the triangle sum theorem. theorems, postulates, & definitions the parallel postulate: given a...
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The Triangle Sum TheoremThe Triangle Sum Theorem
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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.
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1
2
3
EXTERIOR ANGLE
REMOTE INTERIOR
ANGLES
4
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.
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Exterior Angle TheoremExterior 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.
1
2
3
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Through a point not on a given line, there is one and only one line that goes through that pointthat is parallel to the given line.
The Parallel Postulate:The Parallel Postulate:
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The three angles in a triangle add up to be 180º.
The Triangle Sum TheoremThe Triangle Sum Theorem
x + y + z = 180°
x°
y°
z°
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Triangle Sum Theorem ProofTriangle Sum Theorem ProofGIVEN: ∆ABC and BD || ACPROVE: m 1 + ∠ m 2 + ∠ m 3 = 180∠
Statements Reasons
BDAC
4 + 2 + 5 =180
1 4 and 3 5
1 + 2 + 3 = 180
180 in a straight line
Given
Alternate Interior ’s are
Substitution Property
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Find the value of the variable.Find the value of the variable.
a.a. b.b. c.c.x = 25x = 25 x = 9x = 9 x = 47x = 47
d.d. e.e. f.f.x = 36x = 36 x = 25x = 25 x = 45x = 45
g.g. h.h. i.i.x = 15x = 15 x = 134x = 134 x = 75x = 75
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Find the value of the variable(s).Find the value of the variable(s).
a.a. x = 17.5x = 17.5
c.c. x = 53x = 53 d.d. x = 22x = 22
e.e. x = 32x = 32 f.f. x = 30x = 30
b.b. x = 107 y = 87x = 107 y = 87
x
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Find the value of the variable(s).Find the value of the variable(s).
a.a. b.b. c.c.x = 35 y = 37x = 35 y = 37 x = 118 y = 96x = 118 y = 96 x = 85 y = 65x = 85 y = 65
d.d. e.e. f.f.x = 26 y = 64x = 26 y = 64 x = 43 y = 32x = 43 y = 32 x = 62 y = 28x = 62 y = 28
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a. ma. m1 1 b. mb. m22 c. mc. m33= 50= 50 = 130 = 130 = 50 = 50
d. md. m44 e. me. m55 f. mf. m66= 130 = 130 = 40 = 40 = 30 = 30
Find the measure of each numbered angle in the figure.Find the measure of each numbered angle in the figure.
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Find the measure of each numbered angle in the figure.Find the measure of each numbered angle in the figure.
a. ma. m1 1 b. mb. m22 c. mc. m33= 70 = 70 = 110 = 110
d. md. m44 e. me. m55= 102 = 102 = 37 = 37
54 3
21
41 64 29
3238
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Assignment
3.5A and 3.5B3.5A and 3.5BSection 8 - 25Section 8 - 25
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Angles in PolygonsAngles in Polygons
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Objective
• Develop and use formulas for the sums of the measures of interior and exterior angles of a polygon.
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Sum of interior angles in a polygonSum of interior angles in a polygon
We already know that the sum of the interior angles in any triangle is 180°.
a + b + c = 180 °
Do you know the sum of the interior angles for any other polygons?
a b
c
We also know that the sum of the interior angles in any quadrilateral is 360°.
a
bc
d
a + b + c + d = 360 °
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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.
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The number of triangles that a polygon can be divided into is always two less than the number of sides.
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°.The sum of the interior angles in an s-sided polygon is (s – 2) × 180°.
Sum of the interior angles in a polygon
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Interior angles in regular polygonsInterior 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 triangle180° 180° ÷ 3 = 60°
Square2 × 180° = 360° 360° ÷ 4 = 90°
Regular pentagon3 × 180° = 540° 540° ÷ 5 = 108°
Regular hexagon4 × 180° = 720° 720° ÷ 6 = 120°
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Interior and exterior angles in an Interior and exterior angles in an equilateral triangleequilateral triangle
In an equilateral triangle,
60°
60°
Every interior angle measures 60°.
Every exterior angle measures 120°.
120°
120°
60°120°
The sum of the interior angles is 3 × 60° = 180°.
The sum of the exterior angles is 3 × 120° = 360°.
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Interior and exterior angles in a square
In a square,
Every interior angle measures 90°.
Every exterior angle measures 90°.
The sum of the interior angles is 4 × 90° = 360°.
The sum of the exterior angles is 4 × 90° = 360°.
90° 90°
90° 90°
90°
90°
90°
90°
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Interior and exterior angles in a regular pentagon
In a regular pentagon,
Every interior angle measures 108°.
Every exterior angle measures 72°.
The sum of the interior angles is 5 × 108° = 540°.
The sum of the exterior angles is 5 × 72° = 360°.
108°
108° 108°
108° 108°
72°72°
72°
72°
72°
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Interior and exterior angles in a regular Interior and exterior angles in a regular hexagonhexagon
In a regular hexagon,
Every interior angle measures 120°.
Every exterior angle measures 60°.
The sum of the interior angles is 6 × 120° = 720°.
The sum of the exterior angles is 6 × 60° = 360°.
120° 120°
120° 120°
120° 120°
60°
60°
60°
60°
60°
60°
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The sum of exterior angles in a polygonThe 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°.
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• 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.)
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Label the polygons as convex or concave?
• convex • concave • concave • convex
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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.
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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.
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Complete the chart below: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 90540 108 72
720 120 60900
3751 128 4
7
1080 135 45
1260 140 40(n - 2)180
n n360
(n -2 )180
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An exterior angle measure of a regular polygon is given. Find the number of its sides and the measure of each interior angle.
a. 120°
c. 36°
b. 72°
d. 24°
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.
sides845
360
3 sides 5 sides
10 sides 15 sides
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Name the convex polygon whose interior angle measures have each given sum.
a. 540° c. 1800°b. 900° d. 2520°
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. AnglesInteriortheofSum180)2s(
5 sides Pentagon
7 sides Heptagon
12 sidesDodecagon
16 sides 16-gon
720180)2s(
180
720)2s(
42s hexagon6s
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Find the value of each variableFind the value of each variable
a.a. b.b. c.c.a = 18a = 18 r = 15r = 15 y = 22.5y = 22.5 d.d.
e.e. f.f.
n = 24n = 24
x = 27x = 27 s = 18s = 18 g.g. h.h.
i.i.
x = 61.5x = 61.5 x = 72x = 72
x = 124x = 124 j.j. n = 24n = 24 k.k. a = 30a = 30 l.l. x = 90 y = 75x = 90 y = 75z = 120z = 120
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AssignmentAssignment
3.6A and 3.6B3.6A and 3.6BSection 8 - 37Section 8 - 37