Chapter 2 Overview Basic Concepts and Proofs Theorems

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

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 PERPENDICULA R Y-axis 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 PERPENDICULAR – lines, rays, or segments that INTERSECT at right angles OBLIQUE LINES – when DEFINITION S lines, rays, or segments INTERSECT and are NOT PERPENDICULAR 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 CHAIN REASONING Related Vocabulary RIGHT ANGLE NOT PERPENDICULAR H ⊬ SYMBOLS : PERPENDICULAR O Given: OH OK If OH OK , then ∡HOK is a Rt ∡ and if ∡HOK is a Rt ∡, then m∡HOK = 90 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! 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 Right Angles H 90⁰ O Given: m∡ HOK = 90 If m∡HOK = 90 then ∡HOK is a Rt ∡ and if ∡HOK is a Rt ∡, then OH OK 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 90⁰ Right ∡ Perpendicularity, right angles, and H 90⁰ measurements all go together! Right ∡ 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

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 HH (0, 3) COORDINATES G (-4, G 0) x-axis C (-3, -2) 2) (3, -2) 2) DBA(-3, (3, E (0, 0) Could any lines drawn be “oblique lines”? F F (4, 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 8

2. 1 Example Find the area of rectangle PACE Given: AP ‖ to the y-axis CE ‖ to the y-axis Remember an important property of rectangles is that pairs of Area. BOTH = (length)(width) opposite sides are congruent, and: RECT Area. RECT = (7 units)(width) (4 units) If two segments are congruent, Areathe=SAME 28 units 2 then they have 7 measure! RECT Width = |y – y| Length = |x – x| Width = |2 – (-2)| Length = |3 – (-4)| A Width = |2 + 2| C Length = |3 + 4| Length = |7| Width = |4| 4 P E 9

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 10

Chapter 2, Section 2: “Complementary and Supplementary Angles” QUESTION! After studying this SECTION, you should be able to. . . 2. 2 Recognize complementary and supplementary angles If two angles. Related are. Vocabulary COMPLEMENTARY ANGLES, COMPLEMENT (then) are they also ADJACENT ANGLES? - 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 V 75⁰ ’ 40 30⁰ ⁰ 1 8 V 32 N A ” 0 2 1’ ⁰ 4 7 5 60⁰ N A N 11

Chapter 2, Section 2: “Complementary and Supplementary Angles” QUESTION! After studying this SECTION, you should be able to. . . 2. 2 Recognize complementary and supplementary angles If two angles are SUPPLEMENTARY ANGLES, SUPPLEMENT (then) are they also ADJACENT ANGLES? Related Vocabulary - 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 130⁰ R 112⁰ 18’ 40” 95⁰ T A 67⁰ 41’ 20” 50⁰ T R A T P 12

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

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: 14

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 THE ANGLE COMPLEMENT x⁰ (90 – x)⁰ 30⁰ 60⁰ Complements 60⁰ 30⁰ x⁰ Supplements SUPPLEMENT (180 – x)⁰ 150⁰ 30⁰ x⁰ 15

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) ANGLE COMP SUPP x 90 – x 180 – x = 15⁰ 75⁰ 165⁰ 3(90 – x) - 60 180 – x = 270 -3 x -60 x “the angle” 15 180 + 2 x = 210 2 x = 30 90 – 15 x x 15 180 –– 15 x 180 Complement x 15 x = 15 Supplement 16

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! 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 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 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!!! 18

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

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

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 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 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? 120⁰ 160⁰ = 60⁰ 2 G 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 #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 50⁰ G E 50⁰ O What can we conclude about ∡G and ∡M? 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 #6 If angles are complementary to the same angle, then they are What can wecongruent conclude? THEOREM #7 If angles are complementary to congruent angles, then they What canare we congruent conclude? The only difference is the sum! (90 versus 180) 24

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

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 If subtraction a segment is added to twoand congruent 2. 5 Apply the properties of segments angles segments, the sums are congruent. Related Vocabulary (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. AC = BD, because (Commutative Property of Addition!) (7) + (3) = (3) + (7) AB + BC = BC + CD, If two segments have the same measure, they are congruent! 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 50 B C (Commutative Property of Addition!) = then the. Related sums. Vocabulary are congruent. (Addition Property) ⁰ If the ansubtraction angle is properties added to congruent 2. 5 Apply of two segments and angles, Note that we first need to know that two angles are and+then 50 + ∡CBD congruent, 50 that we are adding = ∡CBD m∡ABC + m∡CBD + m∡DBE =them∡CBD SAME angle to both of them. m ∡A m∡ABD = m∡CBE, so If two angles have the same measure, they are congruent! m∡DBE = 50⁰ 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 If an angle is subtracted from two congruent Apply the subtraction properties of segments and angles, the differences are congruent. Related Vocabulary (Subtraction Property) m∡ABD = 80⁰ Note that we first need to know that two angles are that we are subtracting 80 and then - ∡CBD 80 - ∡CBDcongruent, = the from both of them. m∡ABD - m∡CBD = SAME m∡CBEangle - m∡CBD m∡ABC = m∡DBE, so m∡CBE = 80⁰ Ð ABC @ Ð DBE If two angles have the same measure, they are congruent! 28

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 segments are added to congruent the sums are congruent. CF + FG = segments, DE + EH (Addition Property) CG DH, Note=that firstso we need 2 congruent segments, then we need 2 different congruent segments CG ≅ DH to ADD.

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 m∡JIL + m∡LIK = m∡LKI + m∡JKL 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 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 a segment (or angle) is subtracted from congruent segments (or angles), the differences are congruent. (Subtraction Property) QB ≅ RA Note that we need to start with congruent angles or segments and then subtract the same angle or segment from both.

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.

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 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. ∡STW ≅ ∡UVW

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

Theorem: If a segment is added to two congruent segments, the sums are congruent. (Addition Property) Given: Conclusion: Statements Reasons 1. Given PQ @ RS 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. If two segments have the same measure then they are congruent PR @ QS

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

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

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

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

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

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 SUBSTITUTE SUBSTITUTION Theorems Theorem 16 Theorem 17 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 A AB ≅ BC B BC ≅ CD CONCLUSION? AB ≅ CD C D THEOREM: If segments are congruent to the SAME segment, then they are congruent to each other. 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 ∡ 1 ≅ ∡ 2 2 ∡ 2 ≅ ∡ 3 CONCLUSION? 3 1 ∡ 1 ≅ ∡ 3 THEOREM: If angles are congruent to the SAME angle, 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 AB ≅ NM A QR ≅ MP NM ≅ MP CONCLUSION? AB ≅ CD B N R Q M P THEOREM: If segments are congruent to congruent segments, 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 ∡ 7 ≅ ∡ 5 ∡ 6 ≅ ∡ 8 7 5 6 ∡ 5 ≅ ∡ 6 CONCLUSION? ∡ 7 ≅ ∡ 8 8 THEOREM: If angles are congruent to congruent angles, 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 Given: ∡ 1 comps ∡ 2 ≅ ∡ 3 1 2 3 m∡ 1 + m∡ 2 = 90 m∡ 2 ≅ m∡ 3 ∴ m∡ 1 + m∡ 3 = 90 By Substitution Property! 46

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! 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 Name the opposite rays: 3) 2) D 1) A B BA and 48 BC H H C G L K I E J F ED and EF EH and EG IL and IJ

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? 5) Which numbered angle is vertical with ∡ 4? ∡ 3 ∡ 2 6) If m∡ 1 = 65, find the measure of the numbered angles. 49 115° D H 2 65° 3 1 65° 4 G F

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 7) If m∡ 1 = 40, which other numbered angle must be 40°? ∡ 4 50 40° 1 6 2 5 3 4 55°

Self-Check Properties Quiz Questions 1. Given: A E and D are the midpoints of AC and AB, E and AC ≅ AB D F Conclusion: CE ≅ DB C Because? G H B Addition Subtraction Multiplication Division Like divisions of ≅ segs are ≅ Transitive Substitution 51

Self-Check Properties Quiz Questions 2. Given: A FE ≅ FD, and FC ≅ FB E D Conclusion: CD ≅ EB F C Because? G H B ≅SEGS added to ≅SEGS are ≅ Addition Subtraction SEGS Multiplication Division Transitive Substitution 52

Self-Check Properties Quiz Questions 3. Given: A CD bisects ∡ACB, BE bisects ∡ABC, E and ∡ACB ≅ ABC D F C Conclusion: ∡ACD ≅ ∡ABE Because? G H B Addition Subtraction Multiplication Like divisions of ≅ ∡s are ≅ Division Transitive Substitution 53

Self-Check Properties Quiz Questions 4. Given: A CG ≅ GH, BH ≅ GH, E D F C Conclusion: CG ≅ BH Because? G H B Addition Subtraction Multiplication Division If segs are ≅ to SAME seg, Transitive then ≅ to each other Substitution 54

Self-Check Properties Quiz Questions 5. Given: A ∡BCD ≅ ∡CBE , and ∡ACB ≅ ABC E D F C Conclusion: ∡ACD ≅ ∡ABE Because? G H B Addition Subtraction If ≅ ∡s are subtracted from ≅ ∡s, then the like diffs are ≅ Multiplication Division Transitive Substitution 55

Self-Check Properties Quiz Questions 6. Given: A EF = FD, and EF + FB = EB E D Conclusion: FD + FB = EB F = Because? ! C G H B Addition Subtraction Multiplication Division One seg measure can be Transitive Substitution substituted for the other in the EQUATION! 56

Self-Check Properties Quiz Questions 7. Given: A CH ≅ BG E D Conclusion: CG ≅ BH F C Because? G H B Addition Subtraction If the SAME seg is subtracted from ≅SEGS , the like diffs are ≅ Multiplication Division Transitive Substitution 57

Self-Check Properties Quiz Questions 8. Given: A ∡CAB + ∡ACB + ∡ABC = 180° and ∡ACB ≅ ABC E D F C G H B Conclusion: 2(∡ABC) + ∡CAB = 180° Because? Addition Subtraction Multiplication Division One ANGLE measure can be Transitive Substitution substituted for the other in the EQUATION! 58

Self-Check Properties Quiz Questions 9. Given: A CD bisects ∡ACB, BE bisects ∡ABC, E and ∡ACD ≅ ABE D F C Conclusion: ∡ACB ≅ ∡ABC Because? Addition Subtraction Multiplication Division Transitive Substitution G H B Like multiples of ≅ ∡s are ≅ 59

Self-Check Properties Quiz Questions 10. Given: A ∡AFD ≅ ∡AFE , and ∡DFB ≅ EFC E D F C Conclusion: ∡AFC ≅ ∡AFB Because? Addition Subtraction Multiplication Division Transitive Substitution G H B If ≅ ∡s are added to ≅ ∡s, then the like sums are ≅ 60