Geometry

13.1 : Side-Splitting Theorems

p. 774 - 780

mg Idea: When two transversals intersect parallel lines, proportional lengths are created.

If a line is parallel to a side of a triangle and intersectsthe other two sides in distinct points, it splits these sidesinto segments of proportional lengths.

e'Side-S - I ingtTheO em xarnJe.

Example 1Use lenghs as shown in the diagram at the right, in which jÉ ää

a. Find AE. b. Find DE

Solution

a. Since DE Il BC, pu can use fre Side-SplittingTheorem to calculate AE.AE

10

AE =b. The ratio of DE to BC is not the same as the rado of AD to DB or ofAEto

EC because Db and VC are not sides of Similar trianges must

be used. Since DÉ Il ÄC, by AA Similarihu As a

result, the raüos of corresponding lengths are equal.

DE

2.5 •

DE =

15m

10 an

B 18

t19PezL NSldes

Given m Il

Prove

rdpéi6V p I a-theorem

If a line parallel to the bases intersects the legs of atrapezoid and divides those legs into two segments.then the lengths of those segments are proportional.

I Ing

Proof Draw the diagonal ÄPofthe trapezoid, as shown atfre right

By the Triangle Side-SplittingTheorem applied to AACF,

By the Triangle Side-SplittingTheorem applied to AADF,

So, taking tie reciprocal of each side,

Thus, by the Transitive Property of Equality, b

COVERING THE IDEAS

Fill in the Blmk In 1-4, use the figure at the right to find a differentfraction equal to the given fraction.

5. In thetriangledtheright,igCHll

6. In the diagram below, the line segments are parallel to the

of the åiangle. for the lengths x, y, and z

2 2510

14 15

21

Side

13 NT28 mm H / J

30 mm

15 mm

SÖebaSC

7. A såident said, "I have an easy proof of the

Trapezoid Side-Splitting Iheorem, Use the figure

on the bottom of page 777. Start by translafing DF

so that the &anslaåon image of D is A. " finish

this suadent's proof. See margln.

8. In the diagram atthe right, t Il m Il n Il p.

Prove thatv=-z. See margin.

Back to Lesson 13-1

Name

Lesson Master

Objective A

In 1 and 2, find x and y.

Answer Page

Questions on SPUR ObjectivesSee Student Edition pages 819—821 for objectives.

In 3 and 4, determine what segments (if any) must be parallel.

1816 20

15

10

9.6

18 Y

tx)ø

Objective H

set is 7 feet above the ground. "Ille crossbar is 30 inches above the

ground. The legs are 8 feet long. How much of the leg is above

the crossbar, to the nearest inch?

6. In downtown Washington, D.C., Connecticut Avenue,JC-J

New Hampshire Avenue, K Street, and N Street

roughly form a large trapezoid. Assume M Street

runs parallel to K Street. Along New Hampshire Ave., z

the distance from K Street to M Street is 0.30 mile,

and from M Street to N Street is 0.26 mile. Along

Connecticut Ave., the distance from K Street to M L St NWStreet is 0.24 mile. Find the distance along

Connecticut Ave. from M Street to N Street. K St NW

, 30

12

10 L

N St NW

M St NW

Geometry 573

Back to Lesson 13-1 Answer Page

Name

Les son Master Questions on SPUR ObjectivesSee Student Edition pages 819-821 for objectives.

VOCABULARY

In 1 and 2, complete the Side-Splitting Theorem and its converse.

1. If a line is to a side of a triangle and intersects

the other two sides in distinct points, it splits these sides

into segments.

2. If a line intersects OP and OQ in distinct points X and Y so that

then XY is to PQ. -J L a H.8+X

Objective A1.4

3. In AJKLat the right, XYlljR. FindJX, rounded to the ne est tenth. €.(DH 4.72' x

4. In AXYZ at the right, Il ü. Find each missing length.

10

5. In the diagram at the right, h Il j Il k. Find each missing length.

30c. AB = 5y; DE = 12;

6. Given AADM at the right, in which OP Il AD, tell whether each

statement is true orfalse.

z

b. z

574 Geometry

4.8

x 8----

z

h

k

c

x z

O

Classwork

Find x,

3) In AABC, EFI ICE. Find x.

18

x+22

\ 7 X & 30

5) Determine whether BC I IDE.

BD- 9, BA: 27, and CEis one third of

No

2) Find x.

X + 12

4) Determine whether BC I IDE.

AD: 15, DB : 12, AE: 10, and EC- 8

yes Ito--lt6

COMMONLesson 4: Comparing the Ratio Method with the Parallel Method

s.25Date: 1/8/16CORE

0 2014 Common Core, Inc. Some rights reserved. commoncore.org

engageThis work is licensed under a

I Creative Commens Attribution-NgnCommercial-ShareAIike 3.0 Vnpgrted License.

Name: Date :

Geometry M2L4 Side Splitter Theorem HW Period:

1) Find x. 2) Find x.

24

30 10 30 10

30

3) If AD: 24, DB: 27, and EB- 18, find CE 4) Find x, QT, and TPif x + 6,

COMMONlesson 4:

CORE Date:

0 2014 Cornrnon Ccte. regs

SR: 12, 27 and 72: x -4.

1

IYO el SX

1/8/16 engageny s.26Comparing the Ratio Method with the Parallel Method

woo licensed underOjUY.uc.SA J

5) Determine whether Jl<l INM. 6) Determine whether Fäl IDÉ.

18, 30, KM: 21, and ML: 35 AE: 30, AC: 45, and AD is twice DB

Lesson Summary

Alifi€ splits two sides of triangle proportionally if and only if it is

parallel to the thitt$ Side

COMMON Lessm 4: Patellel Method

CORE engage s.27