december 2, 2015 5.1 angles of triangles - mesa public … measure of an exterior angle of a...
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Essential Question
How are the angle measures of a triangle
related?
5.1 Angles of TrianglesDecember 2, 2015
5.1 Angles of TrianglesDecember 2, 2015
Goals – Day 1
Classify triangles by their sides
Classify triangles by their angles
Identify parts of triangles.
Find angle measures in triangles.
5.1 Angles of TrianglesDecember 2, 2015
Triangle
A triangle is a figure formed by three
segments joining three noncollinear points.
A
B
C
This is ABC, which can also be named BCA,
CAB, BAC, CBA, or ACB.
5.1 Angles of TrianglesDecember 3, 2015
Classifying Triangles by Sides
Equilateral
Isosceles
Scalene
5.1 Angles of TrianglesDecember 3, 2015
Classifying Triangles by Angles
Acute
Equiangular
Right
Obtuse
5.1 Angles of TrianglesDecember 2, 2015
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.
5.1 Angles of TrianglesDecember 2, 2015
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.
A
B
C
5.1 Angles of TrianglesDecember 2, 2015
Adjacent Sides
Two sides that share a common vertex are
adjacent sides.
The third side is the opposite side.
R T
AIn RAT, RA and RT are
adjacent sides.
AT is the opposite side from
∠𝑅.
5.1 Angles of TrianglesDecember 2, 2015
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
5.1 Angles of TrianglesDecember 2, 2015
Right Triangle
The LEGS form the right angle.
The third side (opposite the right angle) is
the Hypotenuse.
Leg
Leg
5.1 Angles of TrianglesDecember 2, 2015
Hypotenuse
From the Greek “stretched against”.
Always longer than either leg.
5.1 Angles of TrianglesDecember 2, 2015
What have you learned so far?
In the figure, 𝑀𝑁 ⊥ 𝑄𝑃 and 𝑀𝑃 ≅𝑀𝑄. Complete the following sentence.
1. Name the legs of the
isosceles triangle PMQ.
Segments PM and QM.
P
Q
N M
5.1 Angles of TrianglesDecember 2, 2015
What have you learned so far?
In the figure, 𝑀𝑁 ⊥ 𝑄𝑃 and 𝑀𝑃 ≅𝑀𝑄. Complete the following sentence.
2. Name the base of
isosceles triangle PMQ.
Segment PQ.
P
Q
N M
5.1 Angles of TrianglesDecember 2, 2015
What have you learned so far?
In the figure, 𝑀𝑁 ⊥ 𝑄𝑃 and 𝑀𝑃 ≅𝑀𝑄. Complete the following sentence.
3. Name the hypotenuse of
right triangle PNM.
Segment PM.
P
Q
N M
5.1 Angles of TrianglesDecember 2, 2015
What have you learned so far?
In the figure, 𝑀𝑁 ⊥ 𝑄𝑃 and 𝑀𝑃 ≅𝑀𝑄. Complete the following sentence.
4. Name the legs of right
triangle PNM.
Segments NP and NM.
P
Q
N M
5.1 Angles of TrianglesDecember 2, 2015
What have you learned so far?
In the figure, 𝑀𝑁 ⊥ 𝑄𝑃 and 𝑀𝑃 ≅𝑀𝑄. Complete the following sentence.
5. Name the acute angles
of right triangle QNM.
Q and NMQ
P
Q
N M
5.1 Angles of TrianglesDecember 3, 2015
Example 1
Classify these triangles by its angles and by
its sides.
a. c. b.
125°
Right , Scalene
Obtuse ,
Isosceles
Equiangular, Equilateral
Isosceles , Acute
5.1 Angles of TrianglesDecember 2, 2015
Example 2
Complete the sentence with always,
sometimes, or never.
a. An isosceles triangle is ________ a right
triangle.
b. An obtuse triangle is ________ a right triangle.
c. A right triangle is ________ an equilateral
triangle.
d. A right triangle is ________ an isosceles
triangle.
Sometimes
Never
Never
Sometimes
5.1 Angles of TrianglesDecember 2, 2015
Important Triangle Theorems
5.1 Triangle Sum Theorem
5.2 Exterior Angle Theorem
5.1 Angles of TrianglesDecember 2, 2015
5.1 Triangle Sum Theorem
The sum of the measures of the interior
angles of a triangle is 180°.
A
B
C
mA + mB + mC = 180°
A Proof of the Triangle Sum Thm
1. Given
2. Draw line through point B parallel to AC
2. Parallel Postulate (3.1)
3. m4 + m3 + m5 = 180 3. Def. of Straight Angle
4. Alternate Interior ’s
1. ABC
4. m1 = m4 and m2 = m5
Statements Reasons
Given: ABC
Prove: m1 + m2 + m3 = 180°
December 2, 2015 5.1 Angles of Triangles
1 2
34 5
A
B
C
5. m1 + m3 + m2 = 180 5. Substitution
5.1 Angles of TrianglesDecember 2, 2015
Example 3
Find the measure of 1.
Solution:
m1 + 70 + 32 = 180
m1 + 102 = 180
m1 = 180 – 102
m1 = 78°
70° 32°
1
5.1 Angles of TrianglesDecember 2, 2015
Example 4In MAD:
mM = (2x)°
mA = (3x)°
mD = (4x)
Find the measure of each angle, and classify.
Solution:
2x + 3x + 4x = 180
9x = 180
x = 20
= 2(20) = 40°
= 3(20) = 60°
= 4(20) = 80°
This triangle is acute.
5.1 Angles of TrianglesDecember 2, 2015
Example 5
In RST:
mR=(5x + 10)
mS=(2x + 15)
mT=(3x + 35)
Find the measure of the three angles and
then classify the triangle by angles.
5.1 Angles of TrianglesDecember 2, 2015
Example 5 Solution
(5x + 10) + (2x + 15) + (3x + 35) = 180
10x + 60 = 180
10x = 120
x = 12
mR=(5x + 10) = 5(12) + 10 = 70
mS=(2x + 15) = 2(12) + 15 = 39
mT=(3x + 35) = 3(12) + 35 = 71
ACUTE
5.1 Angles of TrianglesDecember 2, 2015
Your Turn
In ABC:
mA=(x + 30)
mB=x
mC=(x + 60)
Find the measure of the three angles and
then classify the triangle by angles.
5.1 Angles of TrianglesDecember 2, 2015
Your Turn Solution
RIGHT
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°
In ABC:
mA=(x + 30)
mB=x
mC=(x + 60).
5.1 Angles of TrianglesDecember 2, 2015
Your Turn Again.
In ABC:
mA=(6x + 11)
mB=(3x + 2)
mC=(5x - 1)
Find the measure of the three angles and
then classify the triangle by angles.
5.1 Angles of TrianglesDecember 2, 2015
Your Turn Again Solution
ACUTE
6𝑥 + 11 + (3𝑥 + 2) + 5𝑥 − 1 = 180
14𝑥 + 12 = 180
14𝑥 = 168
x = 12
m∠𝐴 = 6(12) + 11 = 83°
m∠𝐵 = 3 12 + 2 = 38°m∠𝐶 = 5 12 − 1 = 59°
In ABC:
mA=(6x + 11)
mB=(3x + 2)
mC=(5x - 1).
Essential Question
How are the angle measures of a triangle
related?
5.1 Angles of TrianglesDecember 2, 2015
5.1 Angles of TrianglesDecember 2, 2015
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
5.1 Angles of TrianglesDecember 2, 2015
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.
5.1 Angles of TrianglesDecember 2, 2015
Corollary to Theorem 5.1
The acute angles of a right triangle are
complementary.
1
2
m1 + m2 + 90 = 180
m1 + m2 = 90
QED
5.1 Angles of TrianglesDecember 2, 2015
Example 1Find X
20°
x°
x = 70°
Since this is a right triangle, the
acute angles are complementary,
and 90 – 20 = 70.
5.1 Angles of TrianglesDecember 2, 2015
Extend the
sides….
1
2
3
1, 2, 3 are INTERIOR ANGLES.
They are INSIDE the triangle.
5.1 Angles of TrianglesDecember 2, 2015
1
2
3
4, 6, 8, 9, 10, and 12 are
EXTERIOR ANGLES.
They are OUTSIDE the triangle.
They are ADJACENT to the interior
angles.
4
6
8 9
10
12
5.1 Angles of TrianglesDecember 2, 2015
1
2
3
5, 7, and 11 are NOT EXTERIOR
ANGLES.
They are simply vertical angles to the
interior angles.
5
7
11
5.1 Angles of TrianglesDecember 2, 2015
It is common (and less confusing) to draw
only one exterior angle at a vertex.
1 2
3
4
5
6
Interior Angles: 1, 2, 3
Exterior Angles: 4, 5, 6
Exterior angles are always supplementary to the interior angles.
5.1 Angles of TrianglesDecember 2, 2015
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.
12
3
m1 = m2 + m3
5.1 Angles of TrianglesDecember 2, 2015
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.
5.1 Angles of TrianglesDecember 2, 2015
5.2 Exterior Angle Thm Proof (Informal)
12
3
4
m2 + m3 + m4 = 180 ( angle sum)
m4 + m1 = 180 (linear pair postulate)
m2 + m3 + m4 = m4 + m1 (substitution)
m2 + m3 = m1 (subtraction)
5.1 Angles of TrianglesDecember 2, 2015
Naming Remote Interior Angles
1 2
3
46
5
8 7
9
For exterior 1, the remote
interior angles
are_____________.6 & 8
5.1 Angles of TrianglesDecember 2, 2015
Naming Remote Interior Angles
1 2
3
46
5
8 7
9
For exterior 4, the remote
interior angles
are_____________.2 & 8
5.1 Angles of TrianglesDecember 2, 2015
Naming Remote Interior Angles
1 2
3
46
5
8 7
9
For exterior 5, the remote
interior angles
are_____________.2 & 8
5.1 Angles of TrianglesDecember 2, 2015
Naming Remote Interior Angles
1 2
3
46
5
8 7
9
For exterior 9, the remote
interior angles
are_____________.2 & 6
5.1 Angles of TrianglesDecember 2, 2015
Naming Remote Interior Angles
1 2
3
46
5
8 7
9
For remote interior angles
6 & 8, the exterior angle
is _____________.1 or 3
5.1 Angles of TrianglesDecember 2, 2015
Naming Remote Interior Angles
1 2
3
46
5
8 7
9
For remote interior angles
2 & 6, the exterior angle
is _____________.7 or 9
5.1 Angles of TrianglesDecember 2, 2015
Naming Remote Interior Angles
1 2
3
46
5
8 7
9
For remote interior angles
2 & 8, the exterior angle
is _____________.4 or 5
5.1 Angles of TrianglesDecember 2, 2015
Example 2
110°1
45°
Find m1.
By Theorem 5.2:
m1 + 45 = 110
m1 = 110 – 45 = 65°
5.1 Angles of TrianglesDecember 2, 2015
Example 3
(x + 15)° (3x – 10)°
45°
Solve for x.
(x + 15) + 45 = 3x – 10
x + 60 = 3x – 10
70 = 2x
x = 35
5.1 Angles of TrianglesDecember 2, 2015
Problems for You
Use the exterior angle theorem!
Write down the equation for each problem
and solve.
5.1 Angles of TrianglesDecember 2, 2015
Your Turn.
1. Find m1
1125
32
Solution:
m1 = 32 + 125
m1 = 157
5.1 Angles of TrianglesDecember 2, 2015
3. Solve for x.
(2x + 30)° 60
110°
Solution:
2x + 30 + 60 = 110
2x + 90 = 110
2x = 20
x = 10
5.1 Angles of TrianglesDecember 2, 2015
4. Solve for x.
(5x) (12x – 4)
(6x + 8)
Solution:
12x – 4 = (6x + 8) + 5x
12x – 4 = 11x + 8
x = 12
5.1 Angles of TrianglesDecember 2, 2015
5. Solve for x.
(3x + 2)
(5x – 10)
(7x + 3)
Solution:
(3x + 2) + (5x – 10) = 7x + 3
8x – 8 = 7x + 3
x = 11
5.1 Angles of TrianglesDecember 2, 2015
A Final Challenge Problem…
Find the measure of each numbered angle.
40°
30°
60°
20°
1
2 3
4 5
6 7
50°
90°
60°
60° 60°
60°
100°
5.1 Angles of TrianglesDecember 2, 2015
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.