Download Section 10.2 Triangles

Section 10.2
Triangles
1.
2.
3.
Triangle
A
Objectives
Solve problems involving angle
relationships in triangles.
Solve problems involving similar
triangles.
Solve problems using the Pythagorean
Theorem.
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Section 10.2
closed geometric figure that has
three sides.
Closed geometric figures
If you start at any point and trace along
the sides, you end up at the starting
point.
closed
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not closed
Section 10.2
Example 1 - Using Angle
Relationships in Triangles
Euclid’s Theorem
Find
the
measure of
angle A for the
triangle ABC.
Euclid’s Theorem: The sum of the
measures of the three interior angles
of any triangle is 180º.
B
Solution:
m∠A + m∠B+ m∠C =180º
m∠A + 120º + 17º = 180º
m∠A + 137º = 180º
m∠A = 180º - 137º = 43º
C
A
∠A + ∠B + ∠C = 180
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o
Section 10.2
3
Example 2 - Using Angle Relationships in
Triangles
50o
2
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Section 10.2
4
Example 2 - Using Angle
Relationships in Triangles
Find
the measures of
angles 3, 4, and 5.
Find the measures of
angles 1 and 2.
50o
39o
39o
3. m∠ 2 = m∠ 3 (vertical angles), m∠ 3 = 51o
Solution:
4. 51o + m∠ 4 + 50o = 180o, m∠4 = 79o
1. m∠1 = 90o (supplementary angles)
5. m∠4 + m∠5 = 180o, 79o + m∠5 = 1800,
2. 90o + m∠ 2 + 39o = 180o, m∠ 2 = 51o
m∠5 = 101o
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Section 10.2
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1
Triangles and Their Characteristics
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Section 10.2
Triangles and Their Characteristics
7
Similar Triangles
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Section 10.2
8
Similar Triangles
Similar figures have the same shape,
but not necessarily the same size.
ln similar triangles, the angles are equal
but the sides may or may not be the same
length.
Corresponding angles are angles that
have the same measure in the two
triangles.
Corresponding sides are the sides
opposite the corresponding angles.
Similar triangles
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Section 10.2
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Similar Triangles (continued)
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Section 10.2
10
Similar Triangles (continued)
4 in.
Triangles ABC and DEF are similar:
Corresponding Angles
Angles A and D
Angles C and F
Angles B and E
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Section 10.2
Corresponding Sides
Sides CB and FE
Sides AB and DE
Sides AC and DF
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Triangles ABC and DEF are similar:
Corresponding Angles
Angles A and D
Angles C and F
Angles B and E
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Section 10.2
Corresponding Sides
Sides CB and FE
Sides AB and DE
Sides AC and DF
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2
Example 3 - Using Similar Triangles
Similar Triangles (continued)
Find the missing length x.
8m
In similar
triangles, the
ratio of the
lengths of
corresponding
sides is constant.
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Section 10.2
length of AB 4
= =2
length of DE 2
12 m
length of AC 6
= =2
length of DF 3
length of CB 8
= =2
length of FE 4
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Pythagorean Theorem
20 m.
Section 10.2
14
The hypotenuse is always the side opposite
the right angle in a right triangle.
Section 10.2
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Example 4
Using the Pythagorean Theorem
Find the length of the
hypotenuse c in this
right triangle:
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Section 10.2
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Example 5
Find the length of side
AC in the right triangle.
C
6 cm.
4 cm.
Solution:
Solution:
Let a = 9 in. and
b = 12 in.
2
2
c = a +b
2
2
2
c = 9 + 12
2
c = 81 + 144
c = 225 = 15 in.
2
2
2
6 = 4 +b
c = 6, a = 4
Rounded to the nearest
tenth.
c = 225
Section 10.2
2
2
2
c = a +b
Let AC = b.
2
2
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30 m
Pythagorean Theorem
The sum of the squares of
the lengths of the legs of a
right triangle equals the
square of the length of the
hypotenuse.
If the legs have lengths a
and b and the hypotenuse
has length c, then
a² + b² = c²
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Solution: 30 = x
12 8
30 x
8⋅ = ⋅8
12 8
20 = x
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Section 10.2
36 = 16 + b 2
2
b = 36 − 16 = 20
b = 20 ≈ 4.5 cm.
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