6.1 Revisit polygon angle sum theorem. Interior and Exterior angles of regular polygons.

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1 G/Honors Geometry February 9 February 26, 2018 Quadrilateral and areas of 2- ate opic ssignment Friday 02/9/ evisit polygon angle sum theorem. Interior and Exterior angles of regular polygons. Pg. 25: #1-9, 11-14, 16, Monday 02/12/18 uesday 02/1/18 Wednesday 02/14/18 hursday 02/15/18 Friday 02/16/18 Monday 02/19/18 uesday 02/20/18 Wednesday 02/21/18 hursday 02/22/18 Friday 02/2/18 hursday 02/26/ tudents will explore the properties of parallelograms. W 6.2:#1-11, Pg 259: 11, tudents will Prove a quadrilateral is a parallelogram in a W 6.: #1-0, Pg 267: 12- variety of ways. 14, tudents will explore the properties of special parallelograms to solve algebraic problems, i.e, hombus, ectangles, quares. Quiz 1 Parallelograms W 6.4: #1-2, Pg 27: 1, and hombus, ectangles, quares. Pg. 279: #1-8, 10-16, 21-2, tudents will explore properties of trapezoids, isosceles trapezoids, and mid segment of a trapezoid. Partner Quiz and 6.5 tudents will explore how quadrilaterals are formed by connecting midpoints. 7.1 tudents will explore quadrilaterals in coordinate plane. 7.1 tudents will explore quadrilaterals in coordinate plane. Quiz rapezoids 7.2 tudents will explore quadrilaterals in coordinate plane o problems 1, 2, from ext book pp and W W 6.6: #1-14. W: #1-14 pgs #1-, 7-9, 1-16, 0-1 Pg. 00: # 18-26, 0 Pg. 04: #1-7, 12, est eview: 6.1 to 7.2 Pg. 288: #1-7. Pg 15: # est quadrilaterals

2 Name ate Period egular Polygons # of ides Name # of riangles One Interior ngle One Exterior ngle riangle 1 4 Quadrilateral 2 5 Pentagon 6 Hexagon 4 8 Octagon 6 9 Nonagon 7 10 ecagon 8 12 odecagon 10 n n-gon n-2 1

3 iscovery Questions: If you knew the sum of the interior angles of any regular n-gon, how could you find the measure of one interior angle? one interior angle = ould you find how many sides a regular polygon had if you knew the measure of one interior angle? Example 1.) Find the measure of one interior angle of a nonagon. Example 2.) Find the measure of one interior angle of a hexagon. Example.) Find the measure of one interior angle of a dodecagon. Example 4.) If one interior angle of a regular polygon is 15, how many sides does it have? Example 5.) If one interior angle of a regular polygon is 108, how many sides does it have? Example 6.) If one interior angle of a regular polygon is approximately 128, how many sides does it have? 2

4 6-2 Notes: Properties of Parallelograms ny four-sided polygon is called a quadrilateral. segment joining any two nonconsecutive vertices is called a diagonal. special kind of quadrilateral in which both pairs of opposite sides are parallel is called a parallelogram (this is the definition of a parallelogram). You can use what you know about parallel lines and transversals to prove some theorems about parallelograms. ef. Parallelogram Opposite sides are parallel Parallelogram Opposite sides are Parallelogram Opposite angles are Parallelogram iagonals bisect each other Parallelogram onsecutive angles are supplementary If each quadrilateral is a parallelogram, find the values of x, y, and z. Y z x Y 44 x z z 7 x Y In parallelogram, m = x and m = 4x Find the measure of angles,,, and. 5. In parallelogram V, diagonals V intersect at Q. If Q = 5x + 1 and Q = x + 15, find Q. and Explain why it is impossible for each figure to be a parallelogram

5 Find x and y so that is a parallelogram. 8. = x + 2y; = ; = 7; = 2x + y 9. m = 2x + 8y; m = 4x + y + 54; m = x + 2y Find x so that is a parallelogram: = x 2 - ; = 2x PQ has vertices P(-4, 7), Q(, 0), (2, -5), and (-5, 2). etermine if PQ is a parallelogram. 4

6 6- Notes: Proving that a Quadrilateral is a Parallelogram You can show that a quadrilateral is a parallelogram if you can show that one of the following is true. ef: oth pairs opp. ides Parallelogram oth pairs of opp. ides Parallelogram iagonals bisect each other Parallelogram oth pairs opp. ngles Parallelogram One pair of opp. ides are both and Parallelogram etermine if each quadrilateral is a parallelogram. Justify your answer etermine whether quadrilateral with the given vertices is a parallelogram. Explain. 4. (2,5), (5,9), (6,), (,-1) 5. (-1,6), (2,-), (5,0), (2,9) 5

7 Geometry Worksheet 6. Parallelograms Name ate Period re the following parallelograms? If yes, why? (use one of the five reasons from section 6.) If no, tell what else would be needed M is the midpoint of and 15. M

8 tate whether the given information is sufficient to support the statement, Quadrilateral is a parallelogram. If the information is sufficient, state the reason. 16. and 2 1 O O = O and O = O 18. and 19. and O = O and and 22. and and and 25. and and and and 29. and 0. is supplementary to is supplementary to 7

9 Geometry Worksheet Name 6.4 ectangles, quares & hombi ate Period 1. In rectangle, = 2x + y, = 5x 2y, = 22, and = 17. Find x and y. E In the diagram for problems 2-7,Q is a rectangle and QZ is a parallelogram. 2. If Q = 2x + 1 and = x 1, find x. Q Z. If m Q = 70, find m QZ. Q Z 4. If m = 5, find m. Z 5. If m Q = m, find m Q. Z Q Q 6. If = x 2 and Q = 4x 6, what is the value of x? Use rectangle UV for questions If m 1 = 0, m 2 = 9. If m 6 = 57, m 4 = 10. If m 8 = 1, m 2 = 11. If m 5 = 16, m = Q Z 7. Z = 6x, ZQ = x + 2y, and = 14 x. Find the values of x and y. Is QZ a special parallelogram? If so, what kind? 6 V 7 5 K 8 1 Q 4 2 U Z 12. is a rhombus. If the perimeter of 1. is a square. If m = x 2 4x, find x. = 68 and = 16, find. 8

10 Use rhombus for problems If m F = 28, m =. 15. If m F = 16x + 6, x =. F 16. If m = 4, m =. 17. If m F = 120 4x, x =. 18. If m = 4x + 6 and m = 12x 18, x =. 19. If m = x 2 6 and m = 5x + 9, x = 20. is a square. = 5x + 2y, = x y, and = 11. Find x and y. 21. contractor is measuring for the foundation of a building that is to be 85 ft by 40 ft. takes and string are placed as shown. he outside corners of the building will be at the points where the strings cross. He then measures and finds WY = 9 ft and XZ = 94 ft. Is WXYZ a rectangle? If not, which way should stakes E and F be moved to made WXYZ a rectangle? G F W 85 ft X E 40 ft H Z Y 22. is a rectangle. Find the length of each diagonal if = 2(x ) and = x is a rectangle. Find each diagonal if c and = 4 c. 9 Given rectangle Q 24. If X Q, find m X. Q X 25. If m Q = 0 and Q = 1, find. 26. If m Q = 45 and Q = 6.2, find Q. 9

11 27. Given rhombus, = 5x + y 1, = 18, = 8x 2y + 2. Find x and y. 28. Given square PQ, = x 2 2x, Q = 4x 5. Find x,, and Q. P Q E etermine whether WXYZ is a parallelogram, a rectangle, a rhombus, or a square for each set of vertices. tate yes or no for each and explain why or why not. how work to support the explanations. For example, if you say the sides are parallel then you need to calculate the slopes. 29. W(5, 6), X(7, 5), Y(9, 9), Z(7, 10) Parallelogram: ectangle: hombus: quare: 0. W(-, -), X(1, -6), Y(5, -), Z(1, 0) Parallelogram: ectangle: hombus: quare: etermine whether EFGH is a parallelogram, a rectangle, a rhombus, or a square for each set of vertices. tate yes or no for each and explain why or why not. how work to support the explanations. For example, if you say the sides are parallel then you need to calculate the slopes. 1. E(0, -), F(-, 0), G(0, ), H(, 0) Parallelogram: hombus: ectangle: quare: 2. E(2, 1), F(, 4), G(7, 2), H(6, -1) Parallelogram: hombus: ectangle: quare: 10

12 Geometry Worksheet rapezoids (6.6) 1. Given: Isosceles trapezoid, m = 0 and m = 85 m 1= m 5= m = Name ate Period m 2= m 6= m = m = m 7= m = m 4 = m 8 = m = 2. Given: Isosceles trapezoid JXVI, m JVI = 42 and m IJV = 65 m 1= m 6= m 11= m 2= m 7= m 12= J X 10 m = m 8= m JIV = I V m 4= m 9= m IJX= m 5 = m 10=. Given: Isosceles trapezoid JXVI, m IXV = 8 and m VJX = 28 m 1= m 6= m 11= m 2= m 7= m 12= J X 10 m = m 8= m IVX = I V m 4= m 9= m VXJ= m 5 = m 10= 11

13 VW 4. is the median of a trapezoid that has bases MN and PO, with V on OM and W on PN. If the vertices of the trapezoid are M(2, 6), N(4, 6), P(10, 0), and O(0, 0), find the coordinates of V and W. VW 5. is the median of a trapezoid MNPO that has bases MN and PO, with V on PM 10), N(9, 10), V(, 7), and W(11, 7), find the coordinates of P and O. and W on ON. If M(5, XY is the median of trapezoid Q in problems XY=18 and = 7. Find Q. 7. = n and Q = 6. Find XY in terms of n. 8. XY=16. Find +Q. Y Y Y Q X 9. X = ½ () and m =10. Find m. Q X 10. = a and Q = 2b. Find XY. Y Q X 11. QX=Y and m XY = 45. Find m. Y Y Q X Q X Q X In problems 12-14, trapezoid is isosceles. Find the variable in each. 12. = x+5 and = x + 1. m =x 7 and m =5x+ 14. = x and = 8x 9. 12

14 NOE: Quadrilaterals from Midpoints E I. Given: Quadrilateral E, F, G, and H are the midpoints of,,, and respectively. H F 1. oes EFGH look like a special kind of quadrilateral? G 2. raw diagonal. and are formed. How is EF related to?. How is HG related to? 4. What kind of quadrilateral is EFGH?_ 5. Why? E II. Quadrilateral is a rhombus. gain E, F, G, and H are midpoints. H F 6. What kind of angle is EHG? 7. What kind of quadrilateral is EFGH? G III. Quadrilateral is a rectangle. gain, E, F, G, and H are midpoints. 8. How are HE and EF related? E (Hint: How are and related?) H F 9. What kind of quadrilateral is EFGH? G 1

15 IV. is a quadrilateral where E, F, G, and H are midpoints. 11. If quadrilateral is a square, then quadrilateral EFGH is a 12. If quadrilateral is a trapezoid, then quadrilateral EFGH is a 1. If quadrilateral is an isosceles trapezoid, then quadrilateral EFGH is a. 14. Given: IOE is an isosceles trapezoid,,, and P are midpoints IO = 14 m I = 70 m = 45 Find: m 1 = m 9 = O P 8 m 2 = m 10 = m = m 11 = I E m 4 = m 12 = m 5 = m 1 = m 6 = m 14 = m 7 = m 15 = m 8 = E = m 16 = = 14

16 onsecutive sides are. Opposite sides are. onsecutive angles are. efined by: Four and four iagonals will. ngles always add up to. ngles: ides: efined by: sides are and. ll parallelograms are. ectangles iagonals: rea: ngles: ides: efined by: rectangle is a where all four angles measure. iagonals: rea: ll rectangles are ; 15

17 hombi (hombus) ngles: ides: efined by: rhombus is a where sides are equal. ll rhombi are. quares iagonals: rea: ngles: ides: efined by: square is a where all four sides are, and all four angles are. ll squares are ; ; rapezoids iagonals: rea: ngles: ides: efined by: trapezoid is a where two opposite sides are and the remaining opposite sides are. Isosceles trapezoids have legs that are. iagonals: rea: 16

18 Geometry G/PreP Unit eview Quadrilaterals Find each of the following values. Use parallelogram GM for problems G = x 10 and GP = x Find x. Name ate Period P 2 G 7 M 2. m GM = 7 and m MG = 95, find m GM.. m GM = 75, find m GM. x = 4. = 2x + y, G = x y, find x and y. y = Use rectangle E for problems If = x 7 and = 2x + 2, find x E If m 2 =, find m If = 2x + 5 and E = 4x 11, find x. 8. If m 1 = x 2 4 and m 8 = x + 52, find x. Use rhombus HOM for problems

19 x= 9. If MO = 24, M = 4x + 2y + 2, and H = 5x y + 14, find x and y. y= M O H 10. If O = 24 and MH = 10, find M. 11. If m 7 = 9, find m 2. Use square QU for problems If U = x and = 5x 4, find x. Q U 1. If m = 6x, find x. 14. If m QU = x 12, find x. Use trapezoid UVW with median XY for problems m V 104 U 16. U = 15, WV =, find ZY. Z Y W 54 V 17. U = x 12, ZY = x + 15, and WV = x 8. Find x. 18

20 Use isosceles trapezoid P for problems Find m Find m Find m. P In problems 21-2, if there is enough information to state that the quadrilateral is a parallelogram give the reason. Write none if there is not enough information to state that the quadrilateral is a parallelogram. 21. E is the midpoint of and. 1 2 E and and 24. he coordinates of the vertices of quadrilateral are (-4, -2), (-1, ), (4, 0), and (1, -5). etermine whether is a parallelogram, a rectangle, a rhombus, or a square. tate yes or no for each and explain why or why not. how work to support the explanations. For example, if you say the sides are parallel then you need to calculate the slopes. Parallelogram: hombus: ectangle: quare: 25. he coordinates of the vertices of quadrilateral PQ are P(4, 4), Q(1, 2), (2, -2), and (5, 0). etermine whether PQ is a parallelogram, a rectangle, a rhombus, or a square. tate yes or no for each and explain why or why not. how work to support the explanations. For example, if you say the sides are parallel then you need to calculate the slopes. Parallelogram: hombus: ectangle: quare: 26. he coordinates of the vertices of quadrilateral WXYZ are W(5, 0), X(6, -8), Y(-1, -4), and Z(-2, 4). etermine whether WXYZ is a parallelogram, a rectangle, a rhombus, or a square. tate yes or no for each and explain why or why not. how work to support the explanations. For example, if you say the sides are parallel then you need to calculate the slopes. Parallelogram: hombus: ectangle: quare: 19

21 27. Find the coordinates of the possible points for the missing vertex in a parallelogram if three of the vertices are (-2, -1, (-1, ), and (4, 1) 28: skip 29. has midpoints, E, and F. If the perimeter of EF is 2, then find the perimeter of. E F 0. skip 1. IOE is an isosceles trapezoid.,,, and P are midpoints. IO = 12 m I = 75 m = 40 1 O Find: m 1 = m 2 = 2 P m = = I E P = 20

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