7-4: Parallel Lines and 7-4: Parallel Lines and Proportional PartsProportional Parts

Expectation:Expectation:G1.1.2: Solve multi-step problems and construct proofs G1.1.2: Solve multi-step problems and construct proofs

involving corresponding angles, alternate interior angles, involving corresponding angles, alternate interior angles, alternate exterior angles, and same-side (consecutive) alternate exterior angles, and same-side (consecutive)

interior angles.interior angles.G1.2.2: Construct and justify arguments and solve multi-G1.2.2: Construct and justify arguments and solve multi-

step problems involving angle measure, side length, step problems involving angle measure, side length, perimeter, and area of all types of triangles.perimeter, and area of all types of triangles.

G2.3.4: Use theorems about similar triangles to solve G2.3.4: Use theorems about similar triangles to solve problems with and without use of coordinates.problems with and without use of coordinates.

Given that ΔXYZ is similar to ΔABC, what is Given that ΔXYZ is similar to ΔABC, what is the sum of the lengths of sides XY and XZ the sum of the lengths of sides XY and XZ in ΔXYZ below?in ΔXYZ below?

A.A. 99B.B. 1212C.C. 1515D.D. 2424E.E. 2727

X

y

Z9

Yz

A

5

C3

B4

Triangle Proportionality TheoremTriangle Proportionality Theorem(Side Splitting Theorem)(Side Splitting Theorem)

If a line is parallel to a side of a triangle and If a line is parallel to a side of a triangle and intersects the other 2 sides in distinct points, then intersects the other 2 sides in distinct points, then it splits those 2 sides into segments of proportional it splits those 2 sides into segments of proportional length. length.

Triangle Proportionality TheoremTriangle Proportionality Theorem

A

ED

C

B

If DE || AC, ADBD

CEBE=

4 common proportions from the 4 common proportions from the side splitting theoremside splitting theorem

w

z

y

x

Determine the value of x below if EB || CD.

A

E B

D C4

x

3

5

In the diagram below, segments that In the diagram below, segments that appear to be parallel are parallel. Solve appear to be parallel are parallel. Solve

for x.for x.

x

15

24

12

In the figure below, segment AB is parallel to In the figure below, segment AB is parallel to segment CD. What is the sum of AC and segment CD. What is the sum of AC and BD?BD?

A.A. 3⅓3⅓B.B. 4⅓4⅓C.C. 5⅓5⅓D.D. 7⅔7⅔E.E. 8⅔8⅔

AC

EDB

3 2

8

5

Converse of Triangle Converse of Triangle Proportionality TheoremProportionality Theorem

If a line intersects two sides of a If a line intersects two sides of a triangle in distinct points and triangle in distinct points and separates the sides into proportional separates the sides into proportional segments, then the line is parallel to segments, then the line is parallel to the third side of the triangle. the third side of the triangle.

A ED

C

B

If AB DE , then AE || BD.=BC DC

Is AC || DE?Is AC || DE?

B6E6

8D

A2

C

True or False? JustifyTrue or False? JustifyIf BE = If BE = 33//55AB, then DC = AB, then DC = 22//55BC.BC.

A

E

BD

C

Midpoint Connector Theorem Midpoint Connector Theorem If a segment’s endpoints are the If a segment’s endpoints are the

midpoints of 2 sides of a triangle, midpoints of 2 sides of a triangle, then it is parallel to and half the then it is parallel to and half the length of the third side. length of the third side.

Midpoint Connector Theorem Midpoint Connector Theorem

AE

DC

B

E and D are the midpoints of their respective sides so ED || AB and

AB = 2ED

Proportional Transversals Proportional Transversals Corollary Corollary

If 3 or more parallel lines intersect 2 If 3 or more parallel lines intersect 2 transversals, then they cut the transversals, then they cut the transversals proportionally.transversals proportionally.

Proportional Transversals Proportional Transversals Corollary Corollary s t

l

m

n

ab

y

z

If l || m and m || n, then . ab

yz=

Find the values of all of the variables below if w + x + y + z = 20

w xy z

6354

Congruent Transversal Segment Congruent Transversal Segment TheoremTheorem

If 3 or more parallel lines cut off congruent segments on one transversal, then they cut congruent segments on every transversal.

Solve for xSolve for x

l

m

n

u t

x12

AssignmentAssignmentpages 367 – 369, # 13 -29 (odds), 39 and 41 (odds)