1. 3.5 The Triangle Sum Theorem

2. Theorems, Postulates, & Definitions
The Parallel Postulate: Given a line and a point not on the line, there is one and only one line that contains the given point and is parallel to the given line.
Triangle Sum Theorem: The sum of the
measures of the angles of a triangle is 180°.
Exterior Angle Theorem: The measure of
an exterior angle of a triangle is equal to the sum
of the measures of the remote interior angles.
Homework

3. REMOTE INTERIOR ANGLES
EXTERIOR ANGLE 2 4 1 3
An exterior angle is formed by one side of a triangle and the extension of another side.
Remote interior angles are those interior angles of a triangle not adjacent to a given exterior angle.
Homework

4. Exterior Angle Theorem
The measure of an exterior angle of a triangle is equal to the sum of the measures of the two remote interior angles. 2 1 3
Homework

5. The Parallel Postulate:
Through a point not on a given line, there is one
and only one line that goes through that point
that is parallel to the given line.
Homework

6. x + y + z = 180° The Triangle Sum Theorem
The three angles in a triangle add up to be 180º. y° x° z°
x + y + z = 180°
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7. Triangle Sum Theorem Proof
GIVEN: ∆ABC and BD || AC
PROVE: m∠1 + m∠2 + m∠3 = 180
Statements
Reasons
1. BD || AC
1. Given
2. 4 + 2 + 5 =180
2. 180 in a straight line
3. 1  4 and 3  5
3. Alternate Interior ’s are 
4. 1 + 2 + 3 = 180
4. Substitution Property
Homework

8. Find the value of the variable.
x = 25 b.
x = 9 c.
x = 47 d.
x = 36 e.
x = 25 f.
x = 45
Homework g.
x = 15 h.
x = 134 i.
x = 75

9. Find the value of the variable(s).
x = 17.5 b.
x = y = 87 x d.
x = 22 c.
x = 53 e.
x = 32 f.
x = 30
Homework

10. Find the value of the variable(s).
x = 35 y = 37 b.
x = y = 96 c.
x = 85 y = 65 d.
x = 26 y = 64 e.
x = 43 y = 32 f.
x = 62 y = 28
Homework

11. Homework Find the measure of each numbered angle in the figure. a. m1
= 50
b. m2
= 130 
c. m3
= 50 
d. m4
= 130 
e. m5
= 40 
f. m6
= 30 
Homework

12. Homework Find the measure of each numbered angle in the figure. 385 4 3 2 1
41
64
29
32
38
a. m1
= 70 
b. m2
= 110 
c. m3
= 46 
d. m4
= 102 
e. m5
= 37 
Homework

13. Assignment
3.5A and 3.5B
Section

14. 3.6 Angles in Polygons

15. Objective
Develop and use formulas for the sums of the measures of interior and exterior angles of a polygon.
Homework

16. Do you know the sum of the interior angles for any other polygons?
Sum of interior angles in a polygon
We already know that the sum of the interior angles in any triangle is 180°. c
a + b + c = 180 ° a b
We also know that the sum of the interior angles in any quadrilateral is 360°. a b c d
a + b + c + d = 360 °
Do you know the sum of the interior angles for any other polygons?
Homework

17. Sum of the interior angles in a polygon
A quadrilateral can be divided into two triangles …
… and a pentagon can be divided into three triangles.
How many triangles can a hexagon be divided into?
A hexagon can be divided into four triangles.
Homework

18. Sum of the interior angles in a polygon
The number of triangles that a polygon can be divided into is always two less than the number of sides.
We can say that:
A polygon with s sides can be divided into (s – 2) triangles.
The sum of the interior angles in a triangle is 180°.
So:
The sum of the interior angles in an s-sided polygon is (s – 2) × 180°.
Homework

19. Interior angles in regular polygons
A regular polygon has equal sides and equal angles.
We can work out the size of the interior angles in a regular polygon as follows:
Name of regular polygon
Sum of the interior angles
Size of each interior angle
Equilateral triangle
180°
180° ÷ 3 =
60°
Square
2 × 180° = 360°
360° ÷ 4 =
90°
Regular pentagon
3 × 180° = 540°
540° ÷ 5 =
108°
Regular hexagon
4 × 180° = 720°
720° ÷ 6 =
120°
Homework

20. Interior and exterior angles in an equilateral triangle
Every interior angle measures 60°.
60°
120°
Every exterior angle measures 120°.
The sum of the interior angles is 3 × 60° = 180°.
120°
60°
60°
120°
The sum of the exterior angles is 3 × 120° = 360°.
Homework

21. Interior and exterior angles in a square
Every interior angle measures 90°.
90°
90°
90°
90°
Every exterior angle measures 90°.
The sum of the interior angles is 4 × 90° = 360°.
90°
90°
90°
The sum of the exterior angles is 4 × 90° = 360°.
90°
Homework

22. Interior and exterior angles in a regular pentagon
Every interior angle measures 108°.
108°
72°
72°
Every exterior angle measures 72°.
108°
108°
72°
The sum of the interior angles is 5 × 108° = 540°.
108°
108°
72°
72°
The sum of the exterior angles is 5 × 72° = 360°.
Homework

23. Interior and exterior angles in a regular hexagon
Every interior angle measures 120°.
60°
120°
120°
60°
Every exterior angle measures 60°.
60°
120°
120°
60°
The sum of the interior angles is 6 × 120° = 720°.
120°
120°
60°
60°
The sum of the exterior angles is 6 × 60° = 360°.
Homework

24. The sum of exterior angles in a polygon
For any polygon, the sum of the exterior angles is 360°.
The sum of the interior angles is (s – 2) × 180°.
Homework

25. Homework A Polygon can either be convex or concave.
If a polygon is convex then no sides go through the interior of the polygon.
(All vertices point outside the polygon.)
If a polygon is concave then it is not convex. A side goes through the interior of the polygon.
(At least one vertex points inside the polygon.)
Homework

26. Label the polygons as convex or concave?
Homework

27. Theorems, Postulates, & Definitions
Sum of the Interior Angles of a Polygon:
The sum, a, of the measures of the interior angles of a polygon with s sides is given by
a = (s – 2)180.
Homework

28. Theorems, Postulates, & Definitions
Sum of the Exterior Angles of a Polygon:
The sum of the measures of the exterior
angles of a polygon is 360.
Homework

29. Complete the chart below:
Regular polygon
Sides
Sum of the Interior Angles
Measure of One Interior Angle
Measure of One Exterior Angle
Triangle 3
Quadrilateral 4
Pentagon 5
Hexagon 6
Heptagon 7
Octagon 8
Nonagon 9
n-gon n
180
60
120
360
90
90
540
108
72
720
120
60  4 7 3 7
51 
900
1080
135
45
1260
140
40
(n - 2)180 n n
360
(n -2 )180
Homework

30. An exterior angle measure of a regular polygon is given
An exterior angle measure of a regular polygon is given. Find the number of its sides and the measure of each interior angle.
Example: The exterior angle measure of a regular polygon is 45. Find the number of sides.
Since there is 360 in the exterior of a figure with any number of sides, divide 360 by 45 to find the number of sides.
a. 120°
3 sides
b. 72°
5 sides
c. 36°
10 sides
d. 24°
15 sides
Homework

31. Name the convex polygon whose interior angle measures have each given sum.
Example: The sum of the interior angle measures of a regular polygon is 720. Find the number of sides.
Use the formula for the sum of the interior angles of a polygon.
a. 540°
b. 900°
c. 1800°
d. 2520°
5 sides
Pentagon
7 sides
Heptagon
12 sides
Dodecagon
16 sides 16-gon
Homework

32. Find the value of each variable
y = 22.5 d. e.
x = 27 f.
s = 18 g.
x = 61.5 h.
x = 72
x = 90 y = 75
z = 120 i.
x = 124 j.
n = 24 k.
a = 30 l.
Homework

33. Assignment
3.6A and 3.6B
Section