1 / 94

6.1 Polygons

6.1 Polygons. Geometry. 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). What is polygon?. Formed by three or more segments (sides).

henrik
Download Presentation

6.1 Polygons

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  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

More Related