Theorems Involving Parallel Lines and

Triangles

Keystone Geometry:

Using Theorems, Midpoints, and Triangles

Theorem:

• If three parallel lines cut off congruent segments on one transversal, then they cut off congruent segments on every

transversal.

• If and AC = CE , then BD = DF.

A B

C D

E F

ABs ruu

||CDs ruu

|| EFs ruu

Theorem:

• A line that contains the midpoint of one side of a triangle and is parallel to another side passes through the midpoint of the third side.

• If M is the midpoint of

XY and , then N is the midpoint of XZ. MNs ruuu

||YZs ru

NM

X

Y Z

The Midsegment

• The segment that joins the midpoints of two sides of a triangle is called the Midsegment. The Midsegment:

Is parallel to the third side. Is half as long as the

third side.

• If M is the midpoint of XY and N is the midpoint of XZ, then MN || YZ and MN = 1/2 YZ.

NM

X

Y Z

are parallel, with AB = BC = CD.

• If QR = 3x and RS = x+12,

then x=____• If PQ = 3x - 9 and QR = 2x - 2,

then x=____• If QR = 20, then QS=____ • If PQ = 6, then PR=____ and PS=____• If PS = 21, then RS=____• If PR = 7x - 2 and RS = 3x + 4, then x=____• If PR = x + 8 and QS = 3x - 2, then x=____

APs ruu

,BQs ruu

,CRs ruu

, and DSs ruu

A P

B Q

C R

D S

6

7

40

12 18

7

10

5

Example: M is the midpoint of XY and N is the midpoint of XZ.

If MN = 6, then YZ =_____.If YZ = 20, then MN =_____.If MN = x and YZ = 3x - 25,Then x =_____, MN = _____, And YZ =_____.If <XMN = 40, then <XYZ =_____. NM

Y

X

Z

12

10

25 25

50

40

They are CorrespondingAngles.