Transcript Triangle Inequalities

MM1G3b
-Understand and use the triangle
inequality, the side-angle inequality, and
the exterior angle inequality.
Important Triangle Facts

A triangle has 6 parts:
 3 sides
 3 angles
The sum of the angles in a triangle is always
180°.
 A triangle is named by its vertices.

Name the triangle and its parts.
∆ ABC
A
AB
C
B
BC
<ACB
<BAC
<CBA
∆ RST
R
AC
RS
S
T
ST
<RST
<TRS
<STR
RT
Classifying
Triangles

Classify triangles by the side lengths.
 Equilateral – all sides are equal
 Isosceles – at least two sides are
 Scalene – no sides are equal

equal
Classify triangles by angle measures.




Right – has one 90⁰ angle
Acute – all angles are less than 90⁰
Obtuse – has one angle greater than 90⁰
Equiangular – all angles are equal
 Do Parts I and II of the Triangle Notes Handout.
Triangle Side Angle Inequalities
Smaller angles are opposite shorter sides.
 Larger angles are opposite longer sides.
Example.

1. In ∆ABC , list the sides in
order from smallest to
largest.
2. In ∆JKL , list the angles
in order from smallest to
largest.
98⁰
47⁰
35⁰
AB, BC, AC
<JLK, <KJL, <JKL
Handout straight edges and compasses
 Construct triangles:

 3”, 4”, 5”
 2”, 3”, 6”
Triangle Inequality Theorem

The sum of any two lengths of any two sides of a
triangle is greater than the length of the third side.
 Could say that the sum of the two shorter sides must be
greater than the longest side.
 If the 3rd side is equal to or less than the sum of the 2 other
sides, then it can not form a triangle.
Examples –
Can these three sides form a triangle?
A. 5, 8, 16
5 + 8 < 16
NO
B. 6, 11, 14 6 + 11 > 14 YES
C. 8, 13, 5
5 + 8 = 13
NO
Triangle Inequality Theorem

A.
If two sides of a triangle are given, describe the possible
lengths of the third side.
2 yd, 6 yd
What possible values would work? Compare the sum of 2 shorter
sides to longest side.
If 3rd side is 1: 1 + 2 > 6 No
If 3rd side is 6:
6 + 2 > 6 Yes
If 3rd side is 2: 2 + 2 > 6 No
If 3rd side is 7:
6 + 2 > 7 Yes
If 3rd side is 3: 3 + 2 > 6 No
If 3rd side is 8:
6 + 2 > 8 No
If 3rd side is 4: 4 + 2 > 6 No
If 3rd side is 9:
6 + 2 > 9 No
If 3rd side is 5: 5 + 2 > 6 Yes
If 3rd side is 10:
6 + 2 > 10 No
So the third side has to be bigger than 4 and less than 8 or 4 < x < 8.
Triangle Inequality Theorem

A.
If two sides of a triangle are given, describe the possible
lengths of the third side.
2 yd, 6 yd
So the third side has to be bigger than 4 and less than 8 or 4 < x < 8.
In other words,
2+x>6
–2
–2
x > 4
or
2+6>x
8>x
or
x< 8
Therefore, the range is going to be x has to greater than the difference
or less than the sum of the two given sides.
Triangle Inequality Theorem

B.
If two sides of a triangle are given, describe the possible
lengths of the third side.
4 in, 12 in
C. 3 ft, 18 ft
4 + x > 12 and 4 + 12 > x
16 > x
–4
–4
x > 8 and x < 16
8 < x < 16
3 + x > 18 and 3 + 18 > x
–3
–3
21 > x
x < 15
x < 21
15 < x < 21
Exterior Angle Inequality Theorem


The measure of an exterior angle of a triangle is greater
than the measure of either of the nonadjacent (remote)
interior angles.
The measure of an exterior angle is the sum of the
remote interior angles.
Example
6
1
2
What relationships do
we know about the
angles listed?
3
4
5
<5 > < 2 <5 > <3
<5 = <2 + <3
<4 > <1 <4 > <2 <4 = <1 + <2
<6 > <1 <6 > <3 <6 = <1 + <3
65°
53
3
(3x – 5)°
What do you know about x?
3x – 5 > 53
3x – 5 > 65
3x – 5 = 53 + 65
3x – 5 = 118
+ 5 = +5
3x = 123
x = 41
Classwork/Homework

Textbook p287 (4-9,13-24 all)