January 11, 2012

1) Write your homework in your agenda: Proportional Parts worksheet 2) Take out your parallel lines worksheet and leave it on your desk.

3) Take out your Cornell Notes from yesterday.

4) Take out your angle quiz.

1) 130 o 2) 128 o 3) 66 o 4) 100 o 5) 90 o 6) 81 o 7) 53 o 8) 58 o Answers to Proving Parallel Lines 9) 50 o 10) 63 o 11) x = 6 12) x = 7 13) x = -7 14) x = 6 15) x = 12 16) x = 8

What is Two Transversal Proportionality Theorem?

 If three or more parallel lines intersect two transversals, then they cut off the transversals proportionally.

A B C D BC EF = CD FG AB AE = AD AG AC AF = BC EF CD AE = FG AB E F G

R 15 P 9 Q 2 1 Two Transversal Proportionality Theorem S 11 T In the diagram  1   2   3, and PQ = 9, QR = 15, and ST = 11. What is the length of TU?

U 3

SOLUTION: Because corresponding angles are congruent, the lines are parallel and you can use Two Transversal Proportionality Theorem

PQ QR = ST TU

Parallel lines divide transversals proportionally.

9 15 = 11 TU

Substitute 9 ● TU = 15 ● 11 Cross Product

9TU = 55 165 3

Divide each side by 9 and simplify.

 So, the length of TU is 55/3 or 18 1/3.

Two-Transversal Proportionality Example Solve for x and y

x

30  15 26 16 .

5

y

 15 26 26x = 15(30) x = 225 13 15y = 16.5(26) y = 28.6

6

Triangle Proportionality Theorem If a line parallel to one side of a triangle intersects the other two sides, then it divides the two side proportionally.

Q S If TU ║ QS, then

RT TQ



RU US

T U R

Converse of the Triangle Proportionality Theorem If a line intersects two sides of a triangle and separates the sides into corresponding segments of proportional lengths, then the line is parallel to the third side. Q S T U If

RT TQ



RU

, then TU ║ QS.

US

R

Example 1 B 8 In the diagram AB ║ ED, BD = 8, DC = 4, and AE = 12. What is the length of EC?

C D 4 E 12 A

BD



AE DC EC

8 8 12

x

 4

EC

 48

EC

 6

Example 2 Given the diagram, determine whether MN ║GH G

LM MG

 56 21  8 3 21 M

LN NH

 48 16  3 1 56 L 48 N 8  3 H 3 1 16

MN is not parallel to GH.

Try This… In the diagram KL ║ MN. Find the values of the variables.

J 9 K 7.5

L 13.5

M 37.5

x y N

Solution J 9 K 7.5

L 13.5

M 37.5

x y To find the value of x, you can set up a proportion.

9 13.5

 37.5



x x

13.5(37.5 – x) = 9x 506.25 – 13.5x = 9x 506.25 = 22.5 x 22.5 = x Write the proportion Cross product Distributive property Add 13.5x to each side.

Divide each side by 22.5

N

Solution J 9 K 7.5

L 13.5

M 37.5

x y To find the value of y, you can set up a proportion.

9  7.5

y

9y = 7.5(22.5) y = 18.75

Write the proportion Cross product property Divide each side by 9.

N

 Try this one too!

PQ



QR PT TS

3 

y

9 20 3(20  

y y

)  9

y y

 9

y

S R 9 Q 20 T 60  12

y y

 5 y 3 P

January 13, 2012

1) Grab a Computation Challenge, keep it face down and put your name on the back 2) Write your homework in your agenda: Study Guide 3) Take out your worksheet and leave it on your desk.

Use proportionality Theorems in Real Life Building Construction: You are insulating your attic, as shown. The vertical 2 x 4 studs are evenly spaced. Explain why the diagonal cuts at the tops of the strips of insulation should have the same length.

Use proportionality Theorems in Real Life  Because the studs AD, BE and CF are each vertical, you know they are parallel to each other. Using Theorem 8.6, you can conclude that

DE = AB EF BC

 Because the studs are evenly spaced, you know that DE = EF. So you can conclude that AB = BC, which means that the diagonal cuts at the tops of the strips have the same lengths.