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LESSON 4.3 – TRIANGLE INEQUALITIES & EXTERIOR ANGLES
Homework: 4.3/ 1-10, 12-16
EXTERIOR ANGLE THEOREM
An exterior angle of a triangle…… is equal in measure to the sum of the measures of its two remote interior angles.
remote interior angles Exterior
angle
EXTERIOR ANGLE THEOREM(YOUR NEW BEST FRIEND)
3
2
1 4
exterior angle
remote interioranglesm<1 + m<2 = m<4
m<BCD = m<A + m<B
m<4= m<1+ m<2
EXTERIOR ANGLE THEOREM
EXAMPLES
m<G + 60˚ = 111˚m<G = 51˚
Remote interior angles
Exterior angle
EXAMPLES
x
82°
30° y
Find x & y
x = 68°
y = 112°
y = 30 + 82y = 112˚
Using Linear pair:180 = 112 + x68˚ = x
Remote interior angles
EXAMPLES
Find m JKM2x – 5 = x + 70 x – 5 = 70 x = 75
m< JKM = 2(75) - 5m< JKM = 150 - 5
m< JKM = 145˚
EXAMPLES
Solve for y in the diagram.
Solve on your own before viewing the
Solution
4y + 35 = 56 + y3y + 35 = 563y = 21
y= 7
SOLUTION
EXAMPLES
Find the measure of in the diagram shown.1
Solve on your own before viewing the
Solution
40 + 3x = 5x - 1040 = 2x - 1050 = 2x
25 = x
Exterior angle:5x – 10 = 5(25) - 10
m < 1= 65
= 125 – 10 = 115m < 1= 180 -
115
SOLUTION
CHECKPOINT: COMPLETE THE EXERCISES.
SOLUTION
Right Scalene triangle
x + 70 = 3x + 10
70 = 2x + 1060 = 2x30 = x
3 (30) + 10 = 100˚
TRIANGLE INEQUALITIES
Make A Triangle
Construct triangle DEF.
D FF E
D E
D FF E
D E
Make A Triangle
Construct triangle DEF.
D E
Make A Triangle
Construct triangle DEF.
D E
Make A Triangle
Construct triangle DEF.
D E
Make A Triangle
Construct triangle DEF.
D E
5 3
13
Q:What’s the problem with this?
A: The shorter segments can’t reach each other to complete the triangle. They don’t add up.
Make A Triangle
Construct triangle DEF.
The sum of the lengths of any two sides of a triangle is greater than the length of the third side.
Triangle Inequality Conjecture
Add the two smallest sides; they MUST be larger than the third side
for the triangle to be formed.
TRIANGLE INEQUALITY CONJECTURE
Given any triangle, if a, b, and c are the lengths of the sides, the following is always true:
a + b > ca + c > bb + c > a
The triangle inequality theorem is very useful when one needs to determine if any 3 given
sides will form of a triangle or not.
In other words, if the 3 conditions above are not met, you can immediately conclude that it is not
a triangle.
EXAMPLEThree segments have lengths: a= 3 cm, b= 6 cm, and c = 4 cm.Can a triangle be formed with these measures?
3 + 6 = 9 and 9 > 4
3 + 4 = 7 and 7 > 6
6 + 4 = 10 and 10 > 3So a triangle can be formed!
EXAMPLE
Three segments have lengths: a= 7 cm, b= 16 cm, and c = 8 cm. Can a triangle be formed with these measures?
7 + 16 = 23 and 23 > 8
7 + 8 = 15 , but 15 < 16. This condition is not met because the sum of these two sides is smaller than the third side
16 + 8 = 24 and 24 > 7
Since one of the conditions is not met, a triangle cannot be formed.
SIMPLY:
If the two smallest side measures do not add up to be greater than the largest side, then the sides do not make a triangle!
If the two smallest side measures add up to be greater than the
largest side, then the sides make a triangle!
Make A TriangleCan the following lengths form a triangle?1.6 mm5 mm10 mm
2.2 ft9 ft13 ft
5.10 mm3 mm6 mm
8.8 m7 m1 m
9.9 mm2 mm10 mm
12.1 mm5 mm3 mm
3.5 cm cm4 cm
√𝟐4. 7 ft
15 ft ft
√𝟏𝟑6.
4 ft7 ft ft
√𝟕
7.10 mm13 mm mm
√𝟓10.
12 mm22 mm mm
√𝟏𝟑
11.5 mm8 mm mm
√𝟏𝟐
In a triangle, the longest side is opposite the largest angle; and the shortest side is opposite the smallest angle.
Side-Angle Conjecture
Side AB is the shortest, because it's across from the smallest angle (40 degrees). Also, the side BC is
the longest because it is across from the largest angle (80 degrees).
Side-Angle
What’s the biggest side?What’s the biggest angle?
What’s the smallest side?What’s the smallest angle?
C
B A
b
a
c
bB
aA
100°
60°
Side-Angle
92°42°
46°
ab
c
Rank the sides from greatest to least.
bca
Rank the angles from greatest to least.CAB
A
CB
7
5
4
Find x.
Practice
25 + x + 15 = 3x - 10
x + 40 = 3x - 10
40 = 2x - 10
50 = 2x
25 = x
3x – 10 3(25) – 10 65°
x + 15 25 + 15 40°
Find x and y.
92 = 50 + x40 = x
92 + y = 180y= 88
Exterior angle Linear pair of angles
Find the measures of <‘s 1, 2, 3, & 4
LP: 92 + <1 = 180<1 = 88
LP: 123 + <2 = 180<2 = 57
EA: <4 = <1 + < 2
<4 = 88 + 57<4 = 145
LP: 145 + <3 = 180
<3 = 35
Find the measure of each numbered angle in the figure.
Exterior Angle TheoremSimplify.
SubstitutionSubtract 70 from each side.
linear pairs are supplementary.
Exterior Angle Theorem
Subtract 64 from each side.
Substitution
Subtract 78 from each side.
If 2 s form a linear pair, they are supplementary.
Substitution
Simplify.
Subtract 143 from each side.
Angle Sum Theorem
Substitution
Simplify.
Answer:
Find the measure of each numbered angle in the figure.
Answer:
YOUR TURN:
