Download 5.1 Angles of Triangles December 2, 2015

December 2, 2015
5.1 Angles of Triangles
Geometry
5.1 Angles of Triangles
Essential Question
How are the angle measures of a triangle
related?
December 2, 2015
5.1 Angles of Triangles
Goals – Day 1
Classify triangles by their sides
 Classify triangles by their angles
 Identify parts of triangles.
 Find angle measures in triangles.

December 2, 2015
5.1 Angles of Triangles
Triangle Symbol

Use the picture  for triangle.
December 2, 2015
5.1 Angles of Triangles
Triangle

A triangle is a figure formed by three
segments joining three noncollinear points.
B
C
A
This is ABC, which can also be named BCA,
CAB, BAC, CBA, or ACB.
December 2, 2015
5.1 Angles of Triangles
Classifying Triangles by Sides
Equilateral 
 Isosceles 
 Scalene 

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5.1 Angles of Triangles
Equilateral Triangle

Three congruent sides.
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5.1 Angles of Triangles
Isosceles Triangle

At least two congruent sides.
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5.1 Angles of Triangles
Scalene Triangle

No congruent sides.
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5.1 Angles of Triangles
Classifying Triangles by Angles
Acute 
 Equiangular 
 Right 
 Obtuse 

December 3, 2015
5.1 Angles of Triangles
Acute Triangle

Three acute angles
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5.1 Angles of Triangles
Equiangular Triangle

Three Congruent Angles
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5.1 Angles of Triangles
Right Triangle

One Right Angle
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5.1 Angles of Triangles
Obtuse Triangle

One Obtuse Angle
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5.1 Angles of Triangles
And to add to the confusion…
An equilateral triangle is also equiangular.
 An equiangular triangle is also acute.
 An equilateral can be considered an
isosceles triangle.
 An equilateral triangle is also acute.

December 2, 2015
5.1 Angles of Triangles
Vertex
Each of the three points joining the sides
of a triangle is a vertex.
 There are three vertices in each triangle.
 Points A, B, and C are the vertices.

B
C
A
December 2, 2015
5.1 Angles of Triangles
Adjacent Sides
Two sides that share a common vertex are
adjacent sides.
 The third side is the opposite side.

A
In RAT, RA and RT are
adjacent sides.
AT is the opposite side from
∠𝑅.
R
December 2, 2015
T
5.1 Angles of Triangles
Isosceles Triangles
(In this case, we consider an isosceles
triangle with only two congruent sides.)
 The congruent sides are the LEGS.
 The third side is the BASE.

Leg
Leg
Base
December 2, 2015
5.1 Angles of Triangles
Right Triangle
The LEGS form the right angle.
 The third side (opposite the right angle) is
the Hypotenuse.

Leg
Leg
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5.1 Angles of Triangles
Hypotenuse
From the Greek “stretched against”.
 Always longer than either leg.

December 2, 2015
5.1 Angles of Triangles
What have you learned so far?
In the figure, 𝑀𝑁 ⊥ 𝑄𝑃 and 𝑀𝑃 ≅
𝑀𝑄. Complete the following sentence.
P
1. Name the legs of the
isosceles triangle PMQ.
N
Segments PM and QM.
Q
December 2, 2015
5.1 Angles of Triangles
M
What have you learned so far?
In the figure, 𝑀𝑁 ⊥ 𝑄𝑃 and 𝑀𝑃 ≅
𝑀𝑄. Complete the following sentence.
P
2. Name the base of
isosceles triangle  PMQ.
N
Segment PQ.
Q
December 2, 2015
5.1 Angles of Triangles
M
What have you learned so far?
In the figure, 𝑀𝑁 ⊥ 𝑄𝑃 and 𝑀𝑃 ≅
𝑀𝑄. Complete the following sentence.
P
3. Name the hypotenuse of
right triangle PNM.
N
Segment PM.
Q
December 2, 2015
5.1 Angles of Triangles
M
What have you learned so far?
In the figure, 𝑀𝑁 ⊥ 𝑄𝑃 and 𝑀𝑃 ≅
𝑀𝑄. Complete the following sentence.
P
4. Name the legs of right
triangle  PNM.
N
Segments NP and NM.
December 2, 2015
5.1 Angles of Triangles
Q
M
What have you learned so far?
In the figure, 𝑀𝑁 ⊥ 𝑄𝑃 and 𝑀𝑃 ≅
𝑀𝑄. Complete the following sentence.
P
5. Name the acute angles
of right triangle  QNM.
N
Q and NMQ
Q
December 2, 2015
5.1 Angles of Triangles
M
Example 1
Classify these triangles by its angles and by
its sides.
a.
c.
b.
125°
Right , Scalene 
December 3, 2015
Obtuse ,
Equiangular, Equilateral 
Isosceles 
Isosceles , Acute 
5.1 Angles of Triangles
Example 2
Complete the sentence with always,
sometimes, or never.
Sometimes a right
a. An isosceles triangle is ________
triangle.
b. An obtuse triangle is ________
a right triangle.
Never
c. A right triangle is ________
an equilateral
Never
triangle.
Sometimes an isosceles
d. A right triangle is ________
triangle.
December 2, 2015
5.1 Angles of Triangles
Important Triangle Theorems
5.1 Triangle Sum Theorem
 5.2 Exterior Angle Theorem

December 2, 2015
5.1 Angles of Triangles
5.1 Triangle Sum Theorem
The sum of the measures of the interior
angles of a triangle is 180°.
B
mA + mB + mC = 180°
A
December 2, 2015
C
5.1 Angles of Triangles
A Proof of the Triangle Sum Thm
B
Given:  ABC
Prove:
4
3
5
m1 + m2 + m3 = 180°
A
Statements
1.  ABC
2. Draw line through point B
parallel to AC
3. m4 + m3 + m5 = 180
1
Reasons
2
C
1. Given
2. Parallel Postulate (3.1)
4. m1 = m4 and m2 = m5
3. Def. of Straight Angle
4. Alternate Interior ’s
5. m1 + m3 + m2 = 180
5. Substitution
December 2, 2015
5.1 Angles of Triangles
Example 3
Find the measure of 1.
Solution:
1
m1 + 70 + 32 = 180
m1 + 102 = 180
70°
m1 = 180 – 102
m1 = 78°
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5.1 Angles of Triangles
32°
Example 4
In MAD:
mM = (2x)° = 2(20) = 40°
mA = (3x)° = 3(20) = 60°
mD = (4x) = 4(20) = 80°
Find the measure of each angle, and classify.
Solution:
2x + 3x + 4x = 180
This triangle is acute.
9x = 180
x = 20
December 2, 2015
5.1 Angles of Triangles
Example 5
In RST:
mR=(5x + 10)
mS=(2x + 15)
mT=(3x + 35)
Find the measure of the three angles and
then classify the triangle by angles.
December 2, 2015
5.1 Angles of Triangles
Example 5 Solution
ACUTE
(5x + 10) + (2x + 15) + (3x + 35) = 180
10x + 60 = 180
10x = 120
x = 12
mR=(5x + 10) = 5(12) + 10 = 70
mS=(2x + 15) = 2(12) + 15 = 39
mT=(3x + 35) = 3(12) + 35 = 71
December 2, 2015
5.1 Angles of Triangles
Your Turn
In ABC:
mA=(x + 30)
mB=x
mC=(x + 60)
Find the measure of the three angles and
then classify the triangle by angles.
December 2, 2015
5.1 Angles of Triangles
Your Turn Solution
In ABC:
mA=(x + 30)
mB=x
mC=(x + 60).
𝑥 + 30 + 𝑥 + 𝑥 + 60 = 180
3𝑥 + 90 = 180
3𝑥 = 90
x = 30
m∠𝐴 = 30 + 30 = 60°
m∠𝐵 = 30°
m∠𝐶 = 30 + 60 = 90°
December 2, 2015
5.1 Angles of Triangles
RIGHT
Your Turn Again.
In ABC:
mA=(6x + 11)
mB=(3x + 2)
mC=(5x - 1)
Find the measure of the three angles and
then classify the triangle by angles.
December 2, 2015
5.1 Angles of Triangles
Your Turn Again Solution




6𝑥 + 11 + (3𝑥 + 2) + 5𝑥 − 1 = 180
In ABC:
mA=(6x + 11)
14𝑥 + 12 = 180
mB=(3x + 2)
14𝑥 = 168
mC=(5x - 1).
x = 12
m∠𝐴 = 6(12) + 11 = 83°
m∠𝐵 = 3 12 + 2 = 38°
m∠𝐶 = 5 12 − 1 = 59°
December 2, 2015
5.1 Angles of Triangles
ACUTE
Assignment
December 2, 2015
5.1 Angles of Triangles
December 2, 2015
5.1 Angles of Triangles
Geometry
5.1 Angles of Triangles
Essential Question
How are the angle measures of a triangle
related?
December 2, 2015
5.1 Angles of Triangles
5.1 Day 2
Yesterday:
 The Interior Angle Theorem: the sum of
the interior angles of a triangle is 180°.
Today:
 The Exterior Angle Theorem
December 2, 2015
5.1 Angles of Triangles
But First…
A corollary to the interior angle theorem.
 A corollary is a theorem that can be
proved easily from another theorem.
 Not “big” enough to warrant title of
theorem.
 A corollary follows from a theorem.
December 2, 2015
5.1 Angles of Triangles
Corollary to Theorem 5.1
The acute angles of a right triangle are
complementary.
m1 + m2 + 90 = 180
1
m1 + m2 = 90
QED
2
December 2, 2015
5.1 Angles of Triangles
Example 1
Find X
x = 70°
20°
Since this is a right triangle, the
acute angles are complementary,
and 90 – 20 = 70.
x°
December 2, 2015
5.1 Angles of Triangles
Interior and Exterior Angles
Start with a
triangle…
December 2, 2015
5.1 Angles of Triangles
Extend the
sides….
2
1
3
1, 2, 3 are INTERIOR ANGLES.
They are INSIDE the triangle.
December 2, 2015
5.1 Angles of Triangles
9
8
2
1
4
3
6
10
12
4, 6, 8, 9, 10, and 12 are
EXTERIOR ANGLES.
They are OUTSIDE the triangle.
They are ADJACENT to the interior
angles.
December 2, 2015
5.1 Angles of Triangles
7
2
1
3
5
11
5, 7, and 11 are NOT EXTERIOR
ANGLES.
They are simply vertical angles to the
interior angles.
December 2, 2015
5.1 Angles of Triangles
It is common (and less confusing) to draw
only one exterior angle at a vertex.
Exterior angles are always supplementary to
the interior angles.
6
3
1
2
5
4
Interior Angles: 1, 2, 3
Exterior Angles: 4, 5, 6
December 2, 2015
5.1 Angles of Triangles
5.2 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.
3
2
1
m1 = m2 + m3
December 2, 2015
5.1 Angles of Triangles
Note:
Sometimes (usually) the two nonadjacent
interior angles are referred to as REMOTE
INTERIOR ANGLES. The theorem then
reads:
 An exterior angle of a triangle is equal to
the sum of the two remote interior angles.

December 2, 2015
5.1 Angles of Triangles
5.2 Exterior Angle Thm Proof (Informal)
m2 + m3 + m4 = 180 ( angle sum)
m4 + m1 = 180 (linear pair postulate)
m2 + m3 + m4 = m4 + m1 (substitution)
m2 + m3 = m1 (subtraction)
3
2
December 2, 2015
4
5.1 Angles of Triangles
1
Naming Remote Interior Angles
4
1 2
3
December 2, 2015
5
For exterior 1, the remote
interior angles
6 & 8
are_____________.
6
8
7
9
5.1 Angles of Triangles
Naming Remote Interior Angles
4
1 2
3
December 2, 2015
5
For exterior 4, the remote
interior angles
2 & 8
are_____________.
6
8
7
9
5.1 Angles of Triangles
Naming Remote Interior Angles
4
1 2
3
December 2, 2015
5
For exterior 5, the remote
interior angles
2 & 8
are_____________.
6
8
7
9
5.1 Angles of Triangles
Naming Remote Interior Angles
4
1 2
3
December 2, 2015
5
For exterior 9, the remote
interior angles
2 & 6
are_____________.
6
8
7
9
5.1 Angles of Triangles
Naming Remote Interior Angles
4
1 2
3
December 2, 2015
5
For remote interior angles
6 & 8, the exterior angle
1 or 3
is _____________.
6
8
7
9
5.1 Angles of Triangles
Naming Remote Interior Angles
4
1 2
3
December 2, 2015
5
For remote interior angles
2 & 6, the exterior angle
7 or 9
is _____________.
6
8
7
9
5.1 Angles of Triangles
Naming Remote Interior Angles
4
1 2
3
December 2, 2015
5
For remote interior angles
2 & 8, the exterior angle
4 or 5
is _____________.
6
8
7
9
5.1 Angles of Triangles
Example 2
Find m1.
45°
By Theorem 5.2:
m1 + 45 = 110
1
December 2, 2015
110°
m1 = 110 – 45 = 65°
5.1 Angles of Triangles
Example 3
(x + 15) + 45 = 3x – 10
45°
x + 60 = 3x – 10
70 = 2x
(x + 15)°
(3x – 10)° x = 35
Solve for x.
December 2, 2015
5.1 Angles of Triangles
Problems for You
Use the exterior angle theorem!
 Write down the equation for each problem
and solve.

December 2, 2015
5.1 Angles of Triangles
Your Turn.
1. Find m1
Solution:
m1 = 32 + 125
m1 = 157
32
1
125
December 2, 2015
5.1 Angles of Triangles
Solution:
2. Find m2
m2 + 45 = 165
m2 = 120
45
2
December 2, 2015
165
5.1 Angles of Triangles
3. Solve for x.
Solution:
2x + 30 + 60 = 110
110°
2x + 90 = 110
2x = 20
(2x + 30)° 60
x = 10
December 2, 2015
5.1 Angles of Triangles
Solution:
4. Solve for x.
12x – 4 = (6x + 8) + 5x
12x – 4 = 11x + 8
x = 12
(6x + 8)
(5x)
December 2, 2015
(12x – 4)
5.1 Angles of Triangles
Solution:
5. Solve for x. (3x + 2) + (5x – 10) = 7x + 3
8x – 8 = 7x + 3
(3x + 2)
x = 11
(7x + 3)
(5x – 10)
December 2, 2015
5.1 Angles of Triangles
A Final Challenge Problem…
Find the measure of each numbered angle.
1
50°
60°
4
60°
40°
5
60°
90°
2 60°
3
20°
30°
December 2, 2015
6 100°
7
60°
5.1 Angles of Triangles
Summary
The sum of the interior angles of a triangle
is 180 degrees.
 The acute angles of a right triangle are
complementary.
 An exterior angle is equal to the sum of
the two remote interior angles.

December 2, 2015
5.1 Angles of Triangles
Assignment
December 2, 2015
5.1 Angles of Triangles