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Polygon Properties and Proportions Activity

Engage in a hands-on activity to explore polygon properties, angles, and proportions. Practice solving proportions and learn about the similarities between polygons.

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Polygon Properties and Proportions Activity

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  1. Warm Up • Solve each proportion. • 1. 3 = 15 3. x 20 • 2. x 35 3 4 24 4 y 7 5

  2. ` ACTIVITY 30min • Need glue & scissors • Critical Thinking • Cut out each figure (one at a time) and rearrange pieces to form a square. • Glue pieces in the square provided. Materials Dissections of Squares

  3. Polygons and Similarities 6.1 and Chapter 7

  4. 6.1 Properties and Attributes of Polygons • To identify and name polygons • To find the sum of the measures of interior and exterior angles of convex polygons and measures of interior and exterior angles of regular polygons • To solve problems involving angle measures of polygons

  5. Define Polygons • A closed figure • Formed by segments

  6. Polygons vs. Not Polygons

  7. Regular Polygon • Convex polygon where all sides and angles are congruent.

  8. Types of Polygons • A Convex polygon is a polygon such that no lines containing a side of the polygon contains a point in the interior of the polygon • ~Angles face out • A Concave polygon is a polygon for which there is a line containing a side of the polygon and a point in the interior of the polygon. • ~ Angles cave in

  9. Convex or Concave

  10. Possible Diagonals/Triangles: Find the sum of degrees Determine how many triangles a shape have by drawing diagonals. Each triangle is 180 degrees.

  11. Thm 6-1-1: Polygon Angle Sum Theorem • Formula: S = 180 (n – 2) • S= sum of the measures of interior angles • N = number of sides Sum S = 180(n -2) S = 180(3-2) S = 180(1) S = 180 An interior angle I = S/n I = 180/3 I = 60

  12. Thm6-1-2: Polygon Exterior Angle Sum Theorem • The sum of the measures of the exterior angles is 360° • To find one exterior angle use formula • Formula: E = 360/ n • E= measure of exterior angles • N = number of sides E = 360/n E = 360/3 E = 120

  13. Warm-Up: Polygons • Number of sides: • Name Polygon: • Number of Triangles: • Is polygon convex or concave. • Find sum of the measures of interior angles: • Find the measure of an interior angle • Find measure of an exterior angle: 6 Hexagon (n- 2) = 4 S = 180(n-2) 180(4)= 720 I = Sum n 720 = 120 6 360 = 60 6 E = 360 n

  14. Twitter #polygonsproperties • P. 397 Think and Discuss #1 • Draw a concave pentagon and convex pentagon. Explain the difference between the two figures.

  15. Textbook p. 395

  16. Practice • Geo: Textbook page 398 # 1- 15

  17. Warm-Up: Polygons • Number of sides: • Name Polygon: • Number of Triangles: • Is polygon convex or concave. • Find sum of the measures of interior angles: • Find the measure of an interior angle • Find measure of an exterior angle: 7 Heptagon (n- 2) = 5 S = 180(n-2) 180(5)= 900 I = Sum n 900 = 128.57 7 360 = 51.43 7 E = 360 n

  18. Warm Up: Part 2 • Solve each proportion. • 1. x = 11 2. 13 = 26 3. x-2 = 3 5 35 49 7x x 8 • Cross Multiply • - 35(x) = 11(5) • - 35x = 55 • 35x = 55 • 35 35 • - x = 1.57 • Cross Multiply • - 13(7x) = 49(26) • - 91x = 1274 • 91x = 1274 • 91 91 • - x = 14 • Cross Multiply • - 8(x-2) = 3(x) • - 8x -16 = 3x • 8x -8x -16 = 3x – 8x • - -16 = -5x • -16 = -5x • -5 -5 • - x = 3.2

  19. Practice • Geo: Textbook page 398 # 1- 15

  20. Chapter 7: Connecting Proportion and Similarity • Recognize and use ratios and proportions • Identify similar figures and use the proportions of similar figures to solve problems • Use proportional parts of triangles to solve problems

  21. Exploring Similar Polygons • Book Definition: • Two polygons are similar if and only if their corresponding angles are congruent and the measures of their corresponding sides are proportional. • Symbol: • In simpler terms: • Two polygons with the same shape but are different sizes.

  22. Are two congruent figures similar? • Think about it…. • Discuss… • Yes, congruent figures have congruent angles and sides are proportional at 1:1 ratio

  23. Scale Factor and Dilation • Dilation is a transformation that reduces or enlarges figures • On a camera: zoom-in/ zoom-out • Scale Factor = ______ A 5 D 2.5 4 B 3 C E 10 H 5 dfgff 8 F 6 G 10 = 5 8 = 4 6 = 3 5 2.5

  24. Scale Factor Scale Factor = _____ 4 8 3 12 24 9 • Scale Factor = ______ A 6 D 2 5 B 4 C E 12 H 4 dfgff 8 F 7 G 12 = 4 9 = 3 24 8 12 = 6 8 = 5 7 = 4 4 2

  25. Similar Triangles: Property 1 Side-Side-Side Similarity • __________________________: if _____________________ of one triangle are ____________ to __________________________ of another triangle, then the triangles are ____________. • Ex. the three sides proportional the three corresponding sides similar

  26. Similar Triangles: Property 2 • _________________________: If the measures of ___________ of ___________ are __________________________________ then the triangles are __________. Ex. Angle - Angle Similarity two angles one triangle congruent to 2 angles of the other similar

  27. Similar Triangles: Property 3 Side-Angle-Side Similarity • ________________________: if the measures of __________ of a triangle are ___________ to the measures of _________ of another triangle and the _________ angles are _________, then the triangles are ___________. Ex. two sides proportional two sides included congruent similar

  28. Similarity Ratio:

  29. Triangle Proportionality Theorem • VW = VX WY XZ

  30. Similarity Ratio: • AE = AD = ED AC AB CB

  31. Similarity Ratio: • EC = ED = CD EA EB AB X

  32. Solve for x and y. • 49 x • y+3 • 20 29 • 21

  33. Warm- up: • 1. Draw a convex heptagon. • 2. Draw a concave nonagon • 3. Sketch a regular hexagon. • Sum of interior angles: • An Interior angle: • An Exterior angle: S = 180(6-2) = 720 I = 720/6 = 120 E = 360/6 = 60

  34. Check Practice: Similar Figures • Notes: Sides are proportional & angles are congruent. • P. 520 # 5, 6, 12, 13, 16, 17, 19

  35. Quiz Time 10 mins • Absolutely NO TALKING during quiz. • Take your time • Good LUCK!

  36. Practice • Find a partner. • Complete “How Can You Tell Which End of a Worm Is His Head?” (must complete in pairs). • Each partner complete a side • For each answer, write in letter of question. EACH SIDE IS DIFFERENT! • Each partner must show work to receive FULL CREDIT. Use notebook paper

  37. Chapter 7: Practice

  38. Warm-Up • 2. Are two congruent figures similar? Explain. • 3. Quadrilaterals ABCD and AEFG are similar. AE=22, AB=5x + 4, EF=8 and BC=2x-2. Solve for x. • A E B G F D C • 1. Use the Interior Angle Theorem to find measure of each angle. C • B 3x-12 • 4x-14 • A 7x +5 5x -15 D • 5x • E

  39. Warm -Up 16 22 11 m 3 8 4 4 x

  40. Warm up • 1. Similar? Yes or No • a. 4 6 1.3 1.8 • 1.4 • 7 b. 6 8 15 20 5 12.5 • 2. What is the difference between concave and convex polygons? • 3. regular heptagon, find each. • A. Sum of I = • B. I= • C. E= • D. Sum of E=

  41. Constructions • Materials: paper, ruler and protractor • Construct each (use polygon chart) • Regular Pentagon ~ sides 3 inches • Regular Octagon ~ sides 2 inches • Your Choice ~ Regular Polygon ~ make sides a reasonable measure • To start… find the measure of one interior angle. Draw one measured side with ruler. • Next use protractor to measure angle • Then draw 2nd side • Repeat steps.

  42. Practice: Constructions • Construct each (use polygon chart) • Materials: paper, ruler and protractor • Include the sum of interior angles and the measure of one interior angle • Regular Pentagon ~ sides 3 inches • Regular Octagon ~ sides 2 inches • Your Choice ~ Regular Polygon (4+ sides) ~ make sides a reasonable measure • Bonus: Your Choice ~ Regular Polygon (4+ sides) ~ make sides a reasonable measure • To start… Draw one measured side with ruler. • Next find the measure of one interior angle and use protractor to measure interior angle • Then draw 2nd side • Repeat steps.

  43. Warm up • 1. Similar? Yes or No • a. 4 6 1.3 1.8 • 1.4 • 7 b. 6 8 15 20 5 12.5 • 2. What is the difference between concave and convex polygons? • 3. regular heptagon, find each. • A. Sum of I = • B. I= • C. E= • D. Sum of E=

  44. Bonus • 1. Draw if needed. Joyce sighted to the top of a tree along a stake that she knew to be 3 feet high. If she is standing 2 feet from the stake and 18 feet from the tree, how high is the tree? • 2. Find x. • 2x+32 8x+5 • 10x+2 12x+3 4x+12 4x+2 • 3. The lengths of the sides of a triangle are 3, 4, and 6. If the length of the shortest side of a similar triangle is 5, find the lengths of its other two sides.

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