Extends MAFS.912.G-CO.3.10 Prove theorems …...2018/11/26 · 5 7 Inequalities in Two Triangles...
Transcript of Extends MAFS.912.G-CO.3.10 Prove theorems …...2018/11/26 · 5 7 Inequalities in Two Triangles...
57 Inequalities in TwoTriangles
Objective To apply inequalities in two triangles
Mathematics Florida StandardsExtends MAFS.912.G-CO.3.10 Prove theorems
about triangles
MP1,MP3
Getting Ready! X C A
Try to make theproblem simplerby finding a way tofind the distance x
without measuringdirectly.
Think of a clock or watch that has an hour hand and
a minute hand. As minutes pass, the distance betweenthe tip of the hour hand and the tip of the minutehand changes. This distance is x in the figure at theright. What is the order of the times below fromleast to greatest length of x? How do you know?
1:00, 3:00, 5:00, 8:30, 1:30, 12:20
MATHEMATICAL
PRACTICES jjj Solve It, the hands of the clock and the segment labeled x form a triangle. As thetime changes, the shape of the triangle changes, but the lengths of two of its sides do
not change.
Essential Understanding in triangles that have two pairs of congruent sides,there is a relationship between the included angles and the third pair of sides.
When you close a door, the angle between the
door and the frame (at the hinge) gets smaller.The relationship between the measure of the
hinge angle and the length of the opposite side
is the basis for the SAS Inequality Theorem, also
known as the Hinge Theorem.
Theorem 5-13 The Hinge Theorem (SAS Inequality Theorem)
Theorem
If two sides of one triangle
are congruent to two sides
of another triangle, and
the included angles are not
congruent, then the longer
third side is opposite the
larger included angle.
m^A > m^X
Then...
BOYZ
You will prove Theorem 5-13 in Exercise 25.
332 Chapter 5 Relationships Within Triangles
How do you applythe Hinge Theorem?After you identify theangles included betweenthe pairs of congruentsides, locate the sidesopposite those angles.
For which side
lengths are the sameat Time 1 and Time 2?
The lengths of the chain
and AB do not change.So, AS and 8C are the
same at Time 1 and
Time 2.
Using the Hinge Theorem
Multiple Choice Which of the following
statements must be true?
Ca:> AS< YU
ct:) sk> yu
cc::>sK<Yu
CE) AK= YU
SA = YO and AK = OU, so the triangles have two pairs of congruent sides.
The included angles, LA and LO, are not congruent. Since mLA > mLO,
SK > YU by the Hinge Theorem. Hie correct answer is B.
Got It? 1. a. What inequality relates LN and OQin tlie figure at the right?
b. Reasoning In AABC, AB = 3,BC = 4, and CA = 6. In APQR,
PQ = S,QR^ 5, and RP = 6.
How can you use indirect reasoning
to explain why mLP > mLAl
Time 1 Time 2
Applying the Hinge Theorem
Swing Ride The diagram below shows the position of a
swing at two different times. As the speed of the swing
ride increases, the angle between the chain and AB
increases. Is the rider farther from point A at
Time 1 or Time 2? Explain how the Hinge
Theorem justifies your answer.
The rider is farther from point A at Time 2.
The lengths of AB and BC stay the same
throughout the ride. Since the angle
formed at Time 2 {L2) is greater thanthe angle formed at Time 1 [LI],you can use the Hinge Theorem to
conclude that AC at Time 2 is longer
tlian AC at Time 1.
Lesson 5-7 Inequalities in Two Triangles 333
Gofit? 2. The diagram below shows a pair of scissors in two different positions. Inwhich position is the distance between the tips of the two blades greater?Use the Hinge Theorem to justily your answer.
The Converse of the Hinge Theorem is also true. The proof of the converse is anindirect proof.
Theorem 5-14 Converse of the Hinge Theorem {SSS Inequality)
Theorem if... Then ...
If two sides of one triangle
are congruent to two sides
of another triangle, and
the third sides are not
congruent, then the larger
included angle is opposite
the longer third side.
SO yz mZ^A > mLX
Proof Indirect Proof of the Converse of the Hinge Theorem (SSS Inequality)
Given: AB = ̂,AC = XZ,BO yz
Prove: mAA > mAX
Step 1 Assume temporarily that mAA > mAX.
This means either mAA < mAJC or
mAA = mAX.
Step 2 If mAA < mAX, then EC < FZ by the Hinge Theorem. This contradicts
the given information that EC > YZ. Therefore, the assumption thatmAA < mAX must be false.
If mAA = mAX, then AABC = AXYZ by SAS. If the two triangles arecongruent, then EC — YZ because corresponding parts of congruent
triangles are congruent. This contradicts the given information that EC > YZ.
Therefore, the assumption that mAA = mAX must be false.
Step 3 The temporary assumption that mAA > mAAX is false. Therefore,
mAA > mAJC.
334 Chapter 5 Relationships Within Triangles
I-*" ai rt n
How do you putupper and lowerlimits on the value
of X?
Use the largest possiblevalue oi mz.TUS as the
upper limit for Sx - 20and the smallest possiblevalue ofm/LTUSas the
lower limit for 5x-20.
Using the Converse of the Hinge Theorem
Algebra What is the range of possible values for x?
Step 1 Find an upper limit for the value of x. UT = UR andUS = US, so A TUS and ARUS have two pairs of
congruent sides. RS > TS, so you can use the Converseof the Hinge Theorem to write an inequality.
m/LRUS > ml. TUS Converse of the Hinge Theorem
60 > Sx — 20 Substitute.
80 > 5x Add 20 to each side.
16 > jc Divide each side by 5.
Find a lower limit for the value of x.Step 2
10 '
mATUS >0 The measure of an angle of a triangle is greater than 0.
Sx — 20 > 0 Substitute.
5x > 20 Add 20 to each side.
x> 4 Divide each side by 5.
Rewrite 16 > x and x > 4 as 4 < x < 16.
Got It? 3. What is the range of possible values Tfor X in the figure at the right?
(3x + 18)
Proof
Th'rI.
How do you knowmlBAE > mLBEA7
Use the Transitive
Property of Inequalityon the inequalities InStatements 4 and 6.
Problem 4 Proving Relationships in Triangles
Given: BA = DE, BE > DA
Prove: mABAE > mABEA
Statement Reasons
1) BA = DE 1) Given
2) AE = AE 2) Reflexive Property of Equality
3) BE>DA 3) Given
4) mABAE > mADEA 4) Converse of the Hinge Theorem
5) mADEA = mADEB + mABEA 5) Angle Addition Postulate
6) mADEA > mABEA 6) Comparison Property of Inequality
7) mABAE > mABEA 7) Transitive Property of Inequality
Got It? 4. Given: = 80, O is the midpoint of LAf
Prove: LM>MN
c PowerGeometry.com \ Lesson 5-7 Inequalities in Two Triangles 335
&Lesson Check
Do you know HOW?
Write an inequality relating the givenside lengths or angle measures.
1.FDandjBC B £ ], p
2. mzit/Sr and mAVST
D
14
12
_ . MATHEMATICALDo you UNDERSTAND? PRACTICES
3. Vocabuiary Explain why Hmge Theorem is an
appropriate name for Theorem 5-13.
4. Error Analysis From the
figure at the right, your
friend concludes that
m/LBAD > m/LBCD. How
would you correct your
friend's mistake?
5. Compare and Contrast How are the Hinge Theorem
and the SAS Congruence Postulate similar?
MATHEMATICAL
PRACTICESPractice and Problem-Solving Exercises
Practice Write an inequality relating the given side lengths. If there is not enoughinformation to reach a conclusion, write no conclusion.
6. AB and AD
8. LM and KL
7. PR and RT
^ See Problem 1.
9. yz and UV
10. The diagram below shows a robotic arm in two different positions. In which ^ See Problem 2.position is the tip of the robotic arm closer to the base? Use the Hinge Theorem
to justify your answer.
336 Chapter 5 Relationships Within Triangles
Algebra Find the range of possible values for each variable. ^ See Problem 3.
11.{X - 6)
13.
36
(2X-12)
4X-10
^ Apply
15. Developing Proof Complete the following proof.
Given; Cis the midpoint of BD,
m/LEAC = rri/LAEC,
mZ-BCA > m/LDCE
Prove: AB>ED
Statements Reasons
1) m/.EAC = mZ-AEC 1) Given
2) AC=EC 2) a. J_
3) C is the midpoint of 5D. 3) b. J_
4) IC = ̂ 4) c. J_
5) d- 5) = segments have = length.
6) m/LBCA > m/-DCE 6) e. J_
7) AB>ED 7) f. J_
Copy and complete with > or <. Explain your reasoning.
16. PTBQfi Q
17. mAQTRmm/.RTS
18. PTMRS
^ See Problem 4.
19. a. Error Analysis Your classmate draws the figure at the right. Explainwhy the figure cannot have the labeled dimensions,
b. Open-Ended Describe a way you could change the dimensions tomake the figure possible.
C PowerGeometry.cbm Lesson 5-7 Inequalities in Two Triangles 337
start
20. Think About a Plan Ship A and Ship B leave from the
same point in the ocean. Ship A travels 150 mi due west,
turns 65° toward north, and then travels another 100 mi.
Ship B travels 150 mi due east, turns 70° toward south, V
and then travels another 100 mi. Which ship is farther
from the starting point? Explain.
• How can you use the given angle measures?
• How does the Hinge Theorem help you to solve
this problem?
21. Which of the following lists the segment lengths in order
from least to greatest?
CS> CD, AB, DE, BC, EE
CT) EE, DE, AB, BC, CD
CO BC, DE, EE, AB. CD
cO EE, BC, DE, AB, CD
22. Reasoning The legs of a right isosceles triangle are congruent to the legs of anisosceles triangle with an 80° vertex angle. Which triangle has a greater perimeter?How do you know?
23. Use the figure at the right.Proof Qjygp. is isosceles with vertex AB,
AABE = ACBD,
mAEBD > mAABE
Prove: ED>AE
j^Chaiienqe 24. Coordinate Geometry AABC has vertices A(0, 7), B(-l, —2), C(2, —1), and0(0, 0). Show that mAAOB > mAAOC.
25. Use the plan below to complete a proof of the Hinge Theorem: If two sides of oneProof triangle are congruent to two sides of another triangle and the included angles are
not congruent, then the longer third side is opposite the larger included angle.
Given: AB = XY, BC = YZ, mAB > mAY
Prove: AOXZ.
Plan for proof:
• Copy AABC. Locate point D outside AABC so
that mACBD — mAZYX and BD = YX. Show that
ADBC = tJCYZ.
• Locate point Fon AC, so that BE bisects AABD.
• Show that AABE = ADBE and that AE = DE.
• Show that AC = FC +DF.
• Use the Triangle Inequality Theorem to write an
inequality that relates DC to the lengths of the other
sides of AECD.
• Relate DC and XZ.
H
s
ShipB
X
338 Chapter 5 Relationships Within Triangles
,l©lApply What You've Learned
Look at the trail map from page 283, shown again below. In the Apply What
You've Learned sections in Lessons 5-1 and 5-6, you worked with the first
two segments of the hike from the campground to the waterfall. Now you will
consider the last segment of the hike.
MATHEMATICAL
PRACTICES
MP2
2.5 km
Campground
C
4.2 km
3.5 km 4.8 km
3.1 km
5.1 km
■""..^WaterfaliW
3.2 km
a. Write an inequality that shows an upper bound for the length of LW.Explain your reasoning.
b. Write an inequality that shows a lower bound for the length of LW.Explain your reasoning.
c. Use your answers from parts (a) and (b) to write a compound inequalitythat shows the range of possible lengths of LW.
c PowerGeometry.com Lesson 5-7 Inequalities in Two Triangles 339