Download Unit 5 - Properties of Polygons notes

5.1 Polygon Sum Conjecture
Guided Notes
DISCOVERING GEOMETRY
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: Three different shaped polygons (hexagons) with six sides are drawn below, 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 =
Sum of interior angles =
Sum of interior angles =
Step 2: Draw convex polygons with the given number of sides and fill in chart below.
Number
3
of sides
Sum of
1800
measures
interior
Number
1
of
triangles
formed
by
diagonals
from one
vertex.
4
5
6
7
8
…
Now, let’s make some conjectures.
QUADRILATERAL SUM CONJECTURE – The sum of the measures of four interior angles of any
quadrilateral is______________ degrees.
n
PENTAGON SUM CONJECTURE – the sum of the measures of five interior angles of any pentagon is
_______________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? ______________________________
POLYGON SUM CONJECTURE – The sum of the measures of the “n” interior angles of an “n-gon” is
________________________.
GUIDED PRACTICE: Find the measures of the angles in the quadrilaterals.
1. Find the measure of angle x. __________________
2. Find the measure of the missing interior angle “x” in the polygon below. __________________
3. Find the measure of each missing exterior angle in the regular hexagon. __________________
2. Complete the EXERCISES on pages 259-261 # __________________, 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.
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 _______________ degrees.
Step 3: Using the Linear Pair Conjecture, find the measure of each corresponding exterior angle.
The measure of each corresponding exterior angle is ________________ degrees.
Step 4: Calculate the sum of the measures of the exterior angles. The sum is ___________ degrees.
EXTERIOR ANGLE SUM CONJECTURE: For any polygon, the sum of the measures of a set of exterior
angles is _________________ .
EQUIANGLULAR POLYGON CONJECTURE: You can find the measure of each interior angle of an
equiangular n-gon by using either of these formulas: ________________ or _________________.
GUIDED PRACTICE:
Complete each statement.
1.
The number of triangles formed in an octagon when all the diagonals from one vertex are
drawn is
.
2.
The sum of the measures of the n interior angles of an n-gon is
3.
The sum of the measures of the exterior angles of a 30-gon is
4.
The measure of one angle in a regular decagon is
5.
If the measure of one exterior angle of a regular polygon is 30°, then the polygon has
sides.
Find each lettered angle measure.
6.
a=
b=
c=
d=
.
.
.
7.
m=
__________
n=
__________
p=
__________
r=
__________
s=
__________
t=
__________
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
Picture/Symbol
2. vertex angles
and non-vertex
angles
2. Complete the INVESTIGATION 1 “What are Some Properties of Kites?” below:
3. Complete INVESTIGATION 2 “What are Some Properties of Trapezoids?” below.
GUIDED PRACTICE:
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.
2. Mid-segment of a trapezoid conjecture _____________________________________________________________________________________
_____________________________________________________________________________________
GUIDED PRACTICE:
Find the measures of the angles.
3. Complete EXERCISES on pages 277-280 # ______________________, using separate paper.
5.5 A - Alg Skills: Writing Linear Equations
GEOMETRY
Guided Notes
DISCOVERING
Name _______________________________ Block________
LEQ: How are linear equations written?
1. Study the examples A, B and C below.
GUIDED PRACTICE:
Practice writing linear equations, given the graphs below.
1. Using slope and y-intercept:
2. Using two points:
3. Find the equation for the perpendicular line, given the graph of a line:
2. Complete EXERCISES on pages 289-290 # ____________________, using separate paper
5.5 B - Properties of Parallelograms
LEQ: What are the basic properties of a parallelogram?
1. Complete INVESTIGATION “Four Parallelogram Properties” below.
GUIDED PRACTICE:
1.
2. Complete EXERCISES on pages 283-286 # _____________________, using separate paper.
5.6 Properties of Special Parallelograms
LEQ: How are properties of rectangles, rhombi and squares used to determine special parallelograms?
1. Complete VOCABULARY below. Begin on page 291-294.
Terms
1. rhombus
Definition
Picture/Symbol
2. rectangle
3. square
2. Complete INVESTIGATION 1 “What Can You Draw with the Double-Edged Straightedge?”
3. Do Rhombus Diagonals Have Special Properties? Draw a picture of a rhombus to illustrate this.
Explain:
_____________________________________________________________________________________
_____________________________________________________________________________________
4. Complete INVESTIGATION 3 “Do Rectangle Diagonals Have Special Properties?
GUIDED PRACTICE:
5. Complete EXERCISES on pages 294-297 # ___________________, using separate paper.
5.7 Proving Quadrilateral Properties
LEQ: How do we prove properties of quadrilaterals?
1. What is a “dart”? See page 298.
2. What does QED mean? See page 299.
3. Study the EXAMPLE below. Notice the flow of the proof.
GUIDED PRACTICE:
2. Complete EXERCISES on pages 300-303 # __________________