5_6 Inequalities of Two Triangles.ppt

math 8

1. Holt Geometry
5-6Inequalities in Two Triangles5-6Inequalities in Two Triangles
Holt Geometry
Warm Up
Lesson Presentation
Lesson Quiz

2. Holt Geometry
5-6Inequalities 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
5-6Inequalities in Two Triangles
Apply inequalities in two triangles.
Objective

4. Holt Geometry
5-6Inequalities in Two Triangles

5. Holt Geometry
5-6Inequalities in Two Triangles
Example 1A: Using the Hinge Theorem and Its
Converse
Compare mBACand 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
5-6Inequalities in Two Triangles
Example 1B: Using the Hinge Theorem and Its
Converse
Compare EFand 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
5-6Inequalities in Two Triangles
Example 1C: Using the Hinge Theorem and Its
Converse
Find the range of values for k.
Step 1Compare the side
lengths in ∆MLNand ∆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
5-6Inequalities in Two Triangles
Example 1C Continued
Step 2Since PLNis in a triangle, mPLN> 0°.
Step 3 Combine the two inequalities.
The range of values for kis 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
5-6Inequalities in Two Triangles
Check It Out!Example 1a
Compare mEGHand 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
5-6Inequalities in Two Triangles
Check It Out!Example 1b
Compare BCand 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
5-6Inequalities 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
5-6Inequalities 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
5-6Inequalities 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
5-6Inequalities 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
5-6Inequalities 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
5-6Inequalities 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
5-6Inequalities 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. ΔThm.
1.SRTSTR
TU > RU
SRTSTR
TU > RU
Check It Out!Example 3b
4. mTSU> mRSU

18. Holt Geometry
5-6Inequalities in Two Triangles
Lesson Quiz: Part I
1.Compare mABCand mDEF.
2.Compare PSand QR.
mABC> mDEF
PS< QR

19. Holt Geometry
5-6Inequalities in Two Triangles
Lesson Quiz: Part II
3.Find the range of values for z.
–3 < z< 7

20. Holt Geometry
5-6Inequalities 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: