Download Triangles

10.2
Triangles
Axioms 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.
Theorem
• 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°.
B
A
C
2-Column Proof
B
• Given: ΔABC
Prove: A + ABC + C = 180°
1
2
3
A
C
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
Exterior Angle Theorem
• Theorem: An exterior angle of a triangle equals the
sum of the non-adjacent interior angles.
C
A
B
D
• 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
Base ∠s of Isosceles △
• Theorem: Base ∠s of an isosceles △ are equal.
C
A
B
• Given: Isosceles△ ABC, with AC = BC
• Prove: ∠A = ∠B
Another Proof
• Theorem: An  formed by 2 radii subtending a
chord is 2 x an inscribed  subtending the same
chord.
A
F
d
x
C
c
y
b
B
a
D
• Given: Circle C, with central ∠C and
inscribed ∠D
• Prove: ∠ACB = 2 • ∠ADB
2-Column Proof
• Given: Circle C, with central ∠C and
inscribed ∠D
• Prove: ∠ACB = 2 • ∠ADB
A
d
x
C
Statements
1. Circle C, with central ∠C,
inscribed ∠D
Reasons
1. Given
D
F
y
b
c
a
2. Draw line CD, intersecting 2. Two points determine a
circle at F
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
B
Your Turn
• Find the measures of angles 1 through 5.
• Solution:
1 = 90º
2 = 180 – (43 + 90)
= 180 – (133)
= 47º
3 = 47º
4 = 180 – (47 + 60)
= 180 – (107)
= 73º
5 = 180 – 73
= 107º
Triangles and Their Characteristics
Similar Triangles
B
△ABC ~ △XYZ iff
• Corresponding angles
A
are equal
• Corresponding sides are proportional
C
Y
•
•
∠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.
X
Z
Example
A
25
x
B
C
D
8
12
E
Example
• How can you estimate the height of a building
when you know your own height (on a sunny day).
6
10
--- = -----x
400
6 • 400 = 10 • x
x
x = 240
6
400
10
Your Turn
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 triangle ABC is a right
triangle with hypotenuse c,
then
a2 + b2 = c2
Example
C
Your Turn
B
11
8
Γ C
A
b