Gch5 l6
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This document discusses using inequalities to compare angles and side lengths in two triangles. It begins with examples that use the hinge theorem and its converse to determine relationships between angles and sides. Application examples are provided, including comparing distances traveled from school. Proofs are presented to demonstrate triangle relationships using statements and reasons. The document concludes with a lesson quiz to assess understanding.
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2) One example involves comparing the distances traveled by two people leaving school at different routes to determine who is farther from school.
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Objective To use inequalities involving angles and sides of triangles
In the Solve It, you explored triangles formed by various lengths of board. You may have
noticed that changing the angle formed by two sides of the sandbox changes the length
of the third side.
Essential Understanding Th e angles and sides of a triangle have special
relationships that involve inequalities.
Property Comparison Property of Inequality
If a 5 b 1 c and c . 0, then a . b.
For a neighborhood improvement project, you
volunteer to help build a new sandbox at
the town playground. You have two boards
that will make up two sides of the
triangular sandbox. One is 5 ft long and the
other is 8 ft long. Boards come in the
lengths shown. Which boards can you use
for the third side of the sandbox? Explain.
Inequalities in
One Triangle
5-6
t
t
tt
o
lele
f
Think about
whether the shape
of the triangle
would be easy to
play in.
Dynamic Activity
Triangle
Inequalities
T
A
C T I V I T I
E S T
AAAAAAAA
C
A
CC
I E
SSSSSSSS
DY
NAMIC
Proof of the Comparison Property of Inequality
Given: a 5 b 1 c, c . 0
Prove: a . b
Statements Reasons
1) c . 0 1) Given
2) b 1 c . b 1 0 2) Addition Property of Inequality
3) b 1 c . b 3) Identity Property of Addition
4) a 5 b 1 c 4) Given
5) a . b 5) Substitution
Proof
hsm11gmse_NA_0506.indd 324 3/6/09 11:56:15 AM
http://media.pearsoncmg.com/aw/aw_mml_shared_1/copyright.html
Problem 1
Got It?
Lesson 5-6 Inequalities in One Triangle 325
Th e Comparison Property of Inequality allows you to prove the following corollary to
the Triangle Exterior Angle Th eorem (Th eorem 3-11).
Proof of the Corollary
Given: /1 is an exterior angle of the triangle.
Prove: m/1 . m/2 and m/1 . m/3.
Proof: By the Triangle Exterior Angle Th eorem, m/1 5 m/2 1 m/3. Since
m/2 . 0 and m/3 . 0, you can apply the Comparison Property of
Inequality and conclude that m/1 . m/2 and m/1 . m/3.
Applying the Corollary
Use the fi gure at the right. Why is ml2 S ml3?
In nACD, CB > CD, so by the Isosceles Triangle Th eorem,
m/1 5 m/2. /1 is an exterior angle of nABD, so by the
Corollary to the Triangle Exterior Angle Th eorem, m/1 . m/3.
Th en m/2 . m/3 by substitution.
1. Why is m/5 . m/C?
You can use the corollary to Th eorem 3-11 to prove the following theorem.
Corollary Corollary to the Triangle Exterior Angle Theorem
Corollary
Th e measure of an exterior
angle of a triangle is greater
than the measure of each of
its remote interior angles.
If . . .
/1 is an exterior angle
Then . . .
m/1 . m/2 and
m/1 . m/3
2 1
3
Proof
3
4
1
25
A
CD
B
Theorem 5-10
Theorem
If two sides of a triangle are
not congruent, then the
larger angle lies opposite
the longer side.
If . . .
XZ . XY
Then . . .
m/Y . m/Z
You will prove Theorem 5-10 in Exercise 40.
X
Y
Z
G
U
I
m
C
Th
G
How do you identify
an exterior angle?
An exterior angle
must be form.
Congruent figures 2013
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- 1. Holt Geometry 5-6 Inequalities in Two Triangles5-6 Inequalities in Two Triangles Holt Geometry Warm UpWarm Up Lesson PresentationLesson Presentation Lesson QuizLesson Quiz
- 2. Holt Geometry 5-6 Inequalities in Two Triangles Warm Up 1. Write the angles in order from smallest to largest. 2. The lengths of two sides of a triangle are 12 cm and 9 cm. Find the range of possible lengths for the third side. ∠X, ∠Z, ∠Y 3 cm < s < 21 cm
- 3. Holt Geometry 5-6 Inequalities in Two Triangles Apply inequalities in two triangles. Objective
- 4. Holt Geometry 5-6 Inequalities in Two Triangles
- 5. Holt Geometry 5-6 Inequalities in Two Triangles Example 1A: Using the Hinge Theorem and Its Converse Compare m∠BAC and m∠DAC. Compare the side lengths in ∆ABC and ∆ADC. By the Converse of the Hinge Theorem, m∠BAC > m∠DAC. AB = AD AC = AC BC > DC
- 6. Holt Geometry 5-6 Inequalities in Two Triangles Example 1B: Using the Hinge Theorem and Its Converse Compare EF and FG. By the Hinge Theorem, EF < GF. Compare the sides and angles in ∆EFH angles in ∆GFH. EH = GH FH = FH m∠EHF > m∠GHF m∠GHF = 180° – 82° = 98°
- 7. Holt Geometry 5-6 Inequalities in Two Triangles Example 1C: Using the Hinge Theorem and Its Converse Find the range of values for k. Step 1 Compare the side lengths in ∆MLN and ∆PLN. By the Converse of the Hinge Theorem, m∠MLN > m∠PLN. LN = LN LM = LP MN > PN 5k – 12 < 38 k < 10 Substitute the given values. Add 12 to both sides and divide by 5.
- 8. Holt Geometry 5-6 Inequalities in Two Triangles Example 1C Continued Step 2 Since ∠PLN is in a triangle, m∠PLN > 0°. Step 3 Combine the two inequalities. The range of values for k is 2.4 < k < 10. 5k – 12 > 0 k < 2.4 Substitute the given values. Add 12 to both sides and divide by 5.
- 9. Holt Geometry 5-6 Inequalities in Two Triangles Check It Out! Example 1a Compare m∠EGH and m∠EGF. Compare the side lengths in ∆EGH and ∆EGF. FG = HG EG = EG EF > EH By the Converse of the Hinge Theorem, m∠EGH < m∠EGF.
- 10. Holt Geometry 5-6 Inequalities in Two Triangles Check It Out! Example 1b Compare BC and AB. Compare the side lengths in ∆ABD and ∆CBD. By the Hinge Theorem, BC > AB. AD = DC BD = BD m∠ADB > m∠BDC.
- 11. Holt Geometry 5-6 Inequalities in Two Triangles Example 2: Travel Application John and Luke leave school at the same time. John rides his bike 3 blocks west and then 4 blocks north. Luke rides 4 blocks east and then 3 blocks at a bearing of N 10º E. Who is farther from school? Explain.
- 12. Holt Geometry 5-6 Inequalities in Two Triangles Example 2 Continued The distances of 3 blocks and 4 blocks are the same in both triangles. The angle formed by John’s route (90º) is smaller than the angle formed by Luke’s route (100º). So Luke is farther from school than John by the Hinge Theorem.
- 13. Holt Geometry 5-6 Inequalities in Two Triangles Check It Out! Example 2 When the swing ride is at full speed, the chairs are farthest from the base of the swing tower. What can you conclude about the angles of the swings at full speed versus low speed? Explain. The ∠ of the swing at full speed is greater than the ∠ at low speed because the length of the triangle on the opposite side is the greatest at full swing.
- 14. Holt Geometry 5-6 Inequalities in Two Triangles Example 3: Proving Triangle Relationships Write a two-column proof. Given: Prove: AB > CB Proof: Statements Reasons 1. Given 2. Reflex. Prop. of ≅ 3. Hinge Thm.
- 15. Holt Geometry 5-6 Inequalities in Two Triangles Check It Out! Example 3a Write a two-column proof. Given: C is the midpoint of BD. Prove: AB > ED m∠1 = m∠2 m∠3 > m∠4
- 16. Holt Geometry 5-6 Inequalities in Two Triangles 1. Given 2. Def. of Midpoint 3. Def. of ≅ ∠s 4. Conv. of Isoc. ∆ Thm. 5. Hinge Thm. 1. C is the mdpt. of BD m∠3 > m∠4, m∠1 = m∠2 3. ∠1 ≅ ∠2 5. AB > ED Statements Reasons Proof:
- 17. Holt Geometry 5-6 Inequalities in Two Triangles Write a two-column proof. Given: Prove: m∠TSU > m∠RSU Statements Reasons 1. Given 3. Reflex. Prop. of ≅ 4. Conv. of Hinge Thm. 2. Conv. of Isoc. Δ Thm. 1. ∠SRT ≅ ∠STR TU > RU ∠SRT ≅ ∠STR TU > RU Check It Out! Example 3b 4. m∠TSU > m∠RSU
- 18. Holt Geometry 5-6 Inequalities in Two Triangles Lesson Quiz: Part I 1. Compare m∠ABC and m∠DEF. 2. Compare PS and QR. m∠ABC > m∠DEF PS < QR
- 19. Holt Geometry 5-6 Inequalities in Two Triangles Lesson Quiz: Part II 3. Find the range of values for z. –3 < z < 7
- 20. Holt Geometry 5-6 Inequalities in Two Triangles Statements Reasons 1. Given 2. Reflex. Prop. of ≅ 3. Conv. of Hinge Thm.3. m∠XYW < m∠ZWY Given: Prove: m∠XYW < m∠ZWY 4. Write a two-column proof. Lesson Quiz: Part III Proof: