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Chapter 2 Overview: Basic Concepts and Proofs

Chapter 2 Overview: Basic Concepts and Proofs. Theorems 4 – 18 & more d efinitions, too!. Page 104, Chapter Summary: Concepts and Procedures. After studying this CHAPTER , you should be able to. 2.1 Recognize the need for clarity and concision in proofs.

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Chapter 2 Overview: Basic Concepts and Proofs

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  1. Chapter 2 Overview: Basic Concepts and Proofs Theorems 4 – 18 & more definitions, too!

  2. Page 104, Chapter Summary: Concepts and Procedures After studying this CHAPTER, you should be able to . . . 2.1 Recognize the need for clarity and concision in proofs 2.1 Understand the concept of perpendicularity 2.2 Recognize complementary and supplementary angles 2.3 Follow a five-step procedure to draw logical conclusions 2.4 Prove angles congruent by means of four new theorems 2.5 Apply the addition properties of segments and angles 2.5 Apply the subtraction properties of segments and angles 2.6 Apply the multiplication and division properties of segments and angles 2.7 Apply the transitive properties of angles and segments 2.7 Apply the Substitution Property 2.8 Recognize opposite rays 2.8 Recognize vertical angles

  3. Chapter 2, Section 1: “Perpendicularity” After studying this SECTION, you should be able to . . . 2.1 Recognize the need for clarity and concision in proofs 2.1 Understand the concept of perpendicularity Related Vocabulary COORDINATES OBLIQUE LINES ORIGIN X-axis PERPENDICULAR Y-axis

  4. Chapter 2, Section 1: “Perpendicularity” After studying this SECTION, you should be able to . . . 2.1 Recognize the need for clarity and concision in proofs 2.1 Understand the concept of perpendicularity Related Vocabulary PERPENDICULAR – lines, rays, or segments that INTERSECT at right angles OBLIQUE LINES– when lines, rays, or segments INTERSECT and are NOTPERPENDICULAR DEFINITIONS

  5. Chapter 2, Section 1: “Perpendicularity” After studying this SECTION, you should be able to . . . 2.1 Recognize the need for clarity and concision in proofs 2.1 Understand the concept of perpendicularity CHAIN REASONING Related Vocabulary SYMBOLS: If OH  OK , Given: OH  OK  H PERPENDICULAR RIGHT ANGLE then ∡HOK is a Rt ∡ ⊬ NOT PERPENDICULAR and if ∡HOK is a Rt ∡, then m∡HOK = 90 O K CONDITIONAL If a right angle is created at the intersection of two rays, then the rays are perpendicular! CONVERSE If two rays are perpendicular, then they create a right angle!

  6. Chapter 2, Section 1: “Perpendicularity” After studying this SECTION, you should be able to . . . 2.1 Recognize the need for clarity and concision in proofs 2.1 Understand the concept of perpendicularity CHAIN REASONING Related Vocabulary Given: m∡ HOK = 90 then OH  OK  Right Angles If m∡HOK = 90 H then ∡HOK is a Rt ∡  90⁰ and if ∡HOK is a Rt ∡, 90⁰ O K CONDITIONAL If a right angle is created at the intersection of two rays, then the rays are perpendicular! CONVERSE If two rays are perpendicular, then they create a right angle!

  7. Perpendicularity, right angles, and 90⁰ measurements all go together! Chapter 2, Section 1: “Perpendicularity” After studying this SECTION, you should be able to . . . 2.1 Recognize the need for clarity and concision in proofs 2.1 Understand the concept of perpendicularity 90⁰  H Right ∡ Right ∡ 90⁰  90⁰ O K CONDITIONAL If a right angle is created at the intersection of two rays, then the rays are perpendicular! CONVERSE If two rays are perpendicular, then they create a right angle!

  8. Chapter 2, Section 1: “Perpendicularity” After studying this SECTION, you should be able to . . . 2.1 Recognize the need for clarity and concision in proofs 2.1 Understand the concept of perpendicularity Related Vocabulary y-axis ORIGIN H H (0, 3) COORDINATES COORDINATES COORDINATES COORDINATES Could any lines drawn be “oblique lines”? G (-4, 0) G F C (-3, -2) B (-3, 2) A (3, 2) D (3, -2) F (4, 0) x-axis E (0, 0) Can you name the  lines? Can you name the ‖ lines? J J (0, -3) ‖  parallel Remember: The x-axis is  to the y-axis

  9. 2.1 Example Remember an important property of rectangles is that BOTH pairs of opposite sides are congruent, and: If two segments are congruent, then they have the SAME measure! Find the area of rectangle PACE Given: AP ‖ to the y-axis CE ‖ to the y-axis AreaRECT = (length)(width) AreaRECT = (7 units)(width) (4 units) AreaRECT = 28 units2 7 Width = |y – y| Length = |x – x| Length = |3 – (-4)| Width = |2 – (-2)| A C Length = |3 + 4| Width = |2 + 2| Length = |7| Width = |4| 4 P E

  10. Chapter 2, Section 2: “Complementary and Supplementary Angles” After studying this SECTION, you should be able to . . . 2.2 Recognize complementary and supplementary angles Related Vocabulary COMPLEMENT (NOT the same as: “You look very nice today!”) COMPLEMENTARY ANGLES SUPPLEMENT (NOT THE SAME AS: “Did you take your vitamins today!”) SUPPLEMENTARY ANGLES

  11. QUESTION! If two angles are COMPLEMENTARY ANGLES, (then) are they also ADJACENT ANGLES? Chapter 2, Section 2: “Complementary and Supplementary Angles” After studying this SECTION, you should be able to . . . 2.2 Recognize complementary and supplementary angles Related Vocabulary COMPLEMENT - the NAME given to each of the two angles whose sum equals 90⁰ COMPLEMENTARY ANGLES - two angles whose sum equals a 90⁰ right angle V 15⁰ 30⁰ N V 32⁰18’40” 57⁰41’20” 60⁰ 75⁰ V A N A N

  12. QUESTION! If two angles are SUPPLEMENTARY ANGLES, (then) are they also ADJACENT ANGLES? Chapter 2, Section 2: “Complementary and Supplementary Angles” After studying this SECTION, you should be able to . . . 2.2 Recognize complementary and supplementary angles Related Vocabulary SUPPLEMENT - the NAME given to each of the two angles whose sum equals 180⁰ SUPPLEMENTARY ANGLES - two angles whose sum equals a 180⁰ straight angle 85⁰ R A 130⁰ R 112⁰18’40” 67⁰41’20” 50⁰ 95⁰ T P R A T T

  13. Chapter 2, Section 2: “Complementary and Supplementary Angles” Is the answer reasonable? Is one of the angles 15 more than twice the other? THINK – If two angles are complementary angles, then their sum equals 90! After studying this SECTION, you should be able to . . . 2.2 Recognize complementary and supplementary angles Related Vocabulary The measure of one of two complementary angles is 15 more than twice the other. Find the measure of each angle. x + 2x + 15 = 90  Write equation 3x + 15 = 90  Simplify 3x = 75  Solve for x 50 + 15 75⁰ 2x + 15 x = 25  Substitute YES! 25⁰ x

  14. If a problem contains ONLY complements or ONLY supplements, use the previous method. Begin by drawing a right angle for two complementary angles or a straight angle to model two supplementary angles, and label them according to the information given in the problem! HOWEVER, if a problem refers to BOTH the complement AND the supplement in the same problem , use the NEXT method:

  15. Chapter 2, Section 2: “Complementary and Supplementary Angles” After studying this SECTION, you should be able to . . . 2.2 Recognize complementary and supplementary angles Use the “Boxer” Method to write expressions for each type of angle: Are you wondering, “what is the “Boxer Method”?” Well, first make a “BOX,” and then let “the angle” equal x x⁰ 30⁰ THE ANGLE Complements 60⁰ 60⁰ x⁰ COMPLEMENT (90 – x)⁰ 30⁰ Supplements (180 – x)⁰ 150⁰ SUPPLEMENT x⁰ 150⁰ 30⁰

  16. Chapter 2, Section 2: “Complementary and Supplementary Angles” Example 2.2 Recognize complementary and supplementary angles The measure of the supplement of an angle is 60 less than 3 times the complement of the angle. √ Find the measure of the complement. The measure of the supplement of an angle is 60 less than 3 times the complement (180 – x) = 3(90 – x) - 60 x 180 – x = 270 -3x -60 “the angle” x ANGLE 15 15⁰ 180 + 2x = 210 90 – x 75⁰ 75⁰ COMP 2x = 30 90 – 15 90 – x 180 – x x = 15 x 165⁰ SUPP Complement 15 x Supplement 180 – x 180 –15 15

  17. Chapter 2, Section 3: “Drawing Conclusions” After studying this SECTION, you should be able to . . . 2.3 Follow a five-step procedure to draw logical conclusions Related Vocabulary No NEW vocabulary!

  18. Chapter 2, Section 3: “Drawing Conclusions” After studying this SECTION, you should be able to . . . 2.3 Follow a five-step procedure to draw logical conclusions See very important TABLE on page 72! 5-STEP Procedure for Drawing Conclusions: • 1. MEMORIZE theorems, definitions, and postulates • 2. Look for KEY WORDS and SYMBOLS in the “givens” • 3. Think of all the theorems, definitions, and postulates that involve those keys. • 4. Decide which theorem, definition, or postulate allows you to draw a conclusion • 5. DRAW A CONCLUSION, and give a reason to justify it. NOTE: The “If . . .” part of the reason should match the GIVEN information! AND the “then . . .” part matches the CONCLUSION being justified! CAUTION! Be sure not to reverse that order!!!

  19. Chapter 2, Section 3: “Drawing Conclusions” After studying this SECTION, you should be able to . . . PRACTICE EXAMPLES 1) If B bisects AC, then ____?______ then . . . BC and BD trisect ∡ABE then . . . AB ≅ BC A C B Key info: a point, bisect, and seg B 2) If AB  AC, then _____?_______. then ∡BAC is a Rt ∡ A C Key info: ,, and  A C • If ∡ABC ≅ ∡CBD ≅∡DBE, • then ____?____. D Key info: ∡ ≅ ∡ ≅ ∡ B E

  20. Chapter 2, Section 3: “Drawing Conclusions” After studying this SECTION, you should be able to . . . JUSTIFY your CONCLUSIONS! 1) If B bisects AC, then ____?______ then . . . AB ≅ BC A C B REASON: If a seg is bisected by a point, then the seg is divided into two congruent segs then . . . BC and BD trisect ∡ABE B 2) If AB  AC, then _____?_______. then ∡BAC is a Rt ∡ A C REASON:If two rays are perpendicular, then they form a right angle A C • If ∡ABC ≅ ∡CBD ≅∡DBE, • then ____?____. D REASON:If an angle has been divided into 3 congruent angles, then it was trisected by two rays. B E

  21. Chapter 2, Section 4: “Congruent Supplements and Complements” After studying this SECTION, you should be able to . . . 2.4 Prove angles congruent by means of four new theorems Related Vocabulary No NEW vocabulary! BUT . . . THEOREM #7 THEOREM #6 THEOREM #5 THEOREM #4

  22. Chapter 2, Section 4: “Congruent Supplements and Complements” After studying this SECTION, you should be able to . . . 2.4 Prove angles congruent by means of four new theorems THEOREM #4 If angles are supplementary to the same angle, then they are congruent ∡1 is supplementary to ∡G ∡2 is also supplementary to ∡G What can we conclude about ∡1 and ∡2? = 60⁰ 120⁰ 2 1 60⁰ G

  23. Chapter 2, Section 4: “Congruent Supplements and Complements” After studying this SECTION, you should be able to . . . 2.4 Prove angles congruent by means of four new theorems THEOREM #5 If angles are supplementary to congruent angles, then they are congruent ∡G is supplementary to ∡E ∡O is supplementary to ∡M ∡E ≅ ∡O M 130⁰ 130⁰ 50⁰ 50⁰ O G E What can we conclude about ∡G and ∡M?

  24. Chapter 2, Section 4: “Congruent Supplements and Complements” After studying this SECTION, you should be able to . . . 2.4 Prove angles congruent by means of four new theorems THEOREM #6 If angles are complementary to the same angle, then they are congruent What can we conclude? THEOREM #7 If angles are complementary to congruent angles, then they are congruent What can we conclude? The only difference is the sum! (90 versus 180)

  25. Chapter 2, Section 4: “Congruent Supplements and Complements” After studying this SECTION, you should be able to . . . S Complete a Proof! Given: ? 3 1 ∡1 is comp to ∡4 R T 4 2 ∡2 is comp to ∡3 ? 3) RT bisects ∡SRV RT bisects ∡SRV 6) TR bisects ∡STV TR bisects ∡STV PROVE: V Statements Reasons 1) ∡1 is comp to ∡4 1) Given 2) Given 2) ∡2 is comp to ∡3 3) Given 4) If a ray bis an ∡, it div it into 2 ≅ ∡s 4) ∡3 ≅ ∡4 5) If ∡’s comp ≅ ∡s, then they are ≅ 5) ∡1 ≅ ∡2 6) If an ∡ is div into 2 ≅ ∡s, then it was bisected by a ray!

  26. Chapter 2, Section 5: “Addition and Subtraction Properties” After studying this SECTION, you should be able to . . . 2.5 Apply the addition properties of segments and angles 2.5 Apply the subtraction properties of segments and angles If a segment is added to two congruent segments, the sums are congruent. (Addition Property) • Note that we first need to know that two segments are congruent, and then that we are adding • the SAME segment to both of them. Related Vocabulary AC = BD, because (7) + (3)= (3)+ (7) (Commutative Property of Addition!) AB + BC = BC + CD,  If two segments have the same measure, they are congruent!

  27. Chapter 2, Section 5: “Addition and Subtraction Properties” After studying this SECTION, you should be able to . . . 2.5 Apply the addition properties of segments and angles 2.5 Apply the subtraction properties of segments and angles If an angle is added to two congruent angles, then the sums are congruent. (Addition Property) • Note that we first need to know that two angles are congruent, and then that we are adding • the SAME angle to both of them. Related Vocabulary m∡ABC = 50⁰ (Commutative Property of Addition!) 50 + ∡CBD = ∡CBD + 50 m∡ABC +m∡CBD=m∡CBD+ m∡DBE m∡ABD = m∡CBE, so m∡DBE = 50⁰ If two angles have the same measure, they are congruent!

  28. Ð @ Ð ABC DBE Chapter 2, Section 5: “Addition and Subtraction Properties” After studying this SECTION, you should be able to . . . 2.5 Apply the addition properties of segments and angles 2.5 Apply the subtraction properties of segments and angles If an angle is subtracted from two congruent angles, the differences are congruent. (Subtraction Property) • Note that we first need to know that two angles are congruent, and then that we are subtracting • the SAME angle from both of them. Related Vocabulary m∡ABD = 80⁰ 80 - ∡CBD 80 - ∡CBD = m∡ABD -m∡CBD= m∡CBE -m∡CBD m∡ABC = m∡DBE, so m∡CBE = 80⁰ If two angles have the same measure, they are congruent!

  29. Chapter 2, Section 5: “Addition and Subtraction Properties” After studying this SECTION, you should be able to . . . 2.5 Apply the addition properties of segments and angles 2.5 Apply the subtraction properties of segments and angles CF +FG=DE+ EH CG = DH, so CG ≅ DH If congruent segments are added to congruent segments, the sums are congruent. (Addition Property) Note that first we need 2 congruent segments, then we need 2 different congruent segments to ADD.

  30. Chapter 2, Section 5: “Addition and Subtraction Properties” After studying this SECTION, you should be able to . . . 2.5 Apply the addition properties of segments and angles 2.5 Apply the subtraction properties of segments and angles If congruent angles are added to congruent angles, the sums are congruent. (Addition Property) • Note that first we need 2 congruent angles, then we need to add two different congruent angles m∡JIL+m∡LIK=m∡LKI+ m∡JKL

  31. Chapter 2, Section 5: “Addition and Subtraction Properties” After studying this SECTION, you should be able to . . . 2.5 Apply the addition properties of segments and angles 2.5 Apply the subtraction properties of segments and angles QB ≅ RA If a segment (or angle) is subtracted from congruent segments (or angles), the differences are congruent. (Subtraction Property) Note that we need to start with congruent angles or segments and then subtract the same angle or segment from both.

  32. Chapter 2, Section 5: “Addition and Subtraction Properties” After studying this SECTION, you should be able to . . . 2.5 Apply the addition properties of segments and angles 2.5 Apply the subtraction properties of segments and angles If a segment (or angle) is subtracted from congruent segments (or angles), the differences are congruent. (Subtraction Property) Note that we need to start with congruent angles or segments and then subtract the same angle or segment from both.

  33. Chapter 2, Section 5: “Addition and Subtraction Properties” After studying this SECTION, you should be able to . . . 2.5 Apply the addition properties of segments and angles 2.5 Apply the subtraction properties of segments and angles ∡STW ≅ ∡UVW If congruent segments (or angles) are subtracted from congruent segments (or angles), the differences are congruent. (Subtraction Property) Note that we start with congruent segments or angles, and then subtract congruent segments or angles.

  34. Using the Addition and Subtraction Properties • An addition propertyis used when the segments or angles in the conclusion are greater than those in the given information • A subtraction propertyis used when the segments or angles in the conclusion are smaller than those in the given information.

  35. PQ @ RS PR QS @ Theorem: If a segment is added to two congruent segments, the sums are congruent. (Addition Property) Given: Conclusion: Statements Reasons 1. 1. Given 2. PQ = RS 2. If two segments are congruent, then they have the same measure 3. PQ + QR = RS + QR 3. Additive Property of Equality 4. PR = QS 4. Addition of Segments 5. 5. If two segments have the same measure then they are congruent

  36. How to use this theorem in a proof: Given: Conclusion: ? ? 1. Given 2. If a segment is subtracted from congruent segments, then the resulting segments are congruent. (Subtraction)

  37. Multiplication Property • If segments (or angles) are congruent, then their like multiples are congruent. Example: If B, C, F, and G are trisection points and then by the Multiplication Property. A B C D E F G H

  38. Division Property If segments (or angles) are congruent, then their like divisions are congruent. C D S Z A O T G If ∡CAT ≅ ∡DOG, and AS and OZ are angle bisectors then, ∡CAS ≅ ∡DOZ by the division property

  39. Using the Multiplication and Division Properties in Proofs • Look for the DOUBLE USE of the words midpoint, trisects, or bisects in the “Givens.” • Use MULTIPLICATION if what is Given is less than the Conclusion • Use DIVISION if what is Given is greater than the Conclusion

  40. Example Given: O is the midpoint of R is the midpoint of Prove: Statements Reasons M O P N R S 1. MP ≅ NS 1. Given 2. Given 2. O is mdpt of MP 3. A mdpt divides a seg into 2 ≅ segs 3. MO ≅ OP 4. Given 4. R is mdpt of NS 4. Same as #3 5. NR ≅ RS 5. If segs are ≅, then their like divisions are ≅ 5. MO ≅ NR (DIVISION PROPERTY)

  41. Chapter 2, Section 7: “Transitive and Substitution Properties” After studying this SECTION, you should be able to . . . 2.7 Apply the Transitive Property of angles and segments 2.7 Apply the Substitution Property Related Vocabulary Theorems Theorem 16 SUBSTITUTE Theorem 17 SUBSTITUTION

  42. Chapter 2, Section 7: “Transitive and Substitution Properties” After studying this SECTION, you should be able to . . . 2.7 Apply the Transitive Property of angles and segments 2.7 Apply the Substitution Property A B AB ≅ BC BC ≅ CD AB ≅ CD CONCLUSION? C D THEOREM: If segments are congruent to the SAME segment, then they are congruent to each other.

  43. Chapter 2, Section 7: “Transitive and Substitution Properties” After studying this SECTION, you should be able to . . . 2.7 Apply the Transitive Property of angles and segments 2.7 Apply the Substitution Property ∡1 ≅ ∡2 3 2 ∡2 ≅ ∡3 1 ∡1 ≅ ∡3 CONCLUSION? THEOREM: If angles are congruent to the SAME angle, then they are congruent to each other.

  44. Chapter 2, Section 7: “Transitive and Substitution Properties” After studying this SECTION, you should be able to . . . 2.7 Apply the Transitive Property of angles and segments 2.7 Apply the Substitution Property AB ≅ NM R A B Q QR ≅ MP NM ≅ MP N M P AB ≅ CD CONCLUSION? THEOREM: If segments are congruent to congruent segments, then they are congruent to each other.

  45. Chapter 2, Section 7: “Transitive and Substitution Properties” After studying this SECTION, you should be able to . . . 2.7 Apply the Transitive Property of angles and segments 2.7 Apply the Substitution Property ∡7 ≅ ∡5 6 5 ∡6 ≅ ∡8 7 ∡5 ≅ ∡6 ∡7 ≅ ∡8 CONCLUSION? 8 THEOREM: If angles are congruent to congruent angles, then they are congruent to each other.

  46. Chapter 2, Section 7: “Transitive and Substitution Properties” After studying this SECTION, you should be able to . . . 2.7 Apply the Transitive Property of angles and segments 2.7 Apply the Substitution Property Given: ∡1 comps ∡2 1 ∡2 ≅ ∡3 3 2 m∡1 + m∡2 = 90 m∡2 ≅ m∡3 ∴ m∡1 + m∡3 = 90 By Substitution Property!

  47. Chapter 2, Section 8: “Vertical Angles” After studying this SECTION, you should be able to . . . 2.8 Recognize opposite rays 2.8 Recognize Vertical Angles Related Vocabulary Opposite Rays - (definition) – collinear rays that share a common endpoint and extend in opposite directions Vertical Angles- (definition) – two angles whose sides are formed by opposite rays. THEOREM 18 Vertical angles are CONGURENT!

  48. Chapter 2, Section 8: “Vertical Angles” After studying this SECTION, you should be able to . . . 2.8 Recognize opposite rays 2.8 Recognize Vertical Angles Related Vocabulary Opposite Rays - (definition) – collinear rays that share a common endpoint and extend in opposite directions L Name the opposite rays: D 3) H 2) H 1) E K I B G A C F J EH IL ED BA BC EG EF IJ and and and and

  49. Chapter 2, Section 8: “Vertical Angles” After studying this SECTION, you should be able to . . . 2.8 Recognize opposite rays 2.8 Recognize Vertical Angles Related Vocabulary Vertical Angles- (definition) – two angles whose sides are formed by opposite rays. 4) Which numbered angle is vertical with ∡1? ∡3 115° 115° H D 5) Which numbered angle is vertical with ∡4? 2 ∡2 65° 65° F G 1 3 4 6) If m∡1 = 65, find the measure of the numbered angles.

  50. Chapter 2, Section 8: “Vertical Angles” After studying this SECTION, you should be able to . . . 2.8 Recognize opposite rays 2.8 Recognize Vertical Angles Related Vocabulary Vertical Angles- (definition) – two angles whose sides are formed by opposite rays. 7) If m∡3 = 55, which other numbered angle must be 55°? ∡6 40° 7) If m∡1 = 40, which other numbered angle must be 40°? 55° 2 1 3 4 ∡4 6 5

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