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Understanding Polygons: Concavity, Regularity, and Diagonals

Explore properties of polygons through examples and theorems. Learn about concave vs. convex, equilateral, equiangular, and regular polygons. Discover the role of diagonals in polygons. Practice with parallelograms and proving quadrilaterals are parallelograms. Delve into special parallelograms like rhombuses.

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Understanding Polygons: Concavity, Regularity, and Diagonals

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  1. 6.1 Polygons Day 1 Part 1 CA Standards 7.0, 12.0, 13.0

  2. Warmup • Solve for the variables. • 1. 10 + 8 + 16 + A = 36 • 2. 6 + 15 + 9 + 3B = 36 • 3. 10 + 8 + 2X + 2X = 36 • 4. 4R + 10 + 108 + 67 + 3R = 360

  3. What is polygon? • Formed by three or more segments (sides). • Each side intersects exactly two other sides, one at each endpoint. • Has vertex/vertices.

  4. Polygons are named by the number of sides they have. Fill in the blank. Quadrilateral Pentagon Hexagon Heptagon Octagon

  5. Concave vs. Convex • Convex: if no line that contains a side of the polygon contains a point in the interior of the polygon. • Concave: if a polygon is not convex. interior

  6. Example • Identify the polygon and state whether it is convex or concave. Convex polygon Concave polygon

  7. A polygon is equilateral if all of its sides are congruent. • A polygon is equiangular if all of its interior angles are congruent. • A polygon is regular if it is equilateral and equiangular.

  8. Decide whether the polygon is regular. ) )) ) ) ) ) )) ) )

  9. A Diagonal of a polygon is a segment that joins two nonconsecutive vertices. diagonals

  10. Interior Angles of a Quadrilateral Theorem • The sum of the measures of the interior angles of a quadrilateral is 360°. B m<A + m<B + m<C + m<D = 360° C A D

  11. Example • Find m<Q and m<R. x + 2x + 70° + 80° = 360° 3x + 150 ° = 360 ° 3x = 210 ° x = 70 ° Q x 2x° R 80° P 70° m< Q = x m< Q = 70 ° m<R = 2x m<R = 2(70°) m<R = 140 ° S

  12. Find m<A C 65° D 55° 123° B A

  13. Use the information in the diagram to solve for j. 60° + 150° + 3j ° + 90° = 360° 210° + 3j ° + 90° = 360° 300° + 3j ° = 360 ° 3j ° = 60 ° j = 20 60° 150° 3j °

  14. 6.2 Properties of Parallelograms Day 1 Part 2 CA Standards 4.0, 7.0, 12.0, 13.0, 16.0, 17.0

  15. Theorems • If a quadrilateral is a parallelogram, then its opposite sides are congruent. • If a quadrilateral is a parallelogram, then its opposite angles are congruent. Q R S P

  16. Theorems • If a quadrilateral is a parallelogram, then its consecutive angles are supplementary. m<P + m<Q = 180° m<Q + m<R = 180° m<R + m<S = 180° m<S + m<P = 180° Q R S P

  17. Using Properties of Parallelograms • PQRS is a parallelogram. Find the angle measure. • m< R • m< Q Q 70 ° R 70 ° + m < Q = 180 ° m< Q = 110 ° 70° P S

  18. Using Algebra with Parallelograms • PQRS is a parallelogram. Find the value of h. P Q 3h 120° S R

  19. Theorems • If a quadrilateral is a parallelogram, then its diagonals bisect each other. R Q M P S

  20. Using properties of parallelograms • FGHJ is a parallelogram. Find the unknown length. • JH • JK 5 5 F G 3 3 K J H

  21. Examples • Use the diagram of parallelogram JKLM. Complete the statement. LM K L NK <KJM N <LMJ NL MJ J M

  22. Find the measure in parallelogram LMNQ. • LM • LP • LQ • QP • m<LMN • m<NQL • m<MNQ • m<LMQ 18 8 L M 9 110° 10 10 9 P 70° 8 32° 70 ° Q N 18 110 ° 32 °

  23. Pg. 325 # 4 – 20, 24 – 34, 37 – 46 • Pg. 333 # 2 – 39

  24. 6.3 Proving Quadrilaterals are Parallelograms Day 2 Part 1 CA Standards 4.0, 7.0, 12.0, 17.0

  25. Warmup • Find the slope of AB. • A(2,1), B(6,9) m=2 • A(-4,2), B(2, -1) m= - ½ • A(-8, -4), B(-1, -3) m= 1/7

  26. Review

  27. Using properties of parallelograms. • Method 1 Use the slope formula to show that opposite sides have the same slope, so they are parallel. • Method 2 Use the distance formula to show that the opposite sides have the same length. • Method 3 Use both slope and distance formula to show one pair of opposite side is congruent and parallel.

  28. Let’s apply~ • Show that A(2,0), B(3,4), C(-2,6), and D(-3,2) are the vertices of parallelogram by using method 1.

  29. Show that the quadrilateral with vertices A(-3,0), B(-2,-4), C(-7, -6) and D(-8, -2) is a parallelogram using method 2.

  30. Show that the quadrilateral with vertices A(-1, -2), B(5,3), C(6,6), and D(0,7) is a parallelogram using method 3.

  31. Proving quadrilaterals are parallelograms • Show that both pairs of opposite sides are parallel. • Show that both pairs of opposite sides are congruent. • Show that both pairs of opposite angles are congruent. • Show that one angle is supplementary to both consecutive angles.

  32. .. continued.. • Show that the diagonals bisect each other • Show that one pair of opposite sides are congruent and parallel.

  33. Show that the quadrilateral with vertices A(-1, -2), B(5,3), C(6,6), and D(0,7) is a parallelogram using method 3.

  34. Show that A(2,-1), B(1,3), C(6,5), and D(7,1) are the vertices of a parallelogram.

  35. 6.4 Rhombuses, Rectangles, and Squares Day 2 Part 2 CA Standards 4.0, 7.0, 12.0, 17.0

  36. Review • Find the value of the variables. p h 52° (2p-14)° 50° 68° p + 50° + (2p – 14)° = 180° p + 2p + 50° - 14° = 180° 3p + 36° = 180° 3p = 144 ° p = 48 ° 52° + 68° + h = 180° 120° + h = 180 ° h = 60°

  37. Special Parallelograms • Rhombus • A rhombus is a parallelogram with four congruent sides.

  38. Special Parallelograms • Rectangle • A rectangle is a parallelogram with four right angles.

  39. Special Parallelogram • Square • A square is a parallelogram with four congruent sides and four right angles.

  40. Corollaries • Rhombus corollary • A quadrilateral is a rhombus if and only if it has four congruent sides. • Rectangle corollary • A quadrilateral is a rectangle if and only if it has four right angles. • Square corollary • A quadrilateral is a square if and only if it is a rhombus and a rectangle.

  41. Example • PQRS is a rhombus. What is the value of b? Q 2b + 3 = 5b – 6 9 = 3b 3 = b P 2b + 3 R S 5b – 6

  42. Review • In rectangle ABCD, if AB = 7f – 3 and CD = 4f + 9, then f = ___ • 1 • 2 • 3 • 4 • 5 7f – 3 = 4f + 9 3f – 3 = 9 3f = 12 f = 4

  43. Example • PQRS is a rhombus. What is the value of b? Q 3b + 12 = 5b – 6 18 = 2b 9 = b P 3b + 12 R S 5b – 6

  44. Theorems for rhombus • A parallelogram is a rhombus if and only if its diagonals are perpendicular. • A parallelogram is a rhombus if and only if each diagonal bisects a pair of opposite angles. L

  45. Theorem of rectangle • A parallelogram is a rectangle if and only if its diagonals are congruent. A B D C

  46. The diagonals are congruent Both pairs of opposite sides are congruent Both pairs of opposite sides are parallel All angles are congruent All sides are congruent Diagonals bisect the angles Parallelogram Rectangle Rhombus Square Match the properties of a quadrilateral B,D A,B,C,D A,B,C,D B,D C,D C

  47. 6.5 Trapezoid and Kites Day 3 Part 1 CA Standards 4.0, 7.0, 12.0

  48. Which of these sums is equal to a negative number? (4) + (-7) + (6) (-7) + (-4) (-4) + (7) (4) + (7) In the first seven games of the basketball season, Cindy scored 8, 2, 12, 6, 8, 4 and 9 points. What was her mean number of points scored per game? 6 7 8 9 Warmup

  49. Let’s define Trapezoid base A B > leg leg > C D base <D AND <C ARE ONE PAIR OF BASE ANGLES. When the legs of a trapezoid are congruent, then the trapezoid is an isosceles trapezoid.

  50. Isosceles Trapezoid • If a trapezoid is isosceles, then each pair of base angles is congruent. B A D C

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