Essential Question How can you use a flowchart to prove a mathematical statement?

Transcription

1 .6 Proving Geometric Relationships OMMON OR Learning Standard HSG-O..9 MOLING WITH MTHMTIS To be proficient in math, you need to map relationships using such tools as diagrams, two-way tables, graphs, flowcharts, and formulas. ssential Question How can you use a flowchart to prove a mathematical statement? Matching Reasons in a Flowchart Proof Work with a partner. Match each reason with the correct step in the flowchart. = + Prove = = + + = + = + =. Segment ddition Postulate (Post..).. Transitive Property of quality. Subtraction Property of quality Matching Reasons in a Flowchart Proof Work with a partner. Match each reason with the correct step in the flowchart. m = m Prove m = m m = m m = m + m m = m + m m = m + m m + m = m m = m. ngle ddition Postulate (Post..4). Transitive Property of quality. Substitution Property of quality. ngle ddition Postulate (Post..4). F. ommutative Property of ddition ommunicate Your nswer. How can you use a flowchart to prove a mathematical statement? 4. ompare the flowchart proofs above with the two-column proofs in the Section.5 xplorations. xplain the advantages and disadvantages of each. Section.6 Proving Geometric Relationships 05

2 .6 Lesson What You Will Learn ore Vocabulary flowchart proof, or flow proof, p. 06 paragraph proof, p. 08 STUY TIP When you prove a theorem, write the hypothesis of the theorem as the statement. The conclusion is what you must Prove. Write flowchart proofs to prove geometric relationships. Write paragraph proofs to prove geometric relationships. Writing Flowchart Proofs nother proof format is a flowchart proof, or flow proof, which uses boxes and arrows to show the flow of a logical argument. ach reason is below the statement it justifies. flowchart proof of the Right ngles ongruence Theorem is shown in xample. This theorem is useful when writing proofs involving right angles. Theorem Theorem. Right ngles ongruence Theorem ll right angles are congruent. Proof xample, p. 06 Proving the Right ngles ongruence Theorem Use the given flowchart proof to write a two-column proof of the Right ngles ongruence Theorem. and are right angles. Prove Flowchart Proof and are right angles. m = 90, m = 90 m l = m l efinition of right angle Transitive Property of quality Two-olumn Proof. and are right angles.. efinition of congruent angles. m = 90, m = 90. efinition of right angle. m = m. Transitive Property of quality efinition of congruent angles Monitoring Progress. opy and complete the flowchart proof. Then write a two-column proof., Prove, Help in nglish and Spanish at igideasmath.com efinition of lines 06 hapter Reasoning and Proofs

3 Theorems Theorem.4 ongruent Supplements Theorem If two angles are supplementary to the same angle (or to congruent angles), then they are congruent. If and are supplementary and and are supplementary, then. Proof xample, p. 07 (case ); x. 0, p. (case ) Theorem.5 ongruent omplements Theorem If two angles are complementary to the same angle (or to congruent angles), then they are congruent. If 4 and 5 are complementary and 6 and 5 are complementary, then 4 6. Proof x. 9, p. (case ); x., p. (case ) To prove the ongruent Supplements Theorem, you must prove two cases: one with angles supplementary to the same angle and one with angles supplementary to congruent angles. The proof of the ongruent omplements Theorem also requires two cases. Proving a ase of ongruent Supplements Theorem Use the given two-column proof to write a flowchart proof that proves that two angles supplementary to the same angle are congruent. and are supplementary. and are supplementary. Prove Two-olumn Proof. and are supplementary. and are supplementary... m + m = 80, m + m = 80. efinition of supplementary angles. m + m = m + m. Transitive Property of quality 4. m = m 4. Subtraction Property of quality efinition of congruent angles Flowchart Proof and are supplementary. and are supplementary. m + m = 80 efinition of supplementary angles m + m = 80 efinition ii of supplementary angles m + m = m + m Transitive Property of quality m = m Subtraction Property of quality efinition i of congruent angles Section.6 Proving Geometric Relationships 07

4 Writing Paragraph Proofs nother proof format is a paragraph proof, which presents the statements and reasons of a proof as sentences in a paragraph. It uses words to explain the logical flow of the argument. Two intersecting lines form pairs of vertical angles and linear pairs. The Linear Pair Postulate formally states the relationship between linear pairs. You can use this postulate to prove the Vertical ngles ongruence Theorem. Postulate and Theorem Postulate.8 Linear Pair Postulate If two angles form a linear pair, then they are supplementary. and form a linear pair, so and are supplementary and m l + m = 80. Theorem.6 Vertical ngles ongruence Theorem Vertical angles are congruent. 4 Proof xample, p. 08, 4 Use the given paragraph proof to write a two-column proof of the Vertical ngles ongruence Theorem. Proving the Vertical ngles ongruence Theorem STUY TIP In paragraph proofs, transitional words such as so, then, and therefore help make the logic clear. JUSTIFYING STPS You can use information labeled in a diagram in your proof. 5 and 7 are vertical angles. Prove 5 7 Paragraph Proof 5 and 7 are vertical angles formed by intersecting lines. s shown in the diagram, 5 and 6 are a linear pair, and 6 and 7 are a linear pair. Then, by the Linear Pair Postulate, 5 and 6 are supplementary and 6 and 7 are supplementary. So, by the ongruent Supplements Theorem, 5 7. Two-olumn Proof. 5 and 7 are vertical angles... 5 and 6 are a linear pair. 6 and 7 are a linear pair.. 5 and 6 are supplementary. 6 and 7 are supplementary.. efinition of linear pair, as shown in the diagram. Linear Pair Postulate ongruent Supplements Theorem 08 hapter Reasoning and Proofs

5 Monitoring Progress Help in nglish and Spanish at igideasmath.com. opy and complete the two-column proof. Then write a flowchart proof. =, = Prove. =, =.. + = +. ddition Property of quality.. Substitution Property of quality 4. + =, + = Substitution Property of quality Rewrite the two-column proof in xample without using the ongruent Supplements Theorem. How many steps do you save by using the theorem? Using ngle Relationships Find the value of x. SOLUTION TPS and QPR are vertical angles. y the Vertical ngles ongruence Theorem, the angles are congruent. Use this fact to write and solve an equation. m TPS = m QPR efinition of congruent angles 48 = (x + ) Substitute angle measures. 47 = x Subtract from each side. 49 = x ivide each side by. T Q 48 P (x + ) S R So, the value of x is 49. Monitoring Progress Use the diagram and the given angle measure to find the other three angle measures. 4. m = 7 5. m = m 4 = 88 Help in nglish and Spanish at igideasmath.com 4 7. Find the value of w. (5w + ) 98 Section.6 Proving Geometric Relationships 09

6 Using the Vertical ngles ongruence Theorem Write a paragraph proof. 4 Prove 4 Paragraph Proof and 4 are congruent. y the Vertical ngles ongruence Theorem, and 4. y the Transitive Property of ngle ongruence (Theorem.), 4. Using the Transitive Property of ngle ongruence (Theorem.) once more,. Monitoring Progress Help in nglish and Spanish at igideasmath.com 8. Write a paragraph proof. is a right angle. Prove is a right angle. oncept Summary Types of Proofs Symmetric Property of ngle ongruence (Theorem.) Prove Two-olumn Proof... m = m. efinition of congruent angles. m = m. Symmetric Property of quality efinition of congruent angles Flowchart Proof m = m m = m efinition of congruent angles Symmetric Property of quality efinition of congruent angles Paragraph Proof is congruent to. y the definition of congruent angles, the measure of is equal to the measure of. The measure of is equal to the measure of by the Symmetric Property of quality. Then by the definition of congruent angles, is congruent to. 0 hapter Reasoning and Proofs

7 .6 xercises ynamic Solutions available at igideasmath.com Vocabulary and ore oncept heck. WRITING xplain why all right angles are congruent.. VOULRY What are the two types of angles that are formed by intersecting lines? Monitoring Progress and Modeling with Mathematics In xercises 6, identify the pair(s) of congruent angles in the figures. xplain how you know they are congruent. (See xamples,, and.) F G N M G 50 L H S P 50 J K M Q R K M L W H X Y 58 Z 6. is supplementary to. is supplementary to F. J In xercises 4, find the values of x and y. (See xample 4.). (8x + 7) 5y (9x 4). (0x 4) (8y 8) (7y 4) 6y 6(x + ). RROR NLYSIS In xercises 5 and 6, describe and correct the error in using the diagram to find the value of x. (x + 45) (6x + ) (x 40) (9x + ) 4x (7y ) (6y + 8) (6x 6) 4. (5x 5) (5y + 5) (7y 9) (6x + 50) F 5. (x + 45) + (9x + ) = 80 x + 48 = 80 x = In xercises 7 0, use the diagram and the given angle measure to find the other three measures. (See xample.) 7. m = 4 8. m = m = x = 4.5 (x + 45) + (x 40) = 90 5x + 5 = 90 5x = 85 x =.4 0. m 4 = 9 Section.6 Proving Geometric Relationships

8 7. PROOF opy and complete the flowchart proof. Then write a two-column proof. (See xample.) Prove 4 4 l, 4 4 Vertical ngles ongruence Theorem (Theorem.6) 8. PROOF opy and complete the two-column proof. Then write a flowchart proof. (See xample.) is a right angle. is a right angle. Prove. is a right angle. is a right angle... and are complementary.. efinition of complementary angles. and are complementary PROVING THORM opy and complete the paragraph proof for the ongruent omplements Theorem (Theorem.5). Then write a two-column proof. (See xample.) and are complementary. and are complementary. Prove and are complementary, and and are complementary. y the definition of angles, m + m = 90 and = 90. y the, m + m = m + m. y the Subtraction Property of quality,. So, by the definition of. hapter Reasoning and Proofs

9 0. PROVING THORM opy and complete the two-column proof for the ongruent Supplement Theorem (Theorem.4). Then write a paragraph proof. (See xample 5.) and are supplementary. and 4 are supplementary. 4 4 Prove. and are supplementary. and 4 are supplementary. 4. m + m = 80, m + m 4 = = m + m 4. Transitive Property of quality 4. m = m 4 4. efinition of congruent angles 5. m + m = 5. Substitution Property of quality 6. m = m PROOF In xercises 4, write a proof using any format.. QRS and PSR are supplementary. Prove QRL PSR. Prove Q P L R S K M N 4. JK JM, KL ML, J M, K L Prove JM ML and JK KL J K. and are complementary. and 4 are complementary. Prove 4 4 M 5. MKING N RGUMNT You overhear your friend discussing the diagram shown with a classmate. Your classmate claims 4 because they are vertical angles. Your friend claims they are not congruent because he can tell by looking at the diagram. Who is correct? Support your answer with definitions or theorems. 4 L Section.6 Proving Geometric Relationships

10 6. THOUGHT PROVOKING raw three lines all intersecting at the same point. xplain how you can give two of the angle measures so that you can find the remaining four angle measures. 8. WRITING How can you save time writing proofs? 9. MTHMTIL ONNTIONS Find the measure of each angle in the diagram. 7. RITIL THINKING Is the converse of the Linear Pair Postulate (Postulate.8) true? If so, write a biconditional statement. xplain your reasoning. 0y (y + ) (4x ) (7x + 4) 0. HOW O YOU S IT? Use the student s two-column proof. and are supplementary. Prove. l and are supplementary... m l = m. efinition of congruent angles. m l + m = 80. efinition of supplementary angles 4. m l + m = Substitution Property of quality 5. m = Simplify. 6. m = ivision Property of quality 7. m = Transitive Property of quality a. What is the student trying to prove? b. Your friend claims that the last line of the proof should be, because the measures of the angles are both 90. Is your friend correct? xplain. Maintaining Mathematical Proficiency Reviewing what you learned in previous grades and lessons Use the cube. (Section.). Name three collinear points. I. Name the intersection of plane F and plane HG.. Name two planes containing. 4. Name three planes containing point. 5. Name three points that are not collinear. F G 6. Name two planes containing point J. J H 4 hapter Reasoning and Proofs