Download Remote: • Interior

______________________
___________
___________
________
_______ _____________________
___________
___________
_
___________
___________
_________
_____
Triangle Properties
Triangles: Angles/Sides
Vocabulary:
•
Corollary:
•
Exterior Angle:
•
Interior:
•
Remote:
•
Remote Interior Angle:
Theorems:
•
Angle Sum Theorem: Sum of measure of angles of a triangle is 180.
Proof:
Given: AABC
Prove: LC +
/2+ LB
=
180
Statements
1. AABC
2. XY CB
2. Parallel Postulate
3.LlZC
3.
4.Z3ZB
4.
5.
5.
6.LC+Z2+ZB=180
6.
Reasons
I .Given
Definition of supplementary angles
Example:
1.
•
Find the missing angle measures
•
/1=
•
/2=
•
/3=
and provide reason.
No Choice Theorem/3 Angle Theorem: if 2 angles of triangle
another triangle, then the
3(
are
to 2 angles of
angle of the triangles are.
Proof:
Given:
Prove:
LA
LB
LF, LC
LE
Statements
1.LA/F
2./CLD
3.ZA+LB+LC=180
4./D+LE+LF=180
5.
AABFA
6.
7.ZBLE
1
Reasons
1.
2.
3.
4.
5. Substitution Property
6. Substitution Property
7.
Rev A
Triangle Properties
• Exterior Angle Theorem: The measure of the exterior angle of a triangle is equal to the sum
of the measures of the 2 remote interior angles.
Proof:
Given: AABC
Prove: LCBD
=
LA
+
LC
A
Statements
Reasons
1. AABC
1LA+LC+LCBA=180
3.
4.
5./A+/C+/CBA=
LDBC + LCBA
6.LA+/C=LDBC
l.Given
2.
3. if form straight L then supplementary
4. Definition of Supplementary
5.
6.
Example:
2. Find the missing angle measures and provide reason.
• /1=
• /2=
• /3=
• /4=
• /5=
Exterior Angle Inequality Theorem: if L is exterior L of a triangle then its measure is
greater than the measure of its corresponding remote interiorL.
Example:
3. Given: DAS
Prove: /4> /1 and /4> /2
Side Angle Theorem: longest side of a triangle is opposite the largest L in triangle,
Examples: Side Order
4. List the sides in order from least to greatest measure.
a.
b.
C.
S.
2
y
RevA
Triangle Properties
•
Angle Side Theorem: largest
of a triangle is opposite the longest side in triangle.
Examples: Angle Order
5. List the angles in order from least to greatest measure.
b.
a.
R
C.
T
T
59cm
R
cm
13
km
S
y
• Triangle Inequality Theorem: sum of lengths of any 2 sides of triangle is greater than the
length of the
3ft1
side.
Examples
6. Determine whether the given measures can be the lengths of sides of a triangle.
a.2,3,4
b.6,8,14
c.i,2,2
•
Shortest Distance Theorem: L segment from point to line is shortest segment from point to line.
Properties of Inequalities:
Comparison Property
Transitive Property
a < b, a = b or a> b
If a < b and b < c then a < c
Ifa>bandb> cthena>c
Addition and Subtraction Properties
If a> b then a + c > b + c and a c > b—c
If a < b then a + c < b + c and a—c < b c
Multiplication and Division Properties
a h
lfc>Oanda<b,thenac<bcand—<—
-
—
—
C
lfc>Oanda>b,thenac>bcand
C
ab
—>
C
—
C
ab
lfc<Oanda<b, thenac>bc and—>
—
C
.
ltc<Oanda>b,thenac<bcand
C
ab
—<—
C
C
Corollaries:
•
4.1: Acute Ls of right triangle are complementary
•
4.2: There can be at most 1 right or obtuse Z in a triangle.
•
5.1:
--
segment from point to plane is shortest segment from point to plane.
3
RevA
Triangle Properties
Classifying Triangles
Define
•
Acute Triangle:
•
Base:
•
Equiangular Triangle:
•
Equilateral Triangle:
•
Hypotenuse:
•
Isosceles Triangle:
•
Legs:
•
Obtuse Triangle:
•
Right Triangle:
•
Scalene Triangle:
•
Vertex Angle:
Classifying Triangles by Angles
• Acute: all angles are acute
• Obtuse: 1 angle is obtuse
• Right: 1 angle is right
• Equiangular: all angles are congruent
Key
•
•
•
Classifying Triangles by Sides
• Scalene: no sides are congruent
• Isosceles: at least 2 sides are congruent
• Equilateral: all 3 sides are congruent
Concepts:
All equilateral triangles are isosceles but not all isosceles triangles are equilateral.
If triangles are equilateral, then they are also equiangular and vice versa.
If c is the length of the longest side of a triangle then
+
Ifa
<
,
Aisobtnse
b
c
+
Ifa
>
,
Aisacute
b
c
+
Ifa
=
,
Aisright
b
c
0 2
o 2
o 2
Examples: Classifying Triangles
7. Identify the type of triangle based on the following sides: 3, 5, 6.
8. Identify the indicated type of triangle in the figure.
a. scalene triangle
A
b. isosceles triangle
4
RevA
Triangle Properties
Examples: Finding Missing Values
9. Find x & measure of each side of the equilateral ARST if RS
x
+
9, ST 2x and RT = 3x —9.
10. Find the measures of the sides of ADEC and classify by sides. Vertices are: E(-5,3), C(2,2) and D(3,9).
Right Triangles
Identifying Parts of Triangle:
•
•
Legs: 2 sides forming right angle (a, b)
Hypotenuse: side opposite the right angle; longest side of triangle (c)
Leg(a) yptenuse (c)
Leg (b)
Pythagorean Theorem
•
•
•
Pythagorean theorem: a2 + b 2 = c 7
(for right angles)
Used to find the length of a side of a right triangle when the lengths of the other 2 sides are known.
Pythagorean Triples:
3-4-5
5-12-13
7-24-25
8-15-17
9-40-41
Special Right Triangles
Triangle Proofs
Hints for Triangle Proofs
•
•
•
•
•
•
Right Triangle: need I or right angle
Acute Triangle: need to show all 3 angles are <90 each
Obtuse Triangle: need to show that 1 angle is between 90 and 180
Scalene Triangle: need to show that all 3 sides are different
Isosceles: need to show that at least 2 of the sides are congruent
Equilateral: need to show that all 3 sides are congruent
Example: Proofs
11. Given: LCBD 700
Prove: AABC is obtuse
5
RevA