DAY 2 – Exterior Angles and Problem Solving Using Interior Angles of
Polygons
Title
Goals
Measures of Interior and Exterior Angles of Polygons
Teachers will:
▪ Explore the sum of exterior angles.
▪ Problem solve using measures of interior/exterior angles
of polygons:
— Use manipulatives and interior angle measures of
polygons to identify three regular tessellations.
— Solve some problems for missing angles.
Standard
Addressed
Current Standards: 3MG2.1; 4MG3.8; 5.MG2.2; 6MG2.2
Materials for
Teacher
Materials for
Students
Description
Pattern blocks for overhead projector (normal pattern
blocks if using a document camera).
Pattern blocks, lesson handout, scissors.
Reflection
Looking Ahead
This lesson complements the other lesson looking at the
interior angles of polygons.
Common Core Standards: 3.G.CA-1; 4.MD.7; 5.G.4; 5.G.CA-2
We explore the sum of exterior angles of a polygon and
solve some problems involving interior and exterior
angles. In this lesson, we will explore regular and semiregular tessellations. We will use manipulatives to discover
which regular polygons will tessellate and which will not.
We will use geometry and measurement to investigate the
three regular and eight semi-regular tessellations.
Link to text
1
Task 1: Measures of Exterior Angles of Convex Polygons
For each of the following polygons:
a. Identify the exterior angles.
b. Find the sum of the exterior angles.
Use your findings to complete the table below.
Number of
Sides
Sum of the
Measures of
Exterior
Angles
3
4
5
6
…
n
Conjecture: In a convex polygon, the sum of the measures of one set of exterior angles is
2
Task 2: What do the measures of interior angles of regular polygons tell us?
Review — What is regular about this polygon?
a. Name this polygon.
b. Why is the figure above a regular polygon?
c. What is the sum of the interior angle measures in the figure above?
d. Find the measure of one interior angle in the figure.
3
Task 3: Measure of Interior Angles/Sum of Interior Angles of Regular Polygons
Think about it: A floor company advertises that it can cover a kitchen floor with regular
polygon shaped tiles — in fact, any shape you want they will do. Is the company advertising
correctly? Which regular polygons could actually fit around a point leaving no gaps?
1. Complete the table below. Construct a hierarchy (or a Venn diagram) showing the
relationships among the following figure: Polygon, Triangle, Scalene Triangle, Isosceles
Triangle, and Equilateral Triangle.
Number of Angles
3
4
5
6
7
8
9
10
11
12
Name of Polygon
Sum of Interior Angle Measures
Measure of Each Angle
2. Cut out the shapes on the back page and use them to create a tessellating pattern. (Note:
To save time we will use pattern blocks for the triangles, squares, and hexagons).
3. Which regular polygons will tessellate on their own (without any spaces or overlaps)?
4. Are there any mathematical reasons why these are the only shapes that will tessellate?
5. Is it possible to tile the plane using only regular octagons? Why or why not?
4
In the previous questions we explored regular tessellations. A regular tessellation is a
design covering the plane made using one type of regular polygons. A semi-regular
tessellation is made using two or more types of regular polygons. With both regular and
semi-regular tessellations, the arrangement of polygons around every vertex point must be
identical. For example, a regular tessellation made of hexagons would have a vertex
configuration of {6, 6, 6} because three hexagons surround any random vertex.
6. Use pattern blocks to create at least two semi-regular tessellations. Draw your
tessellations bellow.
7. Looking back: Define each of the following terms in your own words.
a. Regular polygon
b. Regular tessellation
c. Semi-regular tessellation
d. Vertex configuration
5
Task 4: Problems involving Interior/Exterior Angles of Polygons
1. Given the diagram below, find mA.
2. Find the measure of 2
3. Find the measures of the unknown angles.
4. Jim is tilling the kitchen with mosaic tile pieces in a pattern like the one below. If he
is using tiles that are only regular polygons, what is the measure of x?
6
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