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5-6 5-6 Inequalities InequalitiesininTwo TwoTriangles Triangles Warm Up Lesson Presentation Lesson Quiz HoltMcDougal GeometryGeometry Holt 5-6 Inequalities in Two Triangles Warm up: Determine if it is possible to construct a triangle with the following vertices 1. (1,1)(2,2)(3,3) 2. No- straight line Yes 3. x=4 A=65 B=73 So… AB, BC, AC Holt McDougal Geometry C=42 5-6 Inequalities in Two Triangles Warm Up 4. Write the angles in order from smallest to largest. X, Z, Y 5. The lengths of two sides of a triangle are 12 cm and 9 cm. Find the range of possible lengths for the third side. 3 cm < s < 21 cm Holt McDougal Geometry 5-6 Inequalities in Two Triangles Objective Apply inequalities in two triangles. Holt McDougal Geometry 5-6 Inequalities in Two Triangles Holt McDougal Geometry 5-6 Inequalities in Two Triangles Example 1: Using the Hinge Theorem and Its Converse Compare mBAC and mDAC. By the Converse of the Hinge Theorem, mBAC > mDAC. Holt McDougal Geometry 5-6 Inequalities in Two Triangles Example 2: Using the Hinge Theorem and Its Converse Compare EF and FG. Compare the sides and angles in ∆EFH angles in ∆GFH. mGHF = 180° – 82° = 98° By the Hinge Theorem, EF < GF. Holt McDougal Geometry 98 5-6 Inequalities in Two Triangles Example 3: 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, mPLN < mMLN 5k – 12 < 38 k < 10 Holt McDougal Geometry Substitute the given values. Add 12 to both sides and divide by 5. 5-6 Inequalities in Two Triangles What else do we know about the mPLN Step 2 Since PLN is in a triangle, mPLN > 0°. 5k – 12 > 0 k < 2.4 Step 3 Combine the two inequalities. The range of values for k is 2.4 < k < 10. Holt McDougal Geometry 5-6 Inequalities in Two Triangles Check It Out! Example 4 Compare mEGH and mEGF. Compare the side lengths in ∆EGH and ∆EGF. FG = HG EG = EG EF > EH By the Converse of the Hinge Theorem, mEGH < mEGF. Holt McDougal Geometry 5-6 Inequalities in Two Triangles Check It Out! Example 5 Compare BC and AB. Compare the side lengths in ∆ABD and ∆CBD. AD = DC BD = BD mADB > mBDC. By the Hinge Theorem, BC > AB. Holt McDougal Geometry 5-6 Inequalities in Two Triangles Example 6: 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 SE. Who is farther from school? Explain. Holt McDougal Geometry 5-6 Inequalities 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. Holt McDougal Geometry 5-6 Inequalities in Two Triangles PROOF #1 Write a two-column proof. Given: C is the midpoint of BD. m1 = m 2 m3 > m 4 Prove: AB > ED Holt McDougal Geometry 5.6 Indirect Proof and Inequalities in Two Triangles 5-6 Inequalities in Two Triangles Proof: Statements 1. C is the mdpt. of BD m3 > m4, m1 = m2 Reasons 1. Given 2. Def. of Midpoint 3. 1 2 3. Def. of s 4. Conv. of Isoc. ∆ Thm. 5. AB > ED Holt McDougal Geometry 5. Hinge Thm. 5-6 Inequalities in Two Triangles Proof #2: Proving Triangle Relationships Write a two-column proof. Given: Prove: AB > CB Proof: Statements Reasons 1. Given 2. Reflex. Prop. of 3. Hinge Thm. Holt McDougal Geometry 5-6 Inequalities in Two Triangles Lesson Quiz 1. Compare mABC and mDEF. mABC > mDEF 2. Compare PS and QR. PS < QR Holt McDougal Geometry 5-6 Inequalities in Two Triangles Lesson Quiz 3. Find the range of values for z. –3 < z < 7 Holt McDougal Geometry 5-6 Inequalities in Two Triangles 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. Holt McDougal Geometry 5-6 Inequalities in Two Triangles Holt McDougal Geometry 5-6 Inequalities in Two Triangles Indirect Proof #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. Holt McDougal Geometry 5-6 Inequalities in Two Triangles Check It Out! Example 1 Continued Step 3 Use direct reasoning to lead to a contradiction. m1 + m2 + m3 = 180° 90° + 90° + m3 = 180° 180° + m3 = 180° m3 = 0° However, by the Protractor Postulate, a triangle cannot have an angle with a measure of 0°. Holt McDougal Geometry 5-6 Inequalities in Two Triangles 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. Holt McDougal Geometry