1

Unit 5 Section 1: Properties and Attributes of Polygons What is a polygon??? Side of the polygon: Vertex of the polygon: Diagonal: Number of Sides Name of Polygon

3 4 5 6 7 8 9 10 12 n

2

Tell whether the figure is a polygon. If it is a polygon, name it by the number of sides. Ex #1A

Ex #1B

Ex #1C

You Try #1a

You Try #1b

You Try #1c

What is a regular polygon?

A polygon is concave if any part of a diagonal contains points in the exterior of the polygon. If no diagonal contains points in the exterior, then the polygon is convex. A regular polygon is always convex.

Tell whether the polygon is regular or irregular. Tell whether it is concave or convex. Ex #2A

Ex #2B

Ex #2C

3

Tell whether the polygon is regular or irregular. Tell whether it is concave or convex. You Try #2a

You Try #2b

Sum of the interior angles…….

Ex #3A Find the sum of the interior angle measures of a convex heptagon.

4

Ex #3B Find the measure of each interior angle of a regular 16-gon. Step 1 Find the sum of the interior angle measures. Step 2 Find the measure of one interior angle. Ex #3C Find the measure of each interior angle of pentagon ABCDE. You Try #3 Find the measure of each interior angle of a regular decagon.

Ex #4A Find the measure of each exterior angle of a regular 20-gon.

5

Ex #4B Find the value of b in polygon FGHJKL.

You Try 4a Find the measure of each exterior angle of a regular dodecagon. You Try 4b Find the value of r in polygon JKLM.

Ex #5 Ann is making paper stars for party decorations. What is the measure of ∠1?

6

7

Unit 5 Section 2: Properties of Parallelograms

A quadrilateral with two pairs of parallel sides is a parallelogram. To write the name of a parallelogram, you use the symbol Y .

Parallelogram ABCD YABCD AB P CD , BC P DA Ex #1: In YCDEF , DE = 74 mm,

DG = 31 mm, and m∠FCD = 42°.

a. Find CF.

b. Find m∠EFC.

c. Find DF.

You Try #1 In YKLMN , LM = 28 in.,

LN = 26 in., and m∠LKN = 74°.

a. Find KN.

b. Find m∠NML.

c. Find LO.

8

Ex #2: WXYZ is a parallelogram. a. Find YZ. b. Find m∠Z. You Try #2 EFGH is a parallelogram. a. Find JG. b. Find FH.

Ex #3: Three vertices of YJKLM are J(3, –8), K(–2, 2), and L(2, 6). Find the coordinates of vertex M.

9

You Try #3 Three vertices of YPQRS are P(–3, –2), Q(–1, 4), and S(5, 0). Find the coordinates of vertex R.

Ex #4:

Given: ABCD is a parallelogram.

Prove: ∆AEB ≅ ∆CED

10

11

Unit 5 Section 3: Conditions for Parallelograms Ex #1: Show that JKLM is a parallelogram for a = 3 and b = 9. Ex #2: Show that PQRS is a parallelogram for x = 10 and y = 6.5. Ex #3: Determine if the quadrilateral must be a parallelogram. Justify your answer. Ex #4: Determine if the quadrilateral must be a parallelogram. Justify your answer.

12

Ex #5: Determine if the quadrilateral must be a parallelogram. Justify your answer. Ex #6: Show that quadrilateral ABCD is a parallelogram by using the definition of parallelogram. A(–2, –1), B(–7, –5), C(-3, -17), D(2, -13). Ex #7: Show that quadrilateral JKLM is a parallelogram by using the definition of parallelogram. J(–1, –6), K(–4, –1), L(4, 5), M(7, -1).

13

Unit 5 Section 4: Special Parallelograms: Rectangle, Rhombus & Square

Ex #1: A woodworker constructs a rectangular picture frame so that JK = 50 cm and JL = 86 cm. Find HM.

Ex #2: The rectangular gate has diagonal braces.

Find HJ. Find HK.

Ex #3: TVWX is a rhombus. Find TV.

Find m∠VZT. Find m∠VTZ.

14

Ex #4: CDFG is a rhombus. Find CD.

Find m∠GCH if m∠GCD = (b + 3)° and m∠CDF = (6b – 40)°

Ex #5: Show that the diagonals of square EFGH are congruent perpendicular bisectors of each other.

15

Unit 5 Section 5: Conditions for Special Parallelograms When you are given a parallelogram with certain properties, you can use the theorems below to determine whether the parallelogram is a rectangle.

Ex #1: A manufacture builds a mold for a desktop so that ,AB CD BC DA≅ ≅ , and m∠ABC=90°. Why must ABCD be a rectangle?

16

Below are some conditions you can use to determine whether a parallelogram is a rhombus.

Ex #2: Determine if the conclusion is valid. If not, tell what additional information is needed to make it valid. Given: ,EF FG EG FH≅ ⊥ Conclusion: EFGH is a rhombus.

17

Ex #3: Determine if the conclusion is valid. If not, tell what additional information is needed to make it valid. Given: , , ,EB BG FB BH EG FH EBF EBH≅ ≅ ≅ ≅V V Conclusion: EFGH is a square. Ex #4: Determine if the conclusion is valid. If not, tell what additional information is needed to make it valid. Given: ∠ABC is a right angle. Conclusion: ABCD is a rectangle. Ex #5: Use the diagonals to determine whether a parallelogram with the given vertices is a rectangle, rhombus, or square. Give all the names that apply. P(–1, 4), Q(2, 6), R(4, 3), S(1, 1)

18

Ex #6: Use the diagonals to determine whether a parallelogram with the given vertices is a rectangle, rhombus, or square. Give all the names that apply. W(0, 1), X(4, 2), Y(3, –2), Z(–1, –3) Ex #7: Use the diagonals to determine whether a parallelogram with the given vertices is a rectangle, rhombus, or square. Give all the names that apply. K(–5, –1), L(–2, 4), M(3, 1), N(0, –4) Ex #8 Use the diagonals to determine whether a parallelogram with the given vertices is a rectangle, rhombus, or square. Give all the names that apply. P(–4, 6) , Q(2, 5) , R(3, –1) , S(–3, 0)

19

Unit 5 Section 6: Properties of Kites and Trapezoids Kite: Ex #1 In kite ABCD, m∠DAB = 54°, and m∠CDF = 52°. Find m∠BCD. Find m∠ABC. Find m∠FDA.

20

Ex #2 In kite PQRS, m∠PQR = 78°, and m∠TRS = 59°. Find m∠QRT.

Find m∠QPS. Find m∠PSR.

Ex #3 Lucy is framing a kite with wooden dowels. She uses two dowels that measure 18 cm, one dowel that measures 30 cm, and two dowels that measure 27 cm. To complete the kite, she needs a dowel to place along KL . She has a dowel that is 36 cm long. About how much wood will she have left after cutting the last dowel?

Ex #4 Daryl is going to make a kite by doubling all the measures in the kite. What is the total amount of binding needed to cover the edges of his kite? How many packages of binding must Daryl buy? One package of binding contains 2 yards, or 72 inches.

21

Trapezoid: Ex #5 Find m∠A.

Ex #6 Find m∠F.

Ex #7 JN = 10.6, and NL = 14.8. Find KM.

Ex #8 Find the value of a so that PQRS is isosceles.

22

Ex #9 AD = 12x – 11, and BC = 9x – 2. Find the value of x so that ABCD is isosceles.

Ex #10 Find the value of x so that PQST is isosceles.

The midsegment of a trapezoid is the segment whose endpoints are the midpoints of the legs. In Lesson 5-1, you studied the Triangle Midsegment Theorem. The Trapezoid Midsegment Theorem is similar to it.

23

Ex #11 Find EF.

Ex #12 Find EH.