2.4 how can i use it? pg. 16 angles in a triangle
TRANSCRIPT
2.4
How Can I Use It?Pg. 16
Angles In a Triangle
2.4 – How Can I Use It?_____________Angles In a Triangle
So far in this chapter, you have investigated the angle relationships created when two lines intersect, forming vertical angles. You have also investigated the relationships created when a transversal intersects two parallel lines. Today you will study the angle relationships that result when three non-parallel lines intersect, forming a triangle.
2.23 – ANGLE RELATIONSHIPSMarcos decided to study the angle relationships in triangles by making a tiling. Find the pattern below.
a. Color in one of the angles with a pen or pencil. Then use the same color to shade every angle on the pattern that is equal to the shaded angle.
b. Repeat this process for the other two angles of the triangle, using a different color for each angle in the triangle. When you are done, every angle in your tiling should be shaded with one of the three colors.
c. Now examine your colored tiling. What relationship can you find between the three different-colored angles? You may want to focus on the angles that form a straight angle. What does this tell you about the angles in a triangle?
"If a polygon is a triangle, then the sum
of the interior angles is ___________"180°
d. Let us see if this works for any triangle. Each team member will cut out a different type of triangle: isosceles, scalene, right, or obtuse. Rip off the angles of the triangle and put them together to form a straight line. Do these three angles all add to 180°?
e. How can you convince yourself that your conjecture is true for all triangles? Match the reasons to the proof below to show that the sum of the interior angles of any triangle adds to 180°.
Statements Reasons1. 1.2. 2.3. 3.4. 4.
a d
c e 180d b e
180a b c
Alternate interior = Alternate interior =Supplementarysubstitution
2.26 – TRIANGLE ANGLE RELATIONSHIPS
Use your proof about the angles in a triangle to find x in each diagram below.
x + 40 + 80 = 180
x + 120 = 180
x = 60°
2x + x + 12 + 96 = 180
3x + 108 = 180
x = 24°3x = 72
Type of ∆ Definition Picture
Equilateral Triangle
CL
AS
SIF
ICA
TIO
N B
Y S
IDE
S
All sides are congruent
Type of ∆ Definition Picture
Isosceles Triangle
CL
AS
SIF
ICA
TIO
N B
Y S
IDE
S
2 sides congruent
base
leg leg
Type of ∆ Definition Picture
Scalene Triangle
CL
AS
SIF
ICA
TIO
N B
Y S
IDE
S
No sides are congruent
Type of ∆ Definition Picture
EquiangularTriangle
CL
AS
SIF
ICA
TIO
N B
Y A
NG
LE
S
All angles are congruent
Type of ∆ Definition Picture
Acute Triangle
CL
AS
SIF
ICA
TIO
N B
Y A
NG
LE
S
All angles are acute
Type of ∆ Definition Picture
Right Triangle
CL
AS
SIF
ICA
TIO
N B
Y A
NG
LE
S
One right angle leg
leghyp
otenuse
Type of ∆ Definition Picture
Obtuse Triangle
CL
AS
SIF
ICA
TIO
N B
Y A
NG
LE
S
One obtuse angle and 2 acute angles
Classify the triangle by the best definition based on sides and angles.
scalene
right
equilateral
equiangular
isosceles
obtuse
2.27 – TRIANGLE SUM THEOREMWhat can the Triangle Angle Sum Theorem help you learn about special triangles?
a. Find the measure of each angle in an equilateral triangle. Justify your conclusion.
180°3
60° 60°
60°
b. Consider the isosceles right triangle at right. Find the measures of all the angles.
180 – 902
45°
45°
c. What if you only know one angle of an isosceles triangle? For example, if what are the measures of the other two angles?
34 ,m A
34°180 – 34
2
73° 73°
d. Use the fact that when the triangle is isosceles, the base angles are congruent to solve for x and y.
3x – 11 = 2x + 11
x – 11 = 11
x = 22
2(22)+11
5555
2y + 110 = 180
2y = 70
y = 35
9x – 8 = 28
9x = 36
x = 4
2.28 –EXTERIOR ANGLE OF TRIANGLES a. Compare the interior and exterior angle of the following triangles at right. What do you notice?
70°
55°
ba
c d
a+b+c = 180
Exterior Angle of a Triangle
c+d = 180a+b+c = c+d
a+b = d
b. Find the measure of the exterior angles.
50 + 70 = x120 = x
5x + 45 = 6x + 3045 = x + 30
15 = x
6(15) + 30
120°
2.29 – ANGLE AND LINE RELATIONSHIPS Use your knowledge of angle relationships to answer the questions below. a. In the diagram at right, what is the sum of the angles x and y?
180°
b. While looking at the diagram below, Rianna exclaimed, "I think something is wrong with this diagram." What do you think she is referring to? Be prepared to share your ideas.
E
0°
Yes,
112° + 68° = 180°
d. Write a conjecture based on your conclusion to this problem.
"If the measures of same-side interior angles are __________________, then the lines are _____________."
supplementaryparallel
e. State the angle relationship shown. Then find the value of x that makes the lines parallel.
Alternate interior
4x – 10 = 3x + 13
x – 10 = 13
x = 23
corresponding
8x – 14 = 7x + 2
x – 14 = 2
x = 16
Consecutive interior
20x – 2 + 6x = 18026x – 2 = 180
26x = 182x = 7
Alternate exterior
6x + 17 = 143
6x = 126
x = 21