Learning Target I will apply inequalities in two triangles.

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.

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.

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.

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.

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.

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

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

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

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