Quadrilaterals, Diagonals, and Angles of Polygons.

Quadrilaterals, Diagonals, and Angles of Polygons

Quadrilaterals, Diagonals, and Angles of Polygons

• A Polygon is a simple closed plane figure, having three or more line segments as sides

• A Quadrilateral is any four-sided closed plane figure

• A Diagonal a line segment that connects one vertex to another (but not next to it) on a polygon

Classifying Polygons

Number of Sides

Name of Polygon

Number of Sides

Name of Polygon

3 Triangle 4 Quadrilateral

5 Pentagon 6 Hexagon

7 Heptagon 8 Octagon

9 Nonagon 10 Decagon

Quadrilateral Angles

• We know that the interior angles of a triangle add up to 180 degrees

• How many degrees are in the interior angles of a quadrilateral?

Quadrilateral Angles• If we draw a diagonal from one vertex across to

the opposite vertex, we see that we have formed two triangles

• Therefore, the sum of two triangles will give you the measure of the interior angles of a quadrilateral

• 180 + 180 = 360 degrees!

Quadrilateral Angles Checkpoint

• Find the missing angle of a quadrilateral with the following measures:

m 1 = 117

m 2 = 110

m 3 = 75

m 4 =117 + 110 + 75 + x = 360

302 + x = 360

x = 58

Angles of Polygons Mini-Lab

• Let’s explore this knowledge and how it relates to the angles of other polygons

• Copy and complete the table below:

Number of Sides Sketch of Figure

Number of Triangles

Sum of Angle Measurements

3 1 1(180) = 180

4 2 2(180) = 360

5

6

7

Angles of Polygons Mini-Lab

• Draw a pentagon with diagonals from one vertex to each opposing vertex

Angles of Polygons Mini-Lab

• Let’s explore this knowledge and how it relates to the angles of other polygons

• Copy and complete the table below:

Number of Sides Sketch of Figure

Number of Triangles

Sum of Angle Measurements

3 1 1(180) = 180

4 2 2(180) = 360

5 3 3(180) = 540

6

7

Angles of Polygons Mini-Lab

• Draw a hexagon with diagonals from one vertex to each opposing vertex

Angles of Polygons Mini-Lab

• Let’s explore this knowledge and how it relates to the angles of other polygons

• Copy and complete the table below:

Number of Sides Sketch of Figure

Number of Triangles

Sum of Angle Measurements

3 1 1(180) = 180

4 2 2(180) = 360

5 3 3(180) = 540

6 4 4(180) = 720

7

Angles of Polygons Mini-Lab

• Draw a heptagon with diagonals from one vertex to each opposing vertex

Angles of Polygons Mini-Lab

• Let’s explore this knowledge in how it relates to the angles of other polygons

• Copy and complete the table below:

Number of Sides Sketch of Figure

Number of Triangles

Sum of Angle Measurements

3 1 1(180) = 180

4 2 2(180) = 360

5 3 3(180) = 540

6 4 4(180) = 720

7 5 5(180) = 900

Angles of Polygons Mini-Lab

• What patterns do you see as a result of our experiment?

• The number of triangles in any polygon is always two less than the number of sides.

• Therefore, if n = the number of sides of the polygon; the sum of interior angles of any polygon can be expressed as (n – 2)180!

Angles of Polygons Checkpoint

• Find the sum of the measures of the interior angles of each polygon:

15-gon? 23-gon? 30-gon?(15-sided figure) (23-sided figure) (30-sided figure)

13 x 180 = 2340

Angles of Polygons Checkpoint

• Find the sum of the measures of the interior angles of each polygon:

15-gon? 23-gon? 30-gon?(15-sided figure) (23-sided figure) (30-sided figure)

13 x 180 = 2340 21 x 180 = 3780

Angles of Polygons Checkpoint

• Find the sum of the measures of the interior angles of each polygon:

15-gon? 23-gon? 30-gon?(15-sided figure) (23-sided figure) (30-sided figure)

13 x 180 = 2340 21 x 180 = 3780 28 x 180 = 5040

Regular Polygons• A regular polygon is one that is equilateral

(all sides congruent) and equiangular (all angles congruent)

• Polygons that are not regular are said to be irregular

• If the formula for finding the sum of measures of interior angles of a polygon is (n-2)180, how would you find the measure of each angle of a regular polygon?

( n – 2 )180

n

Regular Polygons Checkpoint

• Find the sum of the measures of the interior angles of each regular polygon and the measure of each individual angle:

15-gon? 23-gon? 30-gon?(15-sided figure) (23-sided figure) (30-sided figure)

13 x 180 = 2340

2340 / 15 = 156

Regular Polygons Checkpoint

• Find the sum of the measures of the interior angles of each regular polygon and the measure of each individual angle:

15-gon? 23-gon? 30-gon?(15-sided figure) (23-sided figure) (30-sided figure)

13 x 180 = 23402340 / 15 = 156

21 x 180 = 3780

3780 / 23 = 164.35

Regular Polygons Checkpoint

• Find the sum of the measures of the interior angles of each regular polygon and the measure of each individual angle:

15-gon? 23-gon? 30-gon?(15-sided figure) (23-sided figure) (30-sided figure)

13 x 180 = 23402340 / 15 = 156

21 x 180 = 37803780 / 23 = 164.35

28 x 180 = 5040

5040 / 30 = 168

Homework

• Skill 4: Polygons (both sides)

• 6-3 Skills Practice: Polygons and Angles

• DUE TOMORROW!