Geo 9 Ch 5 1

5.1 Quadrilaterals Parallelograms/Real World Visual Illusions Def: If a quadrilateral is a parallelogram, then both pair of the opposite sides are parallel. DISCOVER EVERYTHING YOU CAN ABOUT THE PARALLELOGRAM

E

H G

F

1

2

3

4

Geo 9 Ch 5 2

Summary If 1. 2. 3. 4. Examples: Given parallelograms 1. 2. 3.

Prove: SJ QK

1. PQRS PJ RK

50

70 33

x y

45

35

x 2y

3z - 4

15 12

2x + 5y

2x + 2y

3x + 2y + 110

3y + 2

P 1

S K R

Q J

2

Geo 9 Ch 5 3

5-2 Ways to Prove Quadrilaterals are Parallelograms

Theorem 5-4 : ____________________________________________________________ ________________________________________________________________________

Given: TS QR; TQ SR Prove: QRST is a parallelogram.

Theorem 5-5 If one pair of opposite sides of a quadrilateral are both congruent and parallel

then the quad is a parallelogram. Draw a picture with markings.

Theorem 5-6 If both pairs of opposite sides of a quadrilateral are congruent,

then the quad is a parallelogram. Draw a picture with markings.

Theorem 5-7 If the diagonals of a quadrilateral bisect each other,

then the quad is a parallelogram. Draw a picture with markings.

T

R

S

Q

Geo 9 Ch 5 4

Therefore, the 5 ways to prove that a quad is a parallelogram are: 1. Show that both pairs of _____________________________________________________ 2. Show that both pairs of opposite ______________________________________________. 3. Show that one pair of opposite sides are both ___________ and _____________. 4. Show that both pairs of _____________________________________________________. 5. Show that diagonals ________________________. PROOFS 1) Prove: ABCD is a parallelogram

1. 1 3 5ADC 1. Given 2) Prove: AEFD is a parallelogram 1. E is midpoint of AB 1. Given F is midpoint of DC ABCD is a parallelogram

A

C

B

D 1

2

3

4

5

6

A

C

B

D

E

F

Geo 9 Ch 5 5

3) Prove: ABCD is a parallelogram

1. BOC DOA 1. given 4) Prove: AMCN is a parallelogram 1. ABCD is a parallelogram 1. given

sec

sec

5 6

AN bi ts DAB

CM bi ts BCD

B

A

C

D

O

1

A

2

3

4

5

6 B

C D N

M

Could I still do it if I didn’t give you 5 6?

Geo 9 Ch 5 6

5) Prove: AFCE is a parallelogram 1. ABCD is a parallelogram 1. given DE = BF

4

3

A

2

1

B

C D

E

F

Geo 9 Ch 5 7

5-3 Theorems involving Parallel Lines

Theorem 5-8: If two lines are parallel, then the points one line are equidistant from the other line.

Theorem 5-9: If 3 parallel lines cut off congruent segments on one transversal,

then they cut off congruent segments on all transversals.

The midsegment of a triangle is a segment that connects the midpoints of 2 sides of the triangle. Given a

triangle with coordinates A (1, 7) , B (5,3) and C (-1, 1) find the segment that connects that midpoints

of sides AB and AC, label the midpoints M and N, respectively.

(a) Find the length of the midsegment MN and compare it to the length of BC.

(b) What can be said about the lines containing segments BC and MN?

The Midsegment Th:

Geo 9 Ch 5 8

1. Given R, S and T are midpoints of the sides of ABC. Complete the table

AB BC AC SR TR ST

a) 12 14 18

b) 15 22 10

c) 10 9 7.5

2. Given AR BS CT, RS ST Complete: a) If RS = 12 then ST = ______ b) if AB = 8 then AC = ______ c) If AC = 20 then AB = ______ d) If AC = 10x then BC = ______ 3. Given points X, Y and Z are the midpoints of AB, BC and AC. a) If <A = 35, find another angle b) if AB = k then YX = ______ c) If XZ = 2k+3 then BC = __

d) If AB = 9, BC = 8, AC = 6 then the perimeter of XYZ is _______.

e) If the perimeter of XYZ = 24, then the perimeter of ABC = ________.

R S

T A B

C

A

B

C

R

S

T

X Y

Z A B

C

Geo 9 Ch 5 9

5-4 Special Parallelograms A RECTANGLE is a quadrilateral with __________________________________________

A RHOMBUS is a quadrilateral with ____________________________________________

A SQUARE is a quadrilateral with ______________________________________________

PROPERTIES

PROPERTIES

PROPERTIES

Geo 9 Ch 5 10

Property Parallelogram Rectangle Rhombus Square

Opp sides

Opp sides

Opp s

Diag forms 2 s

Diag bisect each other

Diag are

Diag are perp

A diag bisects 2 s

All s are rt s

All sides

Def: If rhombus, then with

adjacent sides congruent.

1.

2.

3.

4.

5.

6.

7.

Def: If rectangle, then with right

angles.

1.

2.

3.

4.

5.

6.

Def: If square, then rectangle that is

a rhombus.

1.

2.

3.

4.

5.

6.

7.

8.

9.

The midpoint of the hypotenuse of a right triangle is

Geo 9 Ch 5 11

Quad ABCD is a rhombus. Find the measure of each angle. (m ADB = 62)

1. m ACD = ______ 2. m DEC = ______ 3. m EDC = ______ 4. m ABC = ______

Quad MNOP is a rectangle. Complete, if m NML = 29. NL = 6

4. m PON = ______ 5. m PMO = ______ 6. PL = ______ 7. MO = ______

ABC is a right triangle. M is the midpoint of AB 9. If AM = 7 then MB = _____, and CM = _____, 10. If AC = 2x, then AM = ______ MB = ______ and MC = ______., What about the angles? Given the right triangle, with W the mp of YZ.

11. if m 2 = m 3 find m 1. 12. If YW = 3x – 2, and WZ = x + 8 find YZ. For 11-13

13. If m 1 = 40, find m 2, m 3, m 4.

A B

C D

E

M N

O P

L

A

B C

M

X

Y

Z

I

2

3 W

4

Geo 9 Ch 5 12

5-5 Trapezoids Warm Up. Start your brains stretching!! Fill in with always, sometimes, or never. 1. A square is ______________ a rhombus 2. The diagonals of a parallelogram ______________ bisect the angles of the parallelogram. 3. A quadrilateral with one pair of sides congruent and one pair parallel is ______________ a parallelogram. 4. The diagonals of a rhombus are ______________ congruent. 5. A rectangle ______________ has consecutive sides congruent. 6. A rectangle ______________ has perpendicular diagonals. 7. The diagonals of a rhombus ______________ bisect each other. 8. The diagonals of a parallelogram are ______________ perpendicular bisectors of each other. A TRAPEZOID is a quadrilateral with only one pair of opposite sides parallel. The parallel sides are called ______________. The other sides are ______________. Draw a picture. An ISOSCELES TRAPEZOID has ____________________________. If you have an isosceles trapezoid, then __________________________________________

Draw auxillary line TX so that TX is parallel to RA T R

A P X

3

2

1

4

Geo 9 Ch 5 13

The median of a trapezoid is the segment that joins the midpoints of the legs. The median of a trapezoid 1. _____________________________________ 2. _____________________________________ In trapezoid ABCD, EF is a median. Complete: 1. If AB = 25, DC = 13, then EF = ________. 2. If AE = 11, FB = 8, then AD = ________ and BC = ________. 3. If AB = 29, and EF = 24 then DC = ________. 4. If AB = 7y + 6, EF = 5y – 3, and DC = y – 5, then y = ________.

A B

C D

E F

A B

C D

E F

Draw a median of a triangle.

Geo 9 Ch 5 14

5. In KLM, HJ, JI and IH are the segments joining the midpoints of the sides.

Find the perimeter of HJI. See any trapezoids? Any isosceles trapezoids?

6. Quad TUNE is an isosceles trapezoid with TU and NE as bases. If m u = 62, find the measures of the other four angles of the trapezoid. 7) DC exceeds AB by 13. Find the bases.

K

L

M

H

I

J 12

12

16

A B

C D

35

Geo 9 Ch 5 15

8) AB = 2x – 5 GC = x + 15 FD = x + 9 Find x 9) 10) Prove: ABCE is a parallelogram 1. Isosceles trapezoid ABCD

CD CE

2

1

3

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B C

D E

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D

E

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C

B

A

10

44

x x

Geo 9 Ch 5 16

GEOM 9 REVIEW SHEET CHAPTER 5 QUADRILATERALS

1. Given AD is an altitude of FED B and C are midpoints of FD and DE AF = 10 FE = 28 AC = 15 BD = 13 FIND: FD = ______ DE = ______ AB = _______

Perim of ABC = ______ Perim of DEF = ______ 2. Give the most descriptive name for quad MNOP given the following:

a) MN || PO; MN PO _____________________________________

b) MN || PO; NO || MP; MO NP _____________________________

c) <M <N <O <P ______________________________________ d) MNOP is a rectangle with MN = NO _________________________ 3. MN is the median of trapezoid ZOID a) The bases of ZOID are ______ and ______ b) If ZO = 8 and MN = 11 then DI = _________ c) If ZO = 8 then TN = __________ 4. Given AB || CD; E and H are midpoints of AD and BC a) AB = 12, CD = 20, EH = ____, EF = ______, FG = ______ GH = ______ b) FG = 3, GH = 5, AB = ______, DC = ______, EF = ______ c) AG = 4x-2, GC = 3x +1, x = ______ d) AB = 3x +1, CD = 5x-1, EH = 6x-10, EH = ______, FG = ______

A

B C

D

E F

Z O

I D

M N T

C

A B

D

E F G H

Geo 9 Ch 5 17

(5) Given: ABCD is a parallelogram.

(a) AB = 5x 10 , CD = 3x + 6 Find x_______

(b) BD = 3x 6 , BE = x + 4 Find x_______

(c) m 1 = (3x 3) , m 2 = (5x)

m 3 =

2

5 5 x Find m ABC_______

(d) AD = 4x y + 10 , AB = 3y x , BC = x + y , CD = x + 2y. Find x_____ , y______ (6) (7)

Given: AB BD , 1 2, Given: AD BC , BE = DE C and E are midpoints , BE = 4

Find: AB_____, AD_____, CE______ Prove: ABCD is a parallelogram (8) (9)

Given: ABCD is a parallelogram Given: AECF is a parallelogram

AE bisects BAD 9 10

CF bisects BCD

Prove: AE = CF Prove: ABCD is a parallelogram

A B

C D

E

1 2

3 4

1

2

A

B C D

E 10

5 6

7 8

9

A B

C D

E

1 2

3 4

10

5 6

7 8

9

A B

C D

E

1 2

3 4

F

11 12

10 5 6

7 8 9 A B

C D

E

1

2

3 4

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Geo 9 Ch 5 18

(10) (11)

Given: AB = BC , 1 2 , BE = DE Given: DE BC , BD = CE

Prove: ABCD is a rhombus Prove: ABC is isosceles (12) (13)

Given: BEDF is a rhombus , 1 5 Given: ABCD is a parallelogram BE = CF , AF = DE

Prove: ABCD is a rhombus Prove: ABCD is a rectangle (14) (15)

Given: AECF is a square , AD = BC Given: AB CD , AD = BC , DE = CE

Prove: ABCD is a parallelogram Prove: 4 5

7 8

A

B

C

D E 1

2 3 4

5 6

A

B C

D E 1 2

3 4

A B

C D

E

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C D

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C D E

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C D

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5 6

7 8 9

10

Geo 9 Ch 5 19

(16)

F

B

A

C

D E

G

H

Given: D,E,G,H are midpoints of

AB, AC, CH, BH

Prove: DEGH is a parallelogram

Geo 9 Ch 5 20

EXTRA PROBLEMS (1) Given the figure to the right,

ABCD is a parallelogram.

(a) m 1 = (9x + 3) , m 2 = (5x 2)

m 6 = (3x + 11)

Find: m 5_____ , m ADC______

(b) DE = 5x 4 , BD = 8x + 3 (c) AD = 2x 8 , AB = 3x + 2y , CD = 5x y , BC = y + 5 Find: x______ Find: x_____ , y_____ (2) Given the figure to the right,

ABCD is a rhombus, m 1 = 54 Find: the measures of the numbered angles (3) Given the figure to the right,

AB AD , CD AD , E and H are midpoints , AB = 12 , AD = 9 , CD = 18 , BH = 6 , BD = 15 Find: EF_______ , FG_______ , GH_______ , CH_______ , AF_______

1 2 3

A B

C D

E

4

5 6 7 8

10

9

D

E

1

A B

C

3

4

5 6 7

8

2

A B

C D

E F G

H

12

18

9

6

Geo 9 Ch 5 21

(4) Given the figure to the right,

ABCD is a rectangle, E , F , G , H are midpoints, AC = 12 Find: BD_______

Perimeter of EFGH_______ (5) (6)

Given: ABCD is a rectangle Given: ABCD is a parallelogram

E , F , G , H are midpoints AE BD , CF BD

Prove: EFGH is a rhombus Prove: AECF is a parallelogram

A B

C D

E

F

G

H

A B

C D

16 17

18

9 10 11 12

2 3

4

5

E

F 1

6

7 8

13

14 15

A B

C D

E

F

G

H

Geo 9 Ch 5 22

QUADRILATERALS

1. ___________________________ 2. ___________________________ 3. ___________________________ 4. __________________________

Geo 9 Ch 5 23

SUPPLEMENTARY PROBLEMS CH 5 QUADRILATERALS

1. Given that P = (-1,-1), Q = (4,3), A = (1,2) and B = (6,k), find the value of k that makes the line AB.

(a) parallel to line PQ ; (b) perpendicular to line PQ.

2. Let A = (-6,-4), B = (1,-1), C = (0,-4), and D= (-7,-7).

a) Show that the opposite sides of the quadrilateral ABCD are parallel. Such a quadrilateral is called a

parallelogram .

b) Find the lengths of all the sides. What is your conclusion?

c) Find the point of intersection of AC and BD (called the diagonals of the parallelogram ) and call it

M. Find AM and MC. What can you conclude?

3. How can one tell whether a given quadrilateral is a parallelogram? Do the converses of the

parallelogram theorems work? Are there any other ways? You might want to draw in an auxiliary line

to help you.

4. Given the points A = (0,0), B = (7,1), and D = (3,4), find coordinates for the point C that makes

quadrilateral ABCD a parallelogram. What if the question requested ABDC instead?

5. The point on a segment AB that is equidistant from A and B is called the midpoint of AB.

For each of the following, find the coordinates for the midpoint of AB.

(a) A = (-1,5) and B = ( 5, -7) (b) A = (m,n) and B = ( k,l)

6. Find the point of intersection of these two lines

2x + 3y = 6 and x – 4y = 2.

7. An equilateral parallelogram is called a rhombus. A square is

a simple example of a rhombus. Show that the lines

3x – 4y = -8, x = 0, 3x – 4y = 12, and x = 4 form the

sides of a rhombus. Support your answer.

Geo 9 Ch 5 24

9. In a right triangle,

(a) if the 2 legs are 3 and 4, what is the hypotenuse?

(b) What if the sides are 6 and 8?

(c) 9 and 12?

(d) 30 and 40?

(e) 36 and 48?

We call patterns of the Pythagorean theorem that are used often triples. Some are

3-4-5 5-12-13 7-24-25 8-15-17.

(f) Could the hypotenuse be 4 when the legs are 3 and 5?

10. Prove that in a rhombus the diagonals create four congruent triangles.

11. The diagram at right shows the graph of 3x + 4y = 12.

The shaded figure is a square, three of whose vertices are on

the coordinate axes. The fourth vertex is on the line. Find

(a) the x- and y- intercepts of the line.

(b) the length of the side of the square.

B

A

C

D

O

AB BC CD AD because

Since B and D are equidistant from A and C, point B and D

Since BD is the perpendicular bisector of AC,

segment ______ segment ____________.

Similarly, since CA is the perpendicular bisector of DB ,

segment _______ segment ____________.

So by ______________ we can say that

Therefore, four congruent triangles are formed.

Geo 9 Ch 5 25

12. A quadrilateral with only one pair of opposite sides parallel is called a trapezoid. The parallel lines are

called bases and the non-parallel sides are called legs. An isosceles trapezoid has congruent legs. Draw

an isosceles trapezoid, TRAP, with the following coordinates;

T = ( 6, 8) R = (1,8) P = (x,y) and A = (-8, 2).

(a) First determine (x,y).

(b) Find the midpoints of the legs, label them M and N

(c) Find TR, MN and PA. Is there a relationship? What is it?

(d) Draw PR. Label the point of intersection with MN as X. Find MX. Notice anything?

(e) Draw TA. Label the point of intersection with MN as Y. Find NY. Notice anything?