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10.2Triangles
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.
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
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
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
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
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
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
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
Triangles and Their Triangles and Their
CharacteristicsCharacteristics
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
ExampleExampleA
E
DC
B
25
8
12
x
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
Your TurnYour Turn
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
ExampleExample
C
Your TurnYour Turn
A C
B
118
b
Γ