POLYGONS POLYGON CLOSED plane figure that is formed by three or more segments called sides. 2.) STTE WHETHER THE POLYGON IS EQUILTERL, REGULR OR EQUINGULR a.) b.) c.) VERTEXThe endpoint of each side of a polygon CONVEX POLYGON polygon is convex if no line that contains a side of the polygon contains a point in the interior of the polygon. CONCVE POLYGON polygon that is not convex DIGONL OF POLYGON segment that joins two nonconsecutive vertices of the polygon 3.) NME THE POLYGON. IS IT CONVEX OR CONCVE? a.) b.) THEOREM The sum of the measures of the interior angles of a quadrilateral is 360 degrees. GENERL FORMUL The sum of the measures of the interior angles of an N-sided polygon is 180(N-2) c.) d.) 1.) DECIDE WHETHER THE FIGURE IS POLYGON OR NOT: a.) b.) e.) f.) c.) d.) g.) h.) e.) f.) 4.) a.) Name the polygon b.) Draw all the diagonals that have vertex as an endpoint. c.) Name all the nonconsecutive angles to L g.) h.) G C F D E
PRLLELOGRMS PRLLELOGRM quadrilateral with both pairs of opposite sides parallel 3.) FIND THE VLUE OF ECH VRILE IN THE PRLLELOGRM: a.) 3y 7 THEOREMS If a quadrilateral is a parallelogram then a) its opposite sides are congruent 5x 3x 18 b) its opposite angles are congruent c) its consecutive angles are supplementary 2y 4 d) its diagonals bisect each other 1.) IS THE FIGURE PRLLELOGRM? a.) b.) (3x 18) 4y b.) (2x 12) 3z c.) 2.) In the parallelogram KLMN, points O, P, Q, and R are midpoints of XN, XK, XL, and XM respectively. Find the following measures: K 61 P 3 O N c.). 7y 2 3x 2 23 5y 4 a.) KN b.) XN c.) KL d.) LN e.) KP f.) KR 68.3 L Q X 5 13 R 10.15 M EXTR CREDIT Solve for x. (3x 41) (2x 7) g.) mlmnl h.) mlnlm i.) mlnml k.) perimeter of KLMN j.) mlxqp 2x (2x 7)
SEE ONE IG FIGURE
When is a quadrilateral a parallelogram? a.) If both pairs of opposite sides of a quadrilateral are congruent b.) If both pairs of opposite angles of a quadrilateral are congruent c.) If an angle of a quadrilateral is supplementary to both of its consecutive angles d.) If the diagonals of a quadrilateral bisect each other e.) If one pair of opposite sides of a quadrilateral are congruent and parallel re you given enough information to determine whether the quadrilateral is a parallelogram? Explain. WYS TO PROVE SHPE IS PRLLELOGRM a.) Show that both pairs of opposite sides are parallel b.) Show that both pairs of opposite sides are congruent c.) Show that both pairs of opposite angles are congruent d.) Show that one angle is supplementary to both consecutive angles e.) Show that the diagonals bisect each other f.) Show that one pair of opposite sides are congruent and parallel 1. 2. 3. 4. 5. 6. 120 60 12 12 60 What additional information is needed in order to prove that quadrilateral CD is a parallelogram? 7. DC 8. DC 9. DC C 10. DE E E 11. mcd md 180 D C What value of x and y will make the polygon a parallelogram? 12. x 2 13. (3x 5) 70 14. x y 12 3x 6 3x 2x (x 3y) 5y y 1
RHOMUSES, RECTNGLES and SQURES a.) quadrilateral is a rhombus if and only if it has four congruent sides b.) quadrilateral is a rectangle if and only if it has four right angles c.) quadrilateral is a square if and only if it is a rhombus and a rectangle ECH SET OF POINTS REPRESENT PRLLELOGRM. PLOT THE POINTS. SKETCH THE PRLLELOGRM and DECIDE IF IT IS RHOMUS, SQURE, RECTNGLE or NEITHER ONE. 7.) P(-2,3) Q(-2,-4) R(2,-4) S(2,3) d.) parallelogram is a rhombus if and only if its diagonals are perpendicular e.) prallelogram is a rhombus if and only if each diagonal bisects a pair of opposite angles f.) parallelogram is a rectangle if and only if its diagonals are congruent DRW SQURE 8.) P(7,-1) Q(3,6) R(-1,-1) S(3,-8) DRW RECTNGLE DRW RHOMUS 9.) P(-4,0) Q(3,7) R(6,4) S(-1,-3) DECIDE WHETHER THE STTEMENT IS SOMETIMES, LWYS or NEVER TRUE 1.) rhombus is equilateral 2.) The diagonals of a rectangle are perpendicular 3.) The opposite angles of a rhombus are supplementary 10.) P(1,1) Q(-2,4) R(-5,1) S(-2,-2) 4.) square is a rectangle 5.) The diagonals of a rectangle bisect each other. 6.) The consecutive angles of a square are supplementary.
TRPEZOIDS TRPEZOID - quadrilateral with exactly one pair of parallel sides. SES OF THE TRPEZOID The parallel sides of a trapezoid SE NGLES The angles that have the base as a side ISOSCELES TRPEZOID Trapezoids which have legs that are congruent MIDSEGMENT OF TRPEZOID The segment that connects the midpoints of its legs KITE quadrilateral that has two pairs of consecutive congruent sides, but opposite sides are not congruent THEOREMS a) If a trapezoid is isosceles, then each pair of base angles is congruent b) If a trapezoid has a pair of congruent base angles, then it is an isosceles trapezoid c) trapezoid is isosceles if and only if its diagonals are congruent d) The midsegment of a trapezoid is parallel to each base and its length is half the sum of the lengths of the bases e) If a quadrilateral is a kite, then its diagonals are perpendicular f) If a quadrilateral is a kite, then exactly one pair of opposite angles is congruent Draw a trapezoid JKLM with JK LM. Match the pair of segments or angles with the term that describes them in trapezoid JKLM. 1. JK and ML 2. MJ and KL 3. ML and KL 4. K and M 5. JL and KM 6. M and L. bases angles. consecutive sides C. opposite angles D. diagonals E. bases F. legs Find the angle measures of CD. 7. 8. 9. 132 D C 112 D C The midsegment of the trapezoid is RT. Find the value of x. D 77 C 10. 7 11. 8 12. R x T R 11 T R 14 x 24 13 x Find the length of the sides to the nearest hundredth or the measure of the angles in kite MTH. T 13. 14. 15. M M 6 5 5 8 T M 27 74 T H 81 118 H H T
Which figure do you think does not belong in a set with the other three? Explain why it does not belong. There may be more than one possible answer. 1 rhombus square trapezoid parallelogram 2 isosceles trapezoid kite rhombus square 3 rhombus trapezoid kite rectangle 4 parallelogram trapezoid rectangle isosceles trapezoid
RES DRW THE INDICTED FIGURE ND WRITE THE FORMUL IN FINDING ITS RE 1.) SQURE 7.) KITE 2.) RECTNGLE 8.) PRLLELOGRM 3.) TRINGLE 9.) RHOMUS 4.) CIRCLE MORE POSTULTES a) If two polygons are congruent, then they have the same area 5.) TRPEZOID b) The area of a region is the sum of the areas of its nonoverlapping parts EXTR CREDIT: FIND THE RE OF THE SHDED REGION: This is a square inscribed in a circle 6.) SQURE 7 in
THE RULE OF PYTHGORS c In any right triangle C, the square of the measure of the length of the hypotenuse a C b (the longest side of the right triangle) is equal to the sum of the squares of the measures of the legs of the right triangle. Mathematically, this is c 2 = a 2 + b 2 You may use a calculator, but show steps needed to prove your answer as in the examples.. Determine if the sides given below belong to a right triangle EXMPLE: 3, 4, 5 14.) 13 y D. rectangle has a length of 28 ft and a diagonal of 35 ft. Draw a sketch, find the width of the rectangle, and then find the area of the rectangle x 4 3 NSWER: Yes, because 5 2 = 3 2 + 4 2 E. Find the area of the triangle, after first finding the missing side 1.) 4, 5, 6 2.) 1.5, 2, 2.5 3.) 6, 8, 10 4.) 9, 40, 41 5.) 10, 20, 30 6.) 1, 2.4, 2.6. For each right triangle, find the missing side. If needed, round your answer to the nearest tenth 7.) a = 2, b = 3, c = F. Each side of a square is 8 inches. Draw a sketch and then find the length of the diagonal of the square. 8.) a = 5, b = 12, c = 9.) a = 6, b= 6, c = 10.) a = 2, b =, c = 5 11.) a =, b = 7, c = 20 12.) a =, b = 20, c = 52 C. Find the length of x and y in the following figures 13.) 5 G. Four squares are lined up horizontally, as shown. The length of a side of the first square is 1. Each square after that has a side twice as long as the side of the previous square. What is the length of? x 12 y 15 EXTR CREDIT: If you plant 1 tree on the first day, 2 on the second, 4 on the third, 8 on the fourth and so on, how many days will it take to plant 125 trees?
POLYGON RES Find the area of the polygon. 1. 2. 3. 3 9 6 6 12 7 4. 5. 11 6. 15 5 14 9 10 7 7. 8. 9. 4 8 4 4 8 12 8 6 Use the Pythagorean Theorem to find the area of the polygon. 10. 8 11. 12. 9 15 10 5 12 3 13. The garage roof shown is made from 14. The Millers have two gardens as shown two isosceles trapezoids and two below. The shaded region represents the lawn isosceles triangles. Find the area of that needs to be fertilized. Find the area the entire roof. of the lawn. 30 ft garden 15 ft 30 ft 15 ft 60 ft 15 ft 10 ft 15 ft 60 ft 10 ft 24 ft