8-4 Triangles
Preview
Warm Up
California Standards
Lesson Presentation
Holt CA Course 1
8-4 Triangles
Warm Up
Solve each equation.
1. 62 + x + 37 = 180 x = 81
2. x + 90 + 11 = 180
x = 79
3. 2x + 18 = 180
x = 81
4. 180 = 3x + 72
x = 36
Holt CA Course 1
8-4 Triangles
California
Standards
MG3.3 Know and understand the
Pythagorean theorem and its converse and use it
to find the length of the missing side of a right
triangle and lengths of other line segments and,
in some situations, empirically verify the
Pythagorean theorem by direct measurement.
Also covered: Review of
6MG2.2
Holt CA Course 1
8-4 Triangles
Vocabulary
Triangle Sum Theorem
acute triangle
right triangle
obtuse triangle
equilateral triangle
isosceles triangle
scalene triangle
Holt CA Course 1
8-4 Triangles
An equilateral triangle has 3 congruent
sides and 3 congruent angles. An isosceles
triangle has at least 2 congruent sides and
2 congruent angles. A scalene triangle has
no congruent sides and no congruent
angles.
Holt CA Course 1
8-4 Triangles
If you tear off two corners of a triangle
and place them next to the third
corner, the three angles seem to form
a straight line. You can also show this
in a drawing.
Holt CA Course 1
8-4 Triangles
Draw a triangle and extend one side.
Then draw a line parallel to the
extended side, as shown.
Two sides of
the triangle
are
transversals to
the parallel
lines.
The three angles in the triangle can be
arranged to form a straight line or 180°.
Holt CA Course 1
8-4 Triangles
An acute triangle has 3 acute angles. A
right triangle has 1 right angle. An obtuse
triangle has 1 obtuse angle.
Holt CA Course 1
8-4 Triangles
Additional Example 1: Finding Angles in Acute,
Right and Obtuse Triangles
A. Find p in the acute triangle.
Triangle Sum
73° + 44° + p° = 180°
Theorem
117 + p = 180
–117
–117
p = 63
Holt CA Course 1
Subtract 117
from both
sides.
8-4 Triangles
Additional Example 1: Finding Angles in Acute,
Right, and Obtuse Triangles
B. Find m in the obtuse triangle.
Triangle Sum
23° + 62° + m° = 180°
Theorem
85 + m = 180
–85
–85
m = 95
Holt CA Course 1
Subtract 85
from both
sides.
62
23
m
8-4 Triangles
Check It Out! Example 1
A. Find a in the acute triangle.
Triangle Sum
88° + 38° + a° = 180°
Theorem
126 + a = 180
–126
–126
a = 54
Holt CA Course 1
Subtract 126
from both
sides.
38°
a°
88°
8-4 Triangles
Check It Out! Example 1
B. Find c in the obtuse triangle.
Triangle Sum
24° + 38° + c° = 180°
Theorem.
24°
62 + c = 180
Subtract 62
–62
–62
from both
c = 118
sides.
Holt CA Course 1
38°
c°
8-4 Triangles
Additional Example 2: Finding Angles in Equilateral,
Isosceles, and Scalene Triangles
A. Find the angle measures in the isosceles
triangle.
62° + t° + t° = 180°
62 + 2t = 180
–62
–62
Triangle Sum Theorem
Simplify.
Subtract 62 from both sides.
2t = 118
2t = 118
2
2
t = 59
Divide both
sides by 2.
The angles labeled t° measure 59°.
Holt CA Course 1
8-4 Triangles
Additional Example 2: Finding Angles in
Equilateral, Isosceles, and Scalene Triangles
B. Find the angle measures in the scalene
triangle.
2x° + 3x° + 5x° = 180°
10x = 180
10
10
Triangle Sum Theorem
Simplify.
Divide both sides by 10.
x = 18
The angle labeled 2x° measures
2(18°) = 36°, the angle labeled
3x° measures 3(18°) = 54°, and
the angle labeled 5x° measures
5(18°) = 90°.
Holt CA Course 1
8-4 Triangles
Check It Out! Example 2
A. Find the angle measures in the isosceles
triangle.
39° + t° + t° = 180° Triangle Sum Theorem
Simplify.
39 + 2t = 180
–39
–39
Subtract 39 from both sides.
2t = 141
2t = 141
2
2
t = 70.5
Divide both
sides by 2
The angles labeled t° measure 70.5°.
Holt CA Course 1
39°
t°
t°
8-4 Triangles
Check It Out! Example 2
B. Find the angle measures in the scalene
triangle.
3x° + 7x° + 10x° = 180° Triangle Sum Theorem
20x = 180
20
20
x=9
Simplify.
Divide both sides by 20.
The angle labeled 3x° measures
3(9°) = 27°, the angle labeled 7x°
measures 7(9°) = 63°, and the
angle labeled 10x° measures
10(9°) = 90°.
3x°
Holt CA Course 1
10x°
7x°
8-4 Triangles
Additional Example 3: Finding Angles in a Triangle
that Meets Given Conditions
The second angle in a triangle is six times
as large as the first. The third angle is half
as large as the second. Find the angle
measures and draw a possible figure.
Let x° = the first angle measure. Then 6x° =
second angle measure, and 1 (6x°) = 3x° =
2
third angle measure.
Holt CA Course 1
8-4 Triangles
Additional Example 3 Continued
The second angle in a triangle is six times
as large as the first. The third angle is half
as large as the second. Find the angle
measures and draw a possible figure.
x° + 6x° + 3x° = 180°
10x = 180
10
10
x = 18
Holt CA Course 1
Triangle Sum Theorem
Simplify.
Divide both sides by 10.
8-4 Triangles
Additional Example 3 Continued
The second angle in a triangle is six times
as large as the first. The third angle is half
as large as the second. Find the angle
measures and draw a possible figure.
x° = 18°
6 • 18° = 108°
3 • 18° = 54°
Holt CA Course 1
The angles measure 18°,
108°, and 54°. The triangle
is an obtuse scalene
triangle.
8-4 Triangles
Check It Out! Example 3 Continued
The second angle in a triangle is three
times larger than the first. The third
angle is one third as large as the second.
Find the angle measures and draw a
possible figure.
x° + 3x° + x° = 180°
5x = 180
5
5
x = 36
Holt CA Course 1
Triangle Sum Theorem
Simplify.
Divide both sides by 5.