1. Five-Minute Check (over Lesson 5–5) Mathematical Practices Then/Now
Theorems: Inequalities in Two Triangles
Example 1: Use the Hinge Theorem and its Converse
Proof: Hinge Theorem
Example 2: Real-World Example: Use the Hinge Theorem
Example 3: Apply Algebra to the Relationships in Triangles
Example 4: Prove Triangle Relationships Using Hinge Theorem
Example 5: Prove Relationships Using Converse of Hinge Theorem
Lesson Menu

2. Determine whether it is possible to form a triangle with side lengths 5, 7, and 8.
A. yes
B. no
5-Minute Check 1

3. Determine whether it is possible to form a triangle with side lengths 4.2, 4.2, and 8.4.
A. yes
B. no
5-Minute Check 2

4. Determine whether it is possible to form a triangle with side lengths 3, 6, and 10.
A. yes
B. no
5-Minute Check 3

5. Find the range for the measure of the third side of a triangle if two sides measure 4 and 13.
B. 6 < n < 16
C. 8 < n < 17
D. 9 < n < 17
5-Minute Check 4

6. Find the range for the measure of the third side of a triangle if two sides measure 8.3 and 15.6.
B. 9.1 < n < 22.7
C. 7.3 < n < 23.9
D. 6.3 < n < 18.4
5-Minute Check 5

7. Write an inequality to describe the length of MN.
___
A. 12 ≤ MN ≤ 19
B. 12 < MN < 19
C. 5 < MN < 12
D. 7 < MN < 12
5-Minute Check 6

8. Mathematical Practices
3 Construct viable arguments and critique the reasoning of others.
1 Make sense of problems and persevere in solving them.
Content Standards
G.CO.10 Prove theorems about triangles. MP

9. You used inequalities to make comparisons in one triangle.
Apply the Hinge Theorem or its converse to make comparisons in two triangles.
Prove triangle relationships using the Hinge Theorem or its converse.
Then/Now

10. Concept

11. A. Compare the measures AD and BD.
Use the Hinge Theorem and Its Converse
A. Compare the measures AD and BD.
In ΔACD and ΔBCD, AC  BC, CD  CD, and mACD > mBCD.
Answer: By the Hinge Theorem, mACD > mBCD, so AD > DB.
Example 1

12. B. Compare the measures mABD and mBDC.
Use the Hinge Theorem and Its Converse
B. Compare the measures mABD and mBDC.
In ΔABD and ΔBCD, AB  CD, BD  BD, and AD > BC.
Answer: By the Converse of the Hinge Theorem, mABD > mBDC.
Example 1

13. A. Compare the lengths of FG and GH.
A. FG > GH
B. FG < GH
C. FG = GH
D. not enough information
Example 1

14. B. Compare mJKM and mKML.
A. mJKM > mKML
B. mJKM < mKML
C. mJKM = mKML
D. not enough information
Example 1

15. Concept

16. Use the Hinge Theorem
HEALTH Doctors use a straight-leg-raising test to determine the amount of pain felt in a person’s back. The patient lies flat on the examining table, and the doctor raises each leg until the patient experiences pain in the back area. Nitan can tolerate the doctor raising his right leg 35° and his left leg 65° from the table. Which leg can Nitan raise higher above the table?
Understand Using the angles given in the problem, you need to determine which leg can be risen higher above the table.
Example 2

17. Plan Draw a diagram of the situation.
Use the Hinge Theorem
Plan Draw a diagram of the situation.
Solve Since Nitan’s legs are the same length and his left leg and the table is the same length in both situations, the Hinge Theorem says his left leg can be risen higher, since 65° > 35°.
Example 2

18. Answer: Nitan can raise his left leg higher above the table.
Use the Hinge Theorem
Answer: Nitan can raise his left leg higher above the table.
Check Nitan’s left leg is pointed 30° more towards the ceiling, so it should be higher that his right leg.
Example 2

19. Meena and Rita are both flying kites in a field near their houses
Meena and Rita are both flying kites in a field near their houses. Both are using strings that are 10 meters long. Meena’s kite string is at an angle of 75° with the ground. Rita’s kite string is at an angle of 65° with the ground. If they are both standing at the same elevation, which kite is higher in the air?
A. Meena’s kite
B. Rita’s kite
Example 2

20. ALGEBRA Find the range of possible values for a.
Apply Algebra to the Relationships in Triangles
ALGEBRA Find the range of possible values for a.
From the diagram we know that
Example 3

21. Converse of the Hinge Theorem
Apply Algebra to the Relationships in Triangles
Converse of the Hinge Theorem
Substitution
Subtract 15 from each side.
Divide each side by 9.
Recall that the measure of any angle is always greater than 0.
Subtract 15 from each side.
Divide each side by 9.
Example 3

22. The two inequalities can be written as the compound inequality
Apply Algebra to the Relationships in Triangles
The two inequalities can be written as the compound
inequality
Example 3

23. Find the range of possible values of n.
C. n > 6
D. 6 < n < 18.3
Example 3

24. Write a two-column proof. Given: JK = HL
Prove Triangle Relationships Using the Hinge Theorem
Write a two-column proof.
Given: JK = HL
mJKH + mHKL < mJHK + mKHL
Prove: JH < KL
Statements
Reasons
1. JK = HL
1. Given
2. HK = HK
2. Reflexive Property
3. mJKH + mHKL < mJHK + mKHL, JH || KL
3. Given
Example 4

25. 4. Alternate Interior angles are
Prove Triangle Relationships Using the Hinge Theorem
Statements
Reasons
4. mHKL = mJHK
4. Alternate Interior angles are
5. mJKH + mJHK < mJHK + mKHL
5. Substitution
6. mJKH < mKHL
6. Subtraction Property of Inequality
7. JH < KL
7. Hinge Theorem
Example 4

26. Which reason correctly completes the following proof
Which reason correctly completes the following proof? Given: Prove: AC > DC
Example 4

27. 3. Angle Addition Postulate
Statements
Reasons 1.
1. Given 2.
2. Reflexive Property
3. mABC = mABD + mDBC
3. Angle Addition Postulate
4. mABC > mDBC
4. Definition of Inequality
5. AC > DC
5. ?
Example 4

28. B. Isosceles Triangle Theorem
A. Substitution
B. Isosceles Triangle Theorem
C. Hinge Theorem
D. none of the above
Example 4

29. Given: SP > ST Prove:
Prove Relationships Using Converse of the Hinge Theorem
Given:
SP > ST
Prove:
Example 5

30. 6. Converse of the Hinge Theorem
Prove Relationships Using Converse of the Hinge Theorem
Answer:
Proof:
Statements
Reasons 1.
1. Given 2.
2. Reflexive Property 3.
3. Given
4. SP > ST
4. Given 5.
5. Substitution 6.
6. Converse of the Hinge Theorem
Example 5

31. Which reason correctly completes the following proof?
Given: X is the midpoint of ΔMCX is isosceles. CB > CM
Prove:
Example 5

32. 1. X is the midpoint of MB; ΔMCX is isosceles 1. Given
Statements
Reasons
4. CB > CM
4. Given
5. mCXB > mCXM
5. ?
1. X is the midpoint of MB; ΔMCX is isosceles
1. Given 2.
2. Definition of midpoint 3.
3. Reflexive Property 6.
6. Definition of isosceles triangle 7.
7. Isosceles Triangle Theorem
8. mCXB > mCMX
8. Substitution
Example 5

33. A. Converse of Hinge Theorem B. Definition of Inequality
C. Substitution
D. none of the above
Example 5