5.6 Inequalities in Two Triangles

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1 5.6 Inequalities in Two Triangles and Indirect Proof Goal p Use inequalities to make comparisons in two triangles. Your Notes VOULRY Indirect Proof THEOREM 5.13: HINGE THEOREM If two sides of one triangle are congruent to two sides of another V 88 triangle, and the included angle of the first is larger than the included R angle of the second, then the third X side of the first is than the WX > third side of the second. THEOREM 5.14: ONVERSE OF THE HINGE THEOREM If two sides of one triangle are congruent to two sides of another triangle, and the third side of the first is longer than the third side of the F second, then the included m > m angle of the first is than the included angle of the second. W T S E opyright Holt Mcougal. ll rights reserved. Lesson 5.6 Geometry Notetaking Guide 139

2 5.6 Inequalities in Two Triangles and Indirect Proof Goal p Use inequalities to make comparisons in two triangles. Your Notes VOULRY Indirect Proof n indirect proof uses a temporary assumption that the desired conclusion is false. y then showing that this assumption leads to a logical impossibility, the original statement is proven true by contradiction. THEOREM 5.13: HINGE THEOREM If two sides of one triangle are congruent to two sides of another triangle, and the included angle of the first is larger than the included angle of the second, then the third side of the first is longer than the third side of the second. V 88 X W R 35 WX > ST T S THEOREM 5.14: ONVERSE OF THE HINGE THEOREM If two sides of one triangle are congruent to two sides of another triangle, and the third side of the first is longer than the third side of the F second, then the included m > m F angle of the first is larger than the included angle of the second E opyright Holt Mcougal. ll rights reserved. Lesson 5.6 Geometry Notetaking Guide 139

3 Your Notes Example 1 Use the onverse of the Hinge Theorem Given that } > }, how does 34 1 compare to 2? Solution 33 You are given that } > } and you know that } > } by the Reflexive Property. ecause 34 > 33, >. So, two sides of n are congruent to two sides of n and the third side in n is. y the onverse of the Hinge Theorem, m > m. 1 2 Example 2 Solve a multi-step problem Travel ar leaves a mall, heads due north for 5 mi and then turns due west for 3 mi. ar leaves the same mall, heads due south for 5 mi and then turns 808 toward east for 3 mi. Which car is farther from the mall? 908 raw a diagram. The distance 3 mi ar driven and the distance back to the mall form two triangles, 5 mi with 5 mile sides and 3 mile sides. Mall dd the third side to the diagram. Use linear pairs to find the 5 mi included angles of and. ecause 1008 > 908, ar is farther mi ar from the mall than ar by the. HOW TO WRITE N INIRET (Proof by ontradiction) Step 1 Identify the statement you want to prove. ssume temporarily that this statement is by assuming that the opposite is. Step 2 Reason logically until you reach a contradiction. Step 3 Point out that the desired conclusion must be because the contradiction proves the temporary assumption. 140 Lesson 5.6 Geometry Notetaking Guide opyright Holt Mcougal. ll rights reserved.

4 Your Notes Example 1 Use the onverse of the Hinge Theorem Given that } > }, how does 34 1 compare to 2? Solution 33 You are given that } > } and you know that } > } by the Reflexive Property. ecause 34 > 33, >. So, two sides of n are congruent to two sides of n and the third side in n is longer. y the onverse of the Hinge Theorem, m 1 > m Example 2 Solve a multi-step problem Travel ar leaves a mall, heads due north for 5 mi and then turns due west for 3 mi. ar leaves the same mall, heads due south for 5 mi and then turns 808 toward east for 3 mi. Which car is farther from the mall? 908 raw a diagram. The distance 3 mi ar driven and the distance back to the mall form two triangles, 5 mi with congruent 5 mile sides and congruent 3 mile sides. Mall dd the third side to the diagram. Use linear pairs to find the 5 mi included angles of 908 and ecause 1008 > 908, ar is farther mi ar from the mall than ar by the Hinge Theorem. HOW TO WRITE N INIRET (Proof by ontradiction) Step 1 Identify the statement you want to prove. ssume temporarily that this statement is false by assuming that the opposite is true. Step 2 Reason logically until you reach a contradiction. Step 3 Point out that the desired conclusion must be true because the contradiction proves the temporary assumption false. 140 Lesson 5.6 Geometry Notetaking Guide opyright Holt Mcougal. ll rights reserved.

5 Your Notes You have reached a contradiction when you have two statements that cannot both be true at the same time. Example 3 Write an indirect proof Write an indirect proof to show that an odd number is not divisible by 6. Given x is an odd number. Prove x is not divisible by 6. Solution Step 1 ssume temporarily that. This means that 5 n for some whole number n. So, multiplying both sides by 6 gives 5. Step 2 If x is odd, then, by definition, x cannot be divided evenly by. However, 5 so 5 5. We know that is a whole number because n is a whole number, so x can be divided evenly by. This contradicts the given statement that. Step 3 Therefore, the assumption that x is divisible by 6 is, which proves that. heckpoint omplete the following exercises. 1. If m > m which is longer, } or }? 2. In Example 2, car leaves the mall and goes 5 miles due west, then turns 858 toward south for 3 miles. Write the cars in order from the car closest to the mall to the car farthest from the mall. Homework 3. Suppose you wanted to prove the statement If x 1 y Þ 5 and y 5 2, then x Þ 3. What temporary assumption could you make to prove the conclusion indirectly? opyright Holt Mcougal. ll rights reserved. Lesson 5.6 Geometry Notetaking Guide 141

6 Your Notes You have reached a contradiction when you have two statements that cannot both be true at the same time. Example 3 Write an indirect proof to show that an odd number is not divisible by 6. Given x is an odd number. Prove x is not divisible by 6. Solution Step 1 ssume temporarily that x is divisible by 6. This means that } x 5 n for some whole number n. 6 So, multiplying both sides by 6 gives x 5 6n. Step 2 If x is odd, then, by definition, x cannot be divided evenly by 2. However, x 5 6n so x } 5 3n. We know that 3n is a 2 5 } 6n 2 Write an indirect proof whole number because n is a whole number, so x can be divided evenly by 2. This contradicts the given statement that x is odd. Step 3 Therefore, the assumption that x is divisible by 6 is false, which proves that x is not divisible by 6. heckpoint omplete the following exercises. 1. If m > m which is longer, } or }? } is longer. Homework 2. In Example 2, car leaves the mall and goes 5 miles due west, then turns 858 toward south for 3 miles. Write the cars in order from the car closest to the mall to the car farthest from the mall. car, car, car 3. Suppose you wanted to prove the statement If x 1 y Þ 5 and y 5 2, then x Þ 3. What temporary assumption could you make to prove the conclusion indirectly? You can temporarily assume that x 5 3. opyright Holt Mcougal. ll rights reserved. Lesson 5.6 Geometry Notetaking Guide 141

7 Words to Review Give an example of the vocabulary word. Midsegment of a triangle oordinate proof Perpendicular bisector Equidistant oncurrent Point of concurrency ircumcenter Incenter 142 Words to Review Geometry Notetaking Guide opyright Holt Mcougal. ll rights reserved.

8 Words to Review Give an example of the vocabulary word. Midsegment of a triangle midsegment oordinate proof type of proof that involves placing geometric figures in a coordinate plane. Perpendicular bisector Equidistant } is a perpendicular bisector of }. oncurrent Point is equidistant from point and point. Point of concurrency The lines are concurrent. ircumcenter Incenter point of concurrency circumcenter incenter 142 Words to Review Geometry Notetaking Guide opyright Holt Mcougal. ll rights reserved.

9 Median of a triangle entroid ltitude of a triangle Orthocenter Indirect proof Review your notes and hapter 5 by using the hapter Review on pages of your textbook. opyright Holt Mcougal. ll rights reserved. Words to Review Geometry Notetaking Guide 143

10 Median of a triangle entroid median centroid W U X T Z Y V ltitude of a triangle Orthocenter altitude orthocenter P Indirect proof To use an indirect proof, assume that the original statement is false and that the opposite is true. If this assumption leads to a contradiction, then the assumption must be false and the original statement must be true. Review your notes and hapter 5 by using the hapter Review on pages of your textbook. opyright Holt Mcougal. ll rights reserved. Words to Review Geometry Notetaking Guide 143

5.3 Use Angle Bisectors of

5.3 Use Angle Bisectors of 5.3 Use ngle isectors of Triangles Goal p Use angle bisectors to find distance relationships. Your Notes VOURY Incenter THEOREM 5.5: NGE ISETOR THEOREM In geometry, distance means the shortest length between