1. Learning Target
I will apply inequalities in two triangles.

2. QUESTION: In the figure below, which side to you think is longer, AB or DE?
Common Sense tells us, but there is a theorem which proves that DE > AB, called the Hinge Theorem.

4. Example 1A: Using the Hinge Theorem and Its Converse
Compare mBAC and mDAC.
Compare the side lengths in ∆ABC and ∆ADC.
AB = AD AC = AC BC > DC
By the Converse of the Hinge Theorem, mBAC > mDAC.

5. Example 1B: Using the Hinge Theorem and Its Converse
Compare EF and FG.
Compare the sides and angles in ∆EFH angles in ∆GFH.
mGHF = 180° – 82° = 98°
EH = GH FH = FH mEHF > mGHF
By the Hinge Theorem, EF < GF.

6. 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.
LN = LN LM = LP MN > PN
By the Converse of the Hinge Theorem, mMLN > mPLN.
5k – 12 < 38
Substitute the given values.
k < 10
Add 12 to both sides and divide by 5.

7. Example 1C Continued
Step 2 Since PLN is in a triangle, mPLN > 0°.
5k – 12 > 0
Substitute the given values.
k < 2.4
Add 12 to both sides and divide by 5.
Step 3 Combine the two inequalities.
The range of values for k is 2.4 < k < 10.

8. Homework: Pg 355 – 356, #9 – 14, 18 – 25.

9. Lesson Quiz: Part I
1. Compare mABC and mDEF.
2. Compare PS and QR.
mABC > mDEF
PS < QR

10. Lesson Quiz: Part II
3. Find the range of values for z.
–3 < z < 7

11. Statements Reasons Lesson Quiz: Part III 4. Write a two-column proof.
Prove: mXYW < mZWY
Given:
Proof:
Statements
Reasons
1. Given
2. Reflex. Prop. of 
3. Conv. of Hinge Thm.
3. mXYW < mZWY