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6.1 Polygons

6.1 Polygons

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6.1 Polygons

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  1. 6.1 Polygons Geometry

  2. Objectives/DFA/HW • Objectives: • You will solve problems using the interior & exterior angle-sum theorems. • DFA: • pp.356-357 #16 & #30 • HW: • pp.356-358 (2-44 even)

  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. Theorem 6-1 – Polygon Angle-Sum Theorem • The sum of the measures of the interior angles of an n-gon is (n-2)180. • Ex. What is the sum of the interior angle measures of a heptagon?

  15. Theorem 6-2 Polygon Exterior Angle-Sum Theorem • The sum of the measures of the exterior angles of polygon, one at each vertex is 360o. • For the petagon • m<1+m<2+m<3+m<4+m<5=360 • Ex. What is the measure of each angle of an octagon.

  16. 6.2 Properties of Parallelograms Geometry Spring 2014

  17. Objective/DFA/HW • Objectives: • You will use properties (angles & sides) of parallelograms & relationships among diagonals to solve problems relating to parallelograms. • DFA: • pp.364 #16 & #22 • HW: • pp.363-366 (2-40 even)

  18. 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

  19. 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

  20. 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

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

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

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

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

  25. 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 °

  26. Find X, Y, & the diagonals X 2x-8 Y+10 Y+2

  27. Theorem 6.7 • If 3 (or more) parallel lines cut off congruent segments on one transversal, then they cut off congruent segments on every transversal.

  28. 6.3 Proving Quadrilaterals are Parallelograms Geometry Spring 2014

  29. Objective/DFA/HW • Objectives: • You will determine whether a quadrialteral is a parallelogram. • DFA: • pp.372 #12 • HW: • pp.372-374 (2-28 even, 36-44 all)

  30. Review

  31. 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.

  32. 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.

  33. 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.

  34. 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.

  35. 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.

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

  37. 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.

  38. Example 4 – p.341 • Show that A(2,-1), B(1,3), C(6,5), and D(7,1) are the vertices of a parallelogram.

  39. Assignments • In class: pp. 342-343 # 1-8 all • Homework: pp.342-344 #10-18 even, 26, 37

  40. 6.4 Rhombuses, Rectangles, and Squares Geometry Spring 2014

  41. Objective/DFA/HW • Objectives: • You will determine whether a parallelogram is a rhombus, rectangle, or a square & you will solve problems using properties of special parallelograms. • DFA: • pp.379 #12 • HW: • pp.379-382 (1-27all)

  42. 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°

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

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

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

  46. 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.

  47. 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

  48. 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

  49. 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

  50. 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