10.2 triangles. axioms and theorems postulate— a statement accepted as true without proof. e.g....

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