chapter 10
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CHAPTER 10. Geometry. 10.2. Triangles. Objectives Solve problems involving angle relationships in triangles. Solve problems involving similar triangles. Solve problems using the Pythagorean Theorem. 3. Triangle. - PowerPoint PPT PresentationTRANSCRIPT
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CHAPTER 10
Geometry
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10.2
Triangles
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Objectives
1. Solve problems involving angle relationships in triangles.
2. Solve problems involving similar triangles.
3. Solve problems using the Pythagorean Theorem.
3
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Triangle
• A closed geometric figure that has three sides, all of which lie on a flat surface or plane.
• Closed geometric figures – If you start at any point and trace along the sides,
you end up at the starting point.
4
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Euclid’s Theorem
Theorem: A conclusion that is proved to be true through deductive reasoning.
Euclid’s assumption: Given a line and a point not on the line, one and only one line can be drawn through the given point parallel to the given line.
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Euclid’s Theorem (cont.)
Euclid’s Theorem: The sum of the measures of the three angles of any triangle is 180º.
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Euclid’s Theorem (cont.)
Proof:m1 = m2 and m3 = m4 (alternate interior angles)Angles 2, 5, and 4 form a straight angle (180º)
Therefore, m1 + m5 + m3 = 180º .
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Two-Column Proof A
Given: ABC△Prove: A + B + ACB = 180°∠ ∠ ∠
Statements
1. Through C, draw line m || to line AB.
2. ∠1 + 2 + 3 = 180°∠ ∠3. ∠1 = A; 3 = B∠ ∠ ∠4. ∠A + ACB + B = 180°∠ ∠
Reasons
1. Parallel postulate
2. Definition of straight angle
3. If 2 || lines, alt. int. ’s are equal∠4. Substitution of equals axiom
1
A B
Cm1
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Theorem: The sum of ’s in a equals 180∠ △ °.
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Exterior ∠ Theorem
Theorem: An exterior angle of a triangle equals the sum of the non-adjacent interior angles.
∠CBD = A + C ∠ ∠Since CBD + ABC = 180 -- def. of supplementary ∠ ∠ s∠
A + C + ABC = 180 -- sum of ∠ ∠ ∠ ’s interior △ s∠ A + C + ABC = CBD + ABC -- axiom∠ ∠ ∠ ∠ ∠ A + C = CBD -- axiom∠ ∠ ∠
1
A
C
B D
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Base s of Isosceles ∠ △
• Theorem: Base s of an isosceles are equal∠ △
• Given: Isosceles ABC, which AC = BC△Prove: A = B∠ ∠
1
A
C
B
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Two-Column Proof B
Given: Circle C with A, B, D on the circleProve: ACB = 2 • ADB∠ ∠
Statements1. Circle C, with central C, ∠
inscribed D∠2. Draw line CD, intersecting circle
at F
3. ∠y = a + b = 2 a∠ ∠ ∠x = c + d = 2 c∠ ∠ ∠ ∠
4. ∠x + y = 2( a + c)∠ ∠ ∠5. ∠ACB = 2 ADB∠
Reasons1. Given
2. 2 points determine a line (postulate)
3. Exterior ; Isosceles angles∠ △
4. Equal to same quantity (axiom)
5. Substitution (axiom)
1
Theorem: An formed by 2 radii subtending a chord is 2 x ∠an inscribed subtending the same chord. ∠ A
B
D
C
F
c
x
b
y
a
d
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Example 1: Using Angle Relationships in Triangles
Find the measure of angle A for the triangle ABC.
Solution:
mA + mB + mC =180º
mA + 120º + 17º = 180º
mA + 137º = 180º
mA = 180º − 137º = 43º
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Example 2: Using Angle Relationships in Triangles
Find the measures of angles 1 through 5.
Solution:m1+ m2 + 43º =180º
90º + m2 + 43º = 180º
m2 + 133º = 180º
m2 = 47º
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Example 2 continued
m3 + m4+ 60º =180º 47º + m4 + 60º = 180ºm4 = 180º − 107º = 73º
m4 + m5 = 180º 73º + m5 = 180°
m5 = 107º
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Triangles and Their Characteristics
15
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Similar Triangles
1
A
B
C
Y
ZX
△ABC ~ XYZ iff△• Corresponding angles
are equal• Corresponding sides are
proportional
• ∠A = X; B = Y; C = Z∠ ∠ ∠ ∠ ∠• AB/XY = BC/YZ = AC/XZ
• If 2 corresponding s of 2 s ∠ △are equal, then s are similar. △
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Example 1
• Given:– AB || DE– AB = 25; CE = 8
• Find AC • Solution• x 25
--- = ----- 8 12
• 8(25) 25 50x = ------- = 2 • ---- = ----- ≈ 16.66 12 3 3
1
A
E
DC
B
25
8
12
x
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Example 2
• How can you estimate the height of a building, if you know your own height (on a sunny day)?
1
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Example 2
1
x
400
610
6 10--- = ------ x 400
6 • 400 = 10 • x
x = 240
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Similar Triangles
• Similar figures have the same shape, but not necessarily the same size.
• In 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.
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Similar Triangles (continued)
CB and FE
Triangles ABC and DEF are similar:
Corresponding Angles Corresponding Sides
Angles A and D Sides
Angles C and F Sides
Angles B and E Sides AB and DE
AC and DF
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Example 3: Using Similar Triangles
Find the missing length x.
Solution: Because the triangles are similar, their corresponding sides are proportional:
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Example 3: Using Similar Triangles
5 7
85 8 7
5 56
5 511.2
xx
x
x
Find the missing length x.
The missing length of x is 11.2 inches.23
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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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Example 5: Using the Pythagorean Theorem
Find the length of the hypotenuse c in this right triangle:
Solution:
Let a = 9 and b = 122 2 2
2 2 2
2
2
9 12
81 144
225
225 15
c a b
c
c
c
c
C
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