
Right Scalene
Example 3.5
Draw a sketch of the triangle described.
1.
Mark the triangle with specific angle measures,
side lengths, or symbols to indicate the necessary
information.
Obtuse Scalene
2.
Right Isosceles
3.
Right Equilateral
(Not Possible)
Theorem 4.2: Exterior Angle Theorem
• The measure of an exterior angle of a
triangle is equal to the sum of the
measures of the two nonadjacent
interior angles.
B
A
C
m A +m B = m C
Example 3.6
Solve for x
6 x  7  2 x  (103  x)
6x  7  x  103
5x  7  103
5x  110
x  22
Exterior Angles Theorem
Combine Like Terms
Subtraction Property
Addition Property
Division Property
Corollary to the Triangle Sum Theorem
• A corollary to a theorem is a statement that
can be proved easily using the original
theorem itself.
– This is treated just like a theorem or a postulate in
proofs.
• The acute angles in a right triangle are
complementary.
A
mA + mB = 90o
B
C
Example 3.7
If you don’t like the Exterior
Angle Theorem, then find
m2 first using the Linear
Pair Postulate.
Find the unknown angle measures.
1.
90o  53o  m1  180o 3.
143o  m1  180o
102o  m2  180o
m1  37 o
m2  78o
2.
90  42  m1  180
o
o
VA
o
132o  m1  180o
m1  48o
90  33  m2  180
o
o
123o  m2  180o
m2  57o
VA
m2  m3  122o
m1  34
o
78o  68o  m1  180o
68o  34o  m2  180o
o
4.
68o  m1  102o
Then find m1 using the
Angle Sum Theorem.
58o  m2  180o
m2  122o
102o  m2  180o
146o  m1  180o
m1  34o
m2  78o
122o  22o  m1  180o 122o  20o  m4  180o
144o  m1  180o
m1  36o
142o  m4  180o
m4  38o
Homework 3.1
• Lesson 3.1 – All Sections
– p1-6
• Due Tomorrow
Lesson 3.2
Inequalities in One Triangle
Lesson 3.2 Objectives
• Order the angles in a triangle from
smallest to largest based on given side
lengths. (G1.2.2)
• Order the side lengths of a triangle from
smallest to largest based on given
angle measures. (G1.2.2)
• Utilize the Triangle Inequality Theorem.
Theorem 5.10: Side Lengths of a Triangle Theorem
• If one side of a triangle is longer than
another side, then the angle opposite
the longer side is larger than the angle
opposite the shorter side.
– Basically, the larger the side, the larger the
angle opposite that side.
2nd Largest
Angle
Smallest
Side
Largest Angle
2nd Longest Side
Smallest
Angle
Theorem 5.11: Angle Measures of a Triangle Theorem
• If one angle of a triangle is larger than
another angle, then the side opposite
the larger angle is longer than the side
opposite the smaller angle.
– Basically, the larger the angle, the larger
the side opposite that angle.
2nd Largest
Angle
Smallest
Side
Largest Angle
2nd Longest Side
Smallest
Angle
Example 3.8
Order the angles from largest to smallest.
2.
1.
B, A, C
Q, P, R
3.
A, C , B
Example 3.9
Order the sides from largest to smallest.
1.
ST , RS , RT
2.
DE , EF , DF
33o
Example 3.10
Order the angles from largest to smallest.
1.
In ABC
AB = 12
BC = 11
C , A, B
AC = 5.8
Order the sides from largest to smallest.
2.
In XYZ
mX = 25o
XY , XZ , YZ
mY = 33o
mZ = 122o
Theorem 5.13: Triangle Inequality
• The sum of the lengths of any two sides
of a triangle is greater than the length of
the third side.
4
3
1
3
6
6
4
2
6
Add each combination of two sides to make sure
that they are longer than the third remaining side.
Example 12
Determine whether the following could be
lengths of a triangle.
a) 6, 10, 15
a) 6 + 10 > 15
10 + 15 > 6
6 + 15 > 10
YES!
b) 11, 16, 32
b) 11 + 16 < 32
NO!
Hint: A shortcut is to make sure that the sum of the two smallest
sides is bigger than the third side.
The other sums will always work.
Homework 3.2
• Lesson 3.2 – Inequalities in One Triangle
– p7-8
• Due Tomorrow
• Quiz Friday, October 15th
Lesson 3.3
Isosceles,
Equilateral,
and
Right Triangles
Lesson 3.3 Objectives
• Utilize the Base Angles Theorem to
solve for angle measures. (G1.2.2)
• Utilize the Converse of the Base Angles
Theorem to solve for side lengths.
(G1.2.2)
• Identify properties of equilateral
triangles to solve for side lengths and
angle measures. (G1.2.2)
Isosceles Triangle Theorems
•Theorem 4.6: Base
Angles Theorem
–If two sides of a
triangle are
congruent, then the
angles opposite them
are congruent.
•Theorem 4.7:
Converse of Base
Angles Theorem
–If two angles of a
triangle are
congruent, then the
sides opposite them
are congruent.
Example 10
Solve for x
Theorem 4.7
Theorem 4.6
4x + 3 = 15
7x + 5 = x + 47
4x = 12
x=3
6x + 5 = 47
6x = 42
x=7
Equilateral Triangles
•Corollary to Theorem 4.6
•Corollary to Theorem 4.7
–If a triangle is
equilateral, then it is
equiangular.
–If a triangle is
equiangular, then it is
equilateral.
Example 11
Solve for x
Corollary to Theorem 4.6
Corollary to Theorem 4.6
In order for a triangle to be
equiangular, all angles must
equal…
It does not matter which two sides you
set equal to each other, just pick the
pair that looks the easiest!
2x + 3 = 4x - 5
3 = 2x - 5
5x = 60
8 = 2x
x = 12
x=4
Homework 3.3
• Lesson 3.3 – Isosceles, Equilateral, and
Right Triangles
– p9-11
• Due Tomorrow
• Quiz Tomorrow
– Friday, October 15th
Lesson 5.3
Medians and Altitudes
of Triangles
Lesson 5.3 Objectives
•
•
•
•
Define a median of a triangle
Identify a centroid of a triangle
Define the altitude of a triangle
Identify the orthocenter of a triangle
Perpendicular Bisector
• A segment, ray, line, or plane that is perpendicular
to a segment at its midpoint is called the
perpendicular bisector.
Triangle Medians
• A median of a triangle is a segment
that does the following:
– Contains one endpoint at a vertex of the
triangle, and
– Contains the other endpoint at the
midpoint of the opposite side of the
triangle.
A
B
D
C
Centroid
Remember: All medians intersect the
midpoint of the opposite side.
• When all three medians are drawn in, they
intersect to form the centroid of a triangle.
– This special point of concurrency is the
balance point for any evenly distributed
triangle.
• In Physics, this is how we locate the center of mass.
Obtuse
Acute
Right
Theorem 5.7:
Concurrency of Medians of a Triangle
• The medians of a triangle intersect at
a point that is two-thirds of the
distance from each vertex to the
midpoint of the opposite side.
– The centroid is 2/3 the distance from
any vertex to the opposite side.
AP = 2/3AE
BP = 2/3BF
CP = 2/3CD
Example 6
S is the centroid of RTW, RS = 4, VW = 6, and TV = 9. Find the
following:
a)
RV
a)
b)
6
RU
b)
6
•
4 is 2/3 of 6
•
c)
Divide 4 by 2 and then muliply by 3. Works everytime!!
SU
c)
d)
2
RW
d)
e)
12
TS
e)
6
•
f)
6 is 2/3 of 9
SV
f)
3
Altitudes
• An altitude of a triangle is the perpendicular
segment from a vertex to the opposite side.
– It does not bisect the angle.
– It does not bisect the side.
• The altitude is often thought of as the height.
• While true, there are 3 altitudes in every triangle
but only 1 height!
Orthocenter
• The three altitudes of a triangle intersect at a
point that we call the orthocenter of the triangle.
• The orthocenter can be located:
– inside the triangle
– outside the triangle, or
– on one side of the triangle
Obtuse
Right
Acute
The orthocenter of a right
triangle will always be located at
the vertex that forms the right
angle.
Theorem 5.8:
Concurrency of Altitudes of a Triangle
• The lines containing the altitudes of
a triangle are concurrent.
Example 7
Is segment BD a median, altitude, or
perpendicular bisector of ABC?
Hint: It could be more than one!
Perpendicular
Bisector
Altitude
Median
None
Homework 3.4
• Lesson 3.4 – Altitudes and Medians
– p12-13
• Due Tomorrow
Lesson 1.7
Intro to Perimeter,
Circumference and
Area
Lesson 1.7 Objectives
• Find the perimeter and area of
common plane figures.
• Establish a plan for problem solving.
Perimeter and Area of a Triangle
• The area of a
triangle is half the
length of the base
• The perimeter can be
times the height of
found by adding the
the triangle.
three sides together.
a of a
– The height
– P=a+b+b
c
triangle
is the
h
• If the third side is
perpendicular
unknown, use the
length from the
Pythagorean Theorembase to the
c opposite vertex of
to solve for the
unknown side.
the triangle.
– a2 + b 2 = c 2
– A = ½bh
• Where a,b are the two
shortest sides and c is
the longest side.
Homework 3.5
• Lesson 3.5 – Area and Perimeter of
Triangles
– p14-15
• Due Tomorrow