Geometry Name: _______________________

**Lesson 11.1: Angle Measures in
Polygons **Date: __________________

**
Essential Question : How is the measure of
an interior angle or an exterior angle of a polygon found?**

Concepts that you need to remember for today’s lesson:

1. What is the sum of the measures of the angles in a triangle? __________

2. What is *inductive reasoning*?
__________________________________________

________________________________________________________________________

3. What is a *regular* *polygon*?
__________________________________________

4. What is the sum of the degrees around a single point? __________

** **

**Complete the following chart: **

Diagram of Number Number of Sum of the measures

__a polygon__: __of sides__:
__triangles inside__: __of the interior angles__:

3 1 180° x 1 = __________

4 2 180° x 2 = __________

____ ____ ___________________

____ ____ ___________________

____ ____ ___________________

____ ____ ___________________

Notice that there is a pattern in the relationship that
exists between the number of sides a shape has and the number of triangles
that can be drawn inside it. If a polygon has *n* sides, how many
triangles can be drawn inside the polygon? ______________

If a polygon has *n* sides, what is **the
sum of the measures of its interior angles**?

**(This is
an important formula!!!)**

What is the formula for finding the ** sum
**of the measures of the interior angles in a polygon with

Example 1: Find the sum of the measures of the interior angles of ...

a. ... a hexagon b. ... a nonagon (9-sided) c. ... a 14 – gon

Since
the angles in a *regular polygon* are congruent, the measure of **one
interior angle of a regular polygon** can be found by dividing the sum of
the measures of the interior angles by the number of angles.

**The formula for the measure of each interior
angle of a regular polygon: **

**(Another
important formula!!!)**

** **

Example 2: Find the measure of one interior angle for ...

a. ... a regular pentagon b. ... a regular octagon c. … a regular 12-gon

Example 3: The measure of each interior angle of a regular polygon is 165°. How many sides does the polygon have?

Consider the following diagram of a convex polygon and
with its **exterior** **angles** shaded.

If we were to cut out the exterior angles we could slide them together at their vertices.

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Notice that when they are brought together, the 3 exterior angles fit together perfectly all the way around a point. How many degrees are there all the way around a point? ______

The diagram can be extended to polygons with different numbers of sides.

**The** **formula for the sum of the measures of
the exterior angles of a polygon:**

**(Again, an important formula!!!)**

Example 4: Find the sum of the measures of the exterior angles for ...

a. ... a quadrilateral b. ... a pentagon c. ... a n-gon

Since
the angles in a *regular polygon* are all congruent, the exterior angles
are also all congruent. This means that the measure of **one exterior angle of
a regular polygon** can be found by dividing the sum of the measures of the
exterior angles by the number of exterior angles.

**The formula for the measure of each exterior
angle of a regular polygon: **

**(Yet another important formula!!!)**

Example 5: Find the measure of each exterior angle for a regular polygon that is ...

a. ... a quadrilateral b. ... a pentagon c. ... a n-gon

Example 6: Find the number of sides the regular polygon has if the measure of one exterior angle is ...

a. ... 60˚ b. ... 30˚