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10.2Triangles
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Axioms and TheoremsAxioms and Theorems• Postulate—A statement accepted as true
without proof.• E.g.
Given a line and a point not on the line, one and only one can be drawn through the point parallel to the given line.
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TheoremTheorem• Theorem—A statement that is proved from
postulates, axioms, and other theorems.• E.g.
The sum of the measures of the three angles of any triangle is 180°.
A C
B
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2-Column Proof2-Column Proof
• Given: ΔABCProve: A + ABC + C = 180°
Statements Reason
1. Draw a line through B parallel to AC.
1. Given a line and a point not on the line, one line can be drawn through the point parallel to the line
2. 1 + 2 + 3 = 180° 2. Definition of a straight angle
3. 1 = A; 3 = C 3. If 2 || lines are cut by a transversal, alt. int. angles are equal.
4. A + ABC + C = 180° 4. Substitution of equals
A C
B
1 2 3
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Exterior Angle Exterior Angle TheoremTheorem
• Theorem: An exterior angle of a triangle equals the sum of the non-adjacent interior angles.
• Given: △ ABC with exterior angle ∠CBD • Prove: ∠CBD = ∠A + ∠C
∠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
A
C
B D
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Base ∠s of Isosceles △ Base ∠s of Isosceles △ • Theorem: Base ∠s of an isosceles △ are equal.
• Given: Isosceles△ ABC, with AC = BC• Prove: ∠A = ∠B
A
C
B
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Another ProofAnother Proof• Theorem: An formed by 2 radii subtending a
chord is 2 x an inscribed subtending the same chord.
• Given: Circle C, with central ∠C and inscribed ∠D
• Prove: ∠ACB = 2 • ∠ADB
C
F
D
B
A
xy
a
d
bc
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2-Column Proof2-Column Proof• Given: Circle C, with central ∠C and
inscribed ∠D• Prove: ∠ACB = 2 • ∠ADB
C
F
D
B
A
xy
a
d
bc
Statements Reasons
1. Circle C, with central ∠C, inscribed ∠D
1. Given
2. Draw line CD, intersecting circle at F
2. Two points determine a line.
3. ∠y = ∠a + ∠b = 2 ∠a∠x = ∠c + ∠d = 2 ∠c
3. Exterior ∠; Isosceles △ angles
4. ∠x + ∠y = 2(∠a + ∠c) 4. Equal to same quantity
5. ∠ACB = 2 ∠ADB
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Your TurnYour Turn• Find the measures of angles 1 through 5.• Solution:
1 = 90º2 = 180 – (43 + 90) = 180 – (133) = 47º3 = 47º4 = 180 – (47 + 60) = 180 – (107) = 73º5 = 180 – 73 = 107º
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Triangles and Their Triangles and Their
CharacteristicsCharacteristics
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Similar TrianglesSimilar Triangles△ABC ~ XYZ iff△
• Corresponding angles are equal
• Corresponding sides are proportional
• ∠A = X; B = Y; C = Z∠ ∠ ∠ ∠ ∠• AB/XY = BC/YZ = AC/XZ
• Theorem:If 2 corresponding s of 2 s are equal, then ∠ △
s are similar. △
A
B
C
X Z
Y
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ExampleExampleA
E
DC
B
25
8
12
x
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ExampleExample• How can you estimate the height of a building
when you know your own height (on a sunny day).
400
610
x
6 10--- = ------ x 400
6 • 400 = 10 • x
x = 240
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Your TurnYour Turn
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Pythagorean TheoremPythagorean 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 triangle ABC is a righttriangle with hypotenuse c,thena2 + b2 = c2
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ExampleExample
C
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Your TurnYour Turn
A C
B
118
b
Γ