1. Quadrilaterals, Diagonals, and Angles of Polygons

2. 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

3. Classifying Polygons Number of Sides Name of Polygon Number of Sides Name of Polygon 3Triangle4Quadrilateral 5Pentagon6Hexagon 7Heptagon8Octagon 9Nonagon10Decagon

4. 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?

5. 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!

6. 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

7. 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 SidesSketch of Figure Number of Triangles Sum of Angle Measurements 311(180) = 180 422(180) = 360 5 6 7

8. Angles of Polygons Mini-Lab Draw a pentagon with diagonals from one vertex to each opposing vertex

9. 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 SidesSketch of Figure Number of Triangles Sum of Angle Measurements 311(180) = 180 422(180) = 360 533(180) = 540 6 7

10. Angles of Polygons Mini-Lab Draw a hexagon with diagonals from one vertex to each opposing vertex

11. 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 SidesSketch of Figure Number of Triangles Sum of Angle Measurements 311(180) = 180 422(180) = 360 533(180) = 540 644(180) = 720 7

12. Angles of Polygons Mini-Lab Draw a heptagon with diagonals from one vertex to each opposing vertex

13. 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 SidesSketch of Figure Number of Triangles Sum of Angle Measurements 311(180) = 180 422(180) = 360 533(180) = 540 644(180) = 720 755(180) = 900

14. 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!

15. 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

16. 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 = 234021 x 180 = 3780

17. 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 = 234021 x 180 = 378028 x 180 = 5040

18. 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

19. 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

20. 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 21 x 180 = 3780 3780 / 23 = 164.35

21. 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 21 x 180 = 3780 3780 / 23 = 164.35 28 x 180 = 5040 5040 / 30 = 168

22. Homework Skill 4: Polygons (both sides) 6-3 Skills Practice: Polygons and Angles DUE TOMORROW!