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What is a vertical angle? What are adjacent angles? What are supplementary angles? What are complementary angles?
EXTERIOR ANGLES
When the sides of a polygon are extended, other angles are formed.
The original angles are the interior angles.
The angles that form linear pairs with the interior angles are the exterior angles.
Angles
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)
The measure of an exterior angle in a triangle is the sum of the measures of the 2 remote interior angles
3
2
1 4
exterior angle
remote interioranglesm<1 + m<2 = m<4
m 1 = m A + ∠ ∠m B ∠
EXTERIOR ANGLE THEOREM
EXAMPLES
m<G = 60 + 69
m<FHG = 180 – 111 = 69 linear pair
EXAMPLES
x
82°
30° y
Find x & y
x = 68°
y = 112°
y = 30 + 82y = 112˚
180 = 30 + 82 + x180 = 112 + x68˚ = x
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= 27
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
m < 1= 5x - 10m < 1= 5(25) - 10m < 1= 125 - 10m < 1= 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.
Make A TriangleCan the following lengths form a triangle?1.4 mm5 mm10 mm
2.2 ft9 ft13 ft
5.10 mm3 mm6 mm
8.8 m7 m1 m
9.9 mm2 mm1 mm
12.1 mm5 mm3 mm
3.5 cm cm4 cm
√𝟐4. 7 ft
15 ft ft
√𝟏𝟑6.
7 ft7 ft ft
√𝟕
7.10 mm13 mm mm
√𝟓10.
12 mm22 mm mm
√𝟏𝟑
11.7 mm8 mm mm
√𝟏𝟐
In a triangle, if one side is longer than another side, then angle opposite the longer side is larger than the other.
Side-Angle Conjecture
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
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