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

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

Objective Apply inequalities in two triangles. Write an indirect proof.

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.

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.

Check It Out! Example 1b Compare BC and AB. Compare the side lengths in ∆ABD and ∆CBD. AD = DC BD = BD mADB > mBDC. By the Hinge Theorem, BC > AB.

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.

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.

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.

Check It Out! Example 3a Write a two-column proof. Given: C is the midpoint of BD. m1 = m2 m3 > m4 Prove: AB > ED

Statements Reasons Proof: 1. C is the mdpt. of BD m3 > m4, 1. Given 2. Def. of Midpoint 3. 1  2 3. Def. of  s 4. Conv. of Isoc. ∆ Thm. 5. AB > ED 5. Hinge Thm.

Check It Out! Example 3b Write a two-column proof. Given: SRT  STR TU > RU Prove: mTSU > mRSU Statements Reasons 1. SRT  STR TU > RU 1. Given 2. Conv. of Isoc. Δ Thm. 3. Reflex. Prop. of  4. mTSU > mRSU 4. Conv. of Hinge Thm.

So far you have written proofs using direct reasoning So far you have written proofs using direct reasoning. You began with a true hypothesis and built a logical argument to show that a conclusion was true. In an indirect proof, you begin by assuming that the conclusion is false. Then you show that this assumption leads to a contradiction. This type of proof is also called a proof by contradiction.

When writing an indirect proof, look for a contradiction of one of the following: the given information, a definition, a postulate, or a theorem. Helpful Hint

Example 1: Writing an Indirect Proof Write an indirect proof that if a > 0, then Step 1 Identify the conjecture to be proven. Given: a > 0 Prove: Step 2 Assume the opposite of the conclusion. Assume

Example 1 Continued Step 3 Use direct reasoning to lead to a contradiction. Given, opposite of conclusion Zero Prop. of Mult. Prop. of Inequality 1  0 Simplify. However, 1 > 0.

Example 1 Continued Step 4 Conclude that the original conjecture is true. The assumption that is false. Therefore

Check It Out! Example 1 Write an indirect proof that a triangle cannot have two right angles. Step 1 Identify the conjecture to be proven. Given: A triangle’s interior angles add up to 180°. Prove: A triangle cannot have two right angles. Step 2 Assume the opposite of the conclusion. An angle has two right angles.

Check It Out! Example 1 Continued Step 3 Use direct reasoning to lead to a contradiction. m1 + m2 + m3 = 180° 90° + 90° + m3 = 180° 180° + m3 = 180° m3 = 0° However, by the Protractor Postulate, a triangle cannot have an angle with a measure of 0°.

Check It Out! Example 1 Continued Step 4 Conclude that the original conjecture is true. The assumption that a triangle can have two right angles is false. Therefore a triangle cannot have two right angles.

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