ANGLES of a CONVEX POLYGONNNNNNNNNN.pptx

ANGLES OF A CONVEX POLYGON

1. Prayer…

2. “NAME ME!” TRIANGLE QUADRILATERAL

3. “NAME ME!” PENTAGON HEXAGON

4. “NAME ME!” HEPTAGON OCTAGON

5. Angles of a Convex Polygon Prepared by: Karlyn B. Ancheta

6. Objectives: At the end of the lesson, the students should be able to to : ∙ inductively derive the relationship between exterior and interior angles of a convex polygon; ∙ determine the sum of the measures of the interior and exterior angles of a polygon.

7. Interior Angle An interior angle is an angle formed by two consecutive sides of polygon. A B C D E F HEXAGON ABCDEF

8. Exterior Angle An exterior angle is an angle formed when one side of a polygon is extended outside. Exterior angle Interior angle

9. Sum of the Interior Angles of a Polygon The sum of all the interior angles of a triangle is ALWAYS 180º. X Y Z Δ XYZ Interior angles ∠X, ∠Y, and ∠Z m ∠X + m∠Y + m∠Z = 180º

10. Sum of the Interior Angles of a Polygon To calculate the sum of the interior angles of a polygon, use the formula below. 180º(n-2) Where n is the number of sides of the polygon.

11. Example 1: Find the sum of the interior angles of a nonagon. n = 9 = 180º (n – 2) = 180º (9 – 2) = 180º (7) = 1,260º sum of the interior angles of a nonagon.

12. Example 2: Find the sum of the interior angles of a 16-sided polygon. n = 16 = 180º (n – 2) = 180º (16 – 2) = 180º (14) = 2520 º sum of the interior angles of a 16-sided polygon.

13. IT’S YOUR TURN!

14. Who can draw or create a pentagon ? How about the hexagon ? A octagon . A triangle . A heptagon .

15. Measure of One Interior Angle of a Regular Polygon To calculate the measure of one interior angle of a regular polygon, use the formula below. 180º(n-2) n Where n is the number of sides of the polygon.

16. Example 1: Find the measure of one interior angle of a regular octagon. n = 8 180º (n – 2) n 180º (n – 2) 8 180º (6) 8 1080º 8 = 135º = = = =

17. Example 2: Determine the measure of one interior angle of a regular dodecagon. n = 12 180º (n – 2) n 180º (n – 2) 12 180º (10) 12 1800º 12 = 150º = = = =

18. Sum of the Exterior Angles of a Polygon The sum of the exterior angles of every polygon is ALWAYS 360º. Use the formula below, find the measure of the exterior angles of a regular polygon. 360º n

19. Example 1: Find the sum of the exterior angles of the following polygons: a. Regular Hexagon 360º 6 = 60º b. Regular Undecagon 360º 11 = 32.72 or 32 8/11

20. Relationship Between Interior and Exterior Angles of a Polygon The sum of the interior angle and its corresponding exterior angle of a polygon is always 180º . These angles form a linear pair because they are adjacent and supplementary.

21. Exterior Angle Theorem This measure of the exterior angle is equal to the sum of the two remote interior angles. a b d c m ∠a + m∠b = m∠d m ∠c + m∠d = 180º

22. Example 1: Find the value of k and x . 62º X 47º K 47º + m ∠K = 180 º m ∠K = 180 º - 47 º m∠K = 133 º 62º + x = K 62º + x = 133º x = 133º - 62º x = 71º = 133º = 71º

23. Example 2: Find the value of x. 65º (2x + 25)º (7x + 5)º 7(17) + 5 119 + 5 = 124 º 2(17) + 25 34 + 25 = 59 65º + 2x + 25 = 7x + 5 2x – 7x = 5 – 90 -5x/-5 = -85/-5 X=17 65º + 59º = 124º = 59º = 124º

24. Example 3: Find the value of m ∠A, m∠B , m∠C , and m∠D . m ∠A + 87 º = 180º m ∠A = 180º - 87º m ∠A = 93º m ∠B + 122 º = 180 º m ∠B = 180 º - 122 º m ∠B = 58 º A C D B 87º 122º

25. Example 3: Find the value of m ∠A, m∠B , m∠C , and m∠D . m ∠C + 87 º = 122º m ∠C = 122º - 87º m ∠C = 35º m ∠C + m ∠D = 180 º 35 º + m ∠D =180 º m ∠D = 145 º A C D B 87º 122º 93º 58º 35º 145º

26. IT’S YOUR TURN!

27. ANSWER ME! Identify what is being asked: What is a polygon that has five-sides? It is a polygon that has seven-sides. It is a polygon that has three-sides. What polygons have six-sides? What polygons have four-sides?

28. Assignment: Give the meaning of the following: Circle 2. Central Angles 3. Inscribed Angles