Download 4.1 Triangles and Angles

Pre-AP Bellwork 10-18
2) Find the value of x for which l ll m
m
l
27x°
19x°
17x°
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3-3 Parallel Lines and
the Triangle angle-sum
theorem
Geometry
Triangle Angle-Sum Theorem
The sum of the angle measures in a
triangle equal 180°
1
2
3
m<1 + m<2 + m<3 = 180°
Names of triangles:
classified by angles
Equiangular
—all angles
congruent
AcuteTriangle
—all angles
acute
Right
Triangle—
one right
angle
Obtuse
Triangle—
one obtuse
angle
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Names of triangles:
classified by sides
Equilateral
—3
congruent
sides
Isosceles
Triangle—at
least 2
congruent
sides
Scalene—
no
congruent
sides
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Classify this triangle by
its angles….
m ABC = 70.26 
m CAB = 41.76 
m BCA = 67.97 
B
C
A
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Exterior Angle Theorem
The measure of an exterior angle in a
triangle is the sum of the measures of
the 2 remote interior angles
remote
interior
angles
exterior
angle
2
1
3
m<4 = m<1 + m<2
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Ex. 2 Finding an Angle
Measure.
Exterior Angle theorem: m1 = m A
+m 1
x + 65 = (2x + 10)
65 = x +10
65
x
55 = x
(2x+10)
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Parts of a triangle
• Each of the three
points joining the
sides of a triangle is a
vertex.(plural:
vertices). A, B and C
are vertices.
• Two sides sharing a
common vertext are
adjacent sides.
• The third is the side
opposite an angle
Side
opposite
A
B
C
adjacent
adjacent
A
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Right Triangle
hypotenuse
• Red represents the
hypotenuse of a
right triangle. The
sides that form
the right angle are
leg
the legs.
leg
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Isosceles Triangles
• An isosceles triangle
can have 3 congruent
sides in which case it
is equilateral. When
an isosceles triangle
has only two congruent
sides, then these two base
sides are the legs of
the isosceles triangle.
The third is the base.
leg
leg
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Identifying the parts of
an isosceles triangle
•
About 7 ft.
5 ft
•
5 ft
Explain why ∆ABC is
an isosceles right
triangle.
In the diagram you
are given that C is
a right angle. By
definition, then
∆ABC is a right
triangle. Because AC
= 5 ft and BC = 5 ft;
AC BC. By
definition, ∆ABC is
also an isosceles
triangle.
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Identifying the parts of
an isosceles triangle
Hypotenuse & Base
•
About 7 ft.
•
5 ft
leg
5 ft
leg
Identify the legs and
the hypotenuse of
∆ABC. Which side is
the base of the
triangle?
Sides AC and BC are
adjacent to the right
angle, so they are
the legs. Side AB is
opposite the right
angle, so it is t he
hypotenuse. Because
AC BC, side AB is
also the base.
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Using Angle Measures of
Smiley faces are
Triangles
interior angles and
hearts represent the
exterior angles
B
A
C
Each vertex has a pair
of congruent exterior
angles; however it is
common to show only
one exterior angle at
each vertex.
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Ex. 3 Finding an Angle
Measure.
Exterior Angle theorem: m1 = m A
+m 1
x + 65 = (2x + 10)
65 = x +10
65
x
55 = x
(2x+10)
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Finding angle measures
• Corollary to the
triangle sum
theorem
• The acute angles
of a right triangle
are complementary.
• m A + m B = 90
2x
x
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Finding angle measures
X + 2x = 90
3x = 90
X = 30
• So m A = 30 and
the m B=60
B
2x
C
x
A
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