holt ca course 1 8-4 triangles vocabulary triangle sum theoremacute triangle right triangleobtuse...
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Holt CA Course 1
8-4 Triangles
Vocabulary
Triangle Sum Theorem acute triangle
right triangle obtuse triangle
equilateral triangle isosceles triangle
scalene triangle midpoint
altitude
Holt CA Course 1
8-4 TrianglesA triangle is a three sided polygon.
What are the different types of triangles?
Equilateral Triangle: All Three Sides Are Congruent.Isosceles Triangle: Two Sides Are Of Equal Length.Scalene Triangle: No Sides Are Congruent Right Triangle: One Angle Is A Right Angle.Acute Triangle: All Angles Are Acute.Obtuse Triangle: One Angle is An Obtuse Angle.
What is the formula for finding the area of a triangle?
A = 1/2 bhb = base, h = height
Holt CA Course 1
8-4 Triangles
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.
73° + 44° + p° = 180°
117 + p = 180
p = 63
–117 –117
Triangle Sum Theorem
Subtract 117 from both sides.
Holt CA Course 1
8-4 Triangles
Additional Example 1: Finding Angles in Acute, Right, and Obtuse Triangles
B. Find m in the obtuse triangle.
23° + 62° + m° = 180°
85 + m = 180
m = 95
–85 –85
Triangle Sum Theorem
Subtract 85 from both sides.
23
62
m
Holt CA Course 1
8-4 Triangles
Check It Out! Example 1
A. Find a in the acute triangle.
88° + 38° + a° = 180°
126 + a = 180
a = 54
–126 –126
88°
38°
a°
Triangle Sum Theorem
Subtract 126 from both sides.
Holt CA Course 1
8-4 Triangles
B. Find c in the obtuse triangle.
24° + 38° + c° = 180°
62 + c = 180
c = 118
–62 –62 c°
24°
38°
Check It Out! Example 1
Triangle Sum Theorem.
Subtract 62 from both sides.
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 TrianglesAdditional Example 2: Finding Angles in Equilateral,
Isosceles, and Scalene Triangles
62° + t° + t° = 180°62 + 2t = 180
2t = 118
–62 –62
A. Find the angle measures in the isosceles triangle.
2t = 1182 2
t = 59
Triangle Sum TheoremSimplify.Subtract 62 from both sides.
Divide both sides by 2.
The angles labeled t° measure 59°.
Holt CA Course 1
8-4 TrianglesAdditional Example 2: Finding Angles in Equilateral,
Isosceles, and Scalene Triangles
2x° + 3x° + 5x° = 180°
10x = 180
x = 18
10 10
B. Find the angle measures in the scalene triangle.
Triangle Sum Theorem
Simplify.Divide both sides by 10.
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 TrianglesCheck It Out! Example 2
39° + t° + t° = 180°39 + 2t = 180
2t = 141
–39 –39
A. Find the angle measures in the isosceles triangle.
2t = 1412 2
t = 70.5
Triangle Sum TheoremSimplify.
Subtract 39 from both sides.
Divide both sides by 2
t°t°
39°
The angles labeled t° measure 70.5°.
Holt CA Course 1
8-4 Triangles
3x° + 7x° + 10x° = 180°
20x = 180
x = 9
20 20
B. Find the angle measures in the scalene triangle.
Triangle Sum Theorem
Simplify.Divide both sides by 20.
3x° 7x°
10x°
Check It Out! Example 2
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°.
Holt CA Course 1
8-4 Triangles
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 (6x°) = 3x° =
third angle measure.
12
Additional Example 3: Finding Angles in a Triangle that Meets Given Conditions
Holt CA Course 1
8-4 Triangles
Additional Example 3 Continued
x° + 6x° + 3x° = 180°
10x = 180 10 10
x = 18
Triangle Sum Theorem
Simplify.Divide both sides by 10.
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.
Holt CA Course 1
8-4 Triangles
x° = 18°
6 • 18° = 108°
3 • 18° = 54°
The angles measure 18°, 108°, and 54°. The triangle is an obtuse scalene triangle.
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.
Holt CA Course 1
8-4 Triangles
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.
Check It Out! Example 3
Let x° = the first angle measure. Then 3x° =
second angle measure, and (3x°) = x° =
third angle measures.
13
Holt CA Course 1
8-4 Triangles
x° + 3x° + x° = 180°
5x = 180 5 5
x = 36
Triangle Sum Theorem
Simplify.Divide both sides by 5.
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.
Holt CA Course 1
8-4 Triangles
x° = 36°
x° = 36°3 • 36° = 108°
The angles measure 36°, 36°, and 108°. The triangle is an obtuse isosceles triangle.
36° 36°
108°
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.
Holt CA Course 1
8-4 Triangles
The midpoint of a segment is the point that divides the segment into two congruent segments. An altitude of a triangle is a perpendicular segment from a vertex of the triangle to the line containing the opposite side.
Holt CA Course 1
8-4 Triangles
In the figure, T is the midpoint of UV and ST is perpendicular to UV. Find the length of ST.
Additional Example 3: Finding the Length of a Line Segment
20 ft
26 ft
S
U
V
T
Step 1 Find the length of TU.
__
__ ______
TU = UV1 2 T is the midpoint
of UV.
= (20) = 101 2
Holt CA Course 1
8-4 Triangles
In the figure, T is the midpoint of UV and ST is perpendicular to UV. Find the length of ST.
Additional Example 3 Continued
Step 2 Use the Pythagorean Theorem. Let ST = a and TU = b.
__ ______
Find the square root.a = 24
a2 + b2 = c2
a2 + 102 = 262
a2 + 100 = 676 –100 –100
a2 = 576
Pythagorean TheoremSubstitute 10 for b and 26 for c.
Simplify the powers. Subtract 100 from each side.
The length of ST is 24 ft, or ST is 24 ft.__
Holt CA Course 1
8-4 Triangles
In the figure, B is the midpoint of DC and AB is perpendicular to DC. Find the length of AB.
Check It Out! Example 3
Step 1 Find the length of BC.
__
______
__
BC = DC1 2 B is the midpoint
of DC.
= (14) = 71 2
14 in
25 in
A
C
D
B
Holt CA Course 1
8-4 TrianglesAdditional Example 3 Continued
Step 2 Use the Pythagorean Theorem. Let AB = a and BC = b.
__ ______
Find the square root.a = 24
a2 + b2 = c2
a2 + 72 = 252
a2 + 49 = 625 –49 –49
a2 = 576
Pythagorean TheoremSubstitute 7 for b and 25 for c.
Simplify the powers.
Subtract 49 from each side.
The length of AB is 24 in, or AB is 24 in.__
In the figure, B is the midpoint of DC and AB is perpendicular to DC. Find the length of AB.
Holt CA Course 1
8-4 TrianglesLesson Quiz: Part I
1. Find the missing angle measure in the acute triangle shown.
2. Find the missing angle measure in the right triangle shown.
38°
55°
Holt CA Course 1
8-4 TrianglesLesson Quiz: Part II
3. Find the missing angle measure in an acute triangle with angle measures of 67° and 63°.
4. Find the missing angle measure in an obtuse triangle with angle measures of 10° and 15°.
50°
155°5. In the figure, M is the midpoint of AB and MD is t perpendicular to AB. Find the length of AB.
____
____
36 m
39 m
DM
A
B
30 m