5.1 Polygon Sum Conjecture
KEY
Guided Notes
GEOMETRY
DISCOVERING
Name_____________________________ Block_______
LEQ: How do you calculate the sum of the interior and exterior angles of a polygon?
1. Complete the INVESTIGATION “Is There a Polygon Sum Formula?” below.
Step 1: Draw three different shaped polygons (hexagons) with six sides, where one is “regular” and two
are irregular. Carefully measure all of the interior angles and then find the sum of the interior angles in
each.
Regular
Irregular
Irregular
Sum of interior angles = 720⁰
Sum of interior angles = 720⁰
Sum of interior angles = 720⁰
Step 2: Draw convex polygons with the given number of sides and fill in chart below.
Number of
sides in
polygon “n”
Sum of
measures
interior
angles
Number of
triangles
formed by
diagonals
from one
vertex.
3
4
5
6
7
8
…
1800
360o
540o
720o
900o
1080o
180(n-2)
1
2
3
4
5
6
(n-2)
Now, let’s make some conjectures.
QUADRILATERAL SUM CONJECTURE – The sum of the measures of four interior angles of any
quadrilateral is 360 degrees.
n
PENTAGON SUM CONJECTURE – the sum of the measures of five interior angles of any pentagon is 540
degrees.
If a polygon has “n” sides it is called an “n-gon.”
Step 3: Draw all diagonals from one vertex of the polygon below. How many triangles are formed?
What is the formula for number of triangles in a polygon? n-2
POLYGON SUM CONJECTURE – The sum of the measures of the “n” interior angles of an “n-gon” is
180(n-2).
GUIDED PRACTICE: Find the measures of the angles in the quadrilaterals.
1. Find the measure of angle x. x = 360 – (56 + 64 + 110) x = 130o
2. Find the measure of the missing interior angle “x” in the polygon below.
Since 180(n-2) = 720
x = 720 – (100 + 140 + 110 + 115 + 135)
x = 120
3. Find the measure of each missing exterior angle in the regular hexagon. Sum of exterior angles
always = 360o, so divide by the number of sides. Each exterior angle = 360o/6 = 60o.
3. Complete the EXERCISES on pages 259-261 # __1-10____________, using separate paper.
5.2 Exterior Angles of a Polygon
LEQ: How do you calculate the sum of the interior and exterior angles of a polygon?
1. Complete the INVESTIGATION “Is There an Exterior Angle Sum?” below.
Step 1: Below is a large polygon. Extend its sides to form a set of exterior angles.
OCTAGON
Step 2: Measure all of the interior angles of the polygon except one. Use the polygon sum conjecture to
calculate the measure of the remaining interior angle.
The measure of the remaining interior angle is Start: 180(n-2)/n
= 180(6)/8 = 135
Step 3: Using the Linear Pair Conjecture, find the measure of each corresponding exterior angle.
The measure of each corresponding exterior angle is 45o (Supplement of 135o) degrees.
Step 4: Calculate the sum of the measures of the exterior angles. The sum is 45o(8) = 360odegrees.
EXTERIOR ANGLE SUM CONJECTURE: For any polygon, the sum of the measures of a set of exterior
angles is360o .
EQUIANGLULAR POLYGON CONJECTURE: You can find the measure of each interior angle of an
equiangular n-gon by using either of these formulas:
180 – (360/n)
or
180(n-2)/n
GUIDED PRACTICE:
Complete each statement.
1.
The number of triangles formed in an octagon when all the diagonals from one vertex are
drawn is
6.
2.
The sum of the measures of the n interior angles of an n-gon is 180(n-2)
3.
The sum of the measures of the exterior angles of a 30-gon is 360o
4.
The measure of one angle in a regular decagon is 180 (n-2)/n = 180(8)/10 =144o .
5.
If the measure of one exterior angle of a regular polygon is 30°, then the polygon has
360/30 = 12 sides.
Find each lettered angle measure.
6.
a = 120o
b = 60o
c = 60o
d = 120o
7.
m=
65o
n=
135o
p=
65o
r=
112.5o
s=
52o
t=
67.5o
2. Complete the EXERCISES on pages 263-265 # ________________, using separate paper.
5.3 Kite and Trapezoid Properties
LEQ: How are the properties of kites and trapezoids determined?
1. Complete the VOCABULARY chart below. Begin on page 268.
Term
1. Kite
Definition
A quadrilateral with two distinct
pairs of congruent consecutive
sides.
Picture/Symbol
A kite has one line of reflectional
symmetry (AC).
BD and AC represent diagonals.
2. vertex angles
and non-vertex
angles
Think of two isosceles triangles
attached at their base and that is
how we name the vertex angles.
The two angles between each
pair of congruent sides is the
vertex angle.
2. Complete the INVESTIGATION 1 “What are Some Properties of Kites?” below:
Non-vertex….congruent
perpendicular
Perpendicular bisector
Vertex, bisected, diagonal
3. Complete INVESTIGATION 2 “What are Some Properties of Trapezoids?” below.
Supplementary
congruent
congruent
GUIDED PRACTICE:
3. x = 180 – 119 = 61
y = 119
4. 2x = 111- (23 + 30)
2x = 111 – (53)
2x = 58
x = 29
4. Complete the EXERCISES on pages 271-274 # ____________________, using separate paper.
5.4 Properties of Mid-segments
LEQ: How does the property of triangle mid-segment extend to trapezoids?
1. Complete INVESTIGATION 1 “Triangle Mid-Segment Properties” below.
four congruent triangles
parallel, half, the third side
2. Mid-segment of a trapezoid conjecture - The mid segment of a trapezoid is parallel to the bases and is
equal in length to the average of the lengths of the bases.
GUIDED PRACTICE:
Find the measures.
40o
120o
10 units
26 units
3. Complete EXERCISES on pages 277-280 # ______________________, using separate paper.