Theorems 4 – 18 & more definitions, too!

Page 104, Chapter Summary: Concepts and Procedures After studying this CHAPTER, you should be able to 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” COORDINATES ORIGIN PERPENDICULAR X-axis Y-axis After studying this SECTION, you should be able to... Related Vocabulary OBLIQUE LINES Recognize the need for clarity and concision in proofs 2.1 Understand the concept of perpendicularity

After studying this SECTION, you should be able to... Related Vocabulary Chapter 2, Section 1: “Perpendicularity” Recognize the need for clarity and concision in proofs 2.1 Understand the concept of perpendicularity DEFINITIONS PERPENDICULAR – lines, rays, or segments that INTERSECT at right angles OBLIQUE LINES – when lines, rays, or segments INTERSECT and are NOT PERPENDICULAR

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

After studying this SECTION, you should be able to... Related Vocabulary Chapter 2, Section 1: “Perpendicularity” Recognize the need for clarity and concision in proofs 2.1 Understand the concept of perpendicularity If a right angle is created at the intersection of two rays, then the rays are perpendicular! CONDITIONAL If two rays are perpendicular, then they create a right angle! CONVERSE H K O Given: m ∡ HOK = 90 then OH  OK then ∡ HOK is a Rt ∡ and if ∡ HOK is a Rt ∡, Ifm ∡ HOK = 90 CHAIN REASONING 90 ⁰  Right Angles  90 ⁰

After studying this SECTION, you should be able to... Chapter 2, Section 1: “Perpendicularity” Recognize the need for clarity and concision in proofs 2.1 Understand the concept of perpendicularity If a right angle is created at the intersection of two rays, then the rays are perpendicular! CONDITIONAL If two rays are perpendicular, then they create a right angle! CONVERSE H K O 90 ⁰  Right ∡  Perpendicularity, right angles, and 90⁰ measurements all go together!

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

Find the area of rectangle PACE 9 Given: AP ‖ to the y-axis CE ‖ to the y-axis C P A E 2.1 Example 4 7 Area RECT = (length)(width) Width = |y – y| Width = |2 – (-2)| Width = |2 + 2| Width = |4| Length = |x – x| Length = |3 – (-4)| Length = |3 + 4| Length = |7| Area RECT = (7 units)(width) Area RECT = 28 units 2 (4 units) 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!

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

After studying this SECTION, you should be able to... Related Vocabulary Chapter 2, Section 2: “Complementary and Supplementary Angles” Recognize complementary and supplementary angles COMPLEMENT COMPLEMENTARY ANGLES - the NAME given to each of the two angles whose sum equals 90 ⁰ - two angles whose sum equals a 90 ⁰ right angle 15 ⁰ 75 ⁰ 30 ⁰ 60 ⁰ 57 ⁰ 41’20” 32 ⁰ 18’40” V NV NAA V N QUESTION! If two angles are COMPLEMENTARY ANGLES, (then) are they also ADJACENT ANGLES ?

After studying this SECTION, you should be able to... Related Vocabulary Chapter 2, Section 2: “Complementary and Supplementary Angles” Recognize complementary and supplementary angles SUPPLEMENT SUPPLEMENTARY ANGLES - the NAME given to each of the two angles whose sum equals 180 ⁰ - two angles whose sum equals a 180 ⁰ straight angle 85 ⁰ 95 ⁰ 130 ⁰ 50 ⁰ 67 ⁰ 41’20”112 ⁰ 18’40” T R T AR A T R P QUESTION! If two angles are SUPPLEMENTARY ANGLES, (then) are they also ADJACENT ANGLES ?

After studying this SECTION, you should be able to... Related Vocabulary Chapter 2, Section 2: “Complementary and Supplementary Angles” x 2x + 15 x + 2x + 15 = Recognize complementary and supplementary angles The measure of one of two complementary angles is 15 more than twice the other. Find the measure of each angle.  Write equation THINK – If two angles are complementary angles, then their sum equals 90!  Simplify 3x + 15 = 90  Solve for x 3x = 75 x = 25  Substitute 25 ⁰ ⁰ Is the answer reasonable? Is one of the angles 15 more than twice the other? YES!

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 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 SUPPLEMENT x⁰x⁰ (90 – x) ⁰ (180 – x) ⁰ x⁰x⁰ x⁰x⁰ 30 ⁰ 60 ⁰ 150 ⁰ 60 ⁰ 150 ⁰ Complements Supplements

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 ANGLE COMP SUPP x 90 – x 180 – x Complement Supplement “the angle” x 90 – x 180 – x (180 – x) =3(90 – x) – x = x x = 210 2x = 30 x = ⁰ 75 ⁰ 165 ⁰ √ – – 15 x x 75 ⁰

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

See very important TABLE on page 72! After studying this SECTION, you should be able to... Chapter 2, Section 3: “Drawing Conclusions” NOTE: The “If...” part of the reason should match the GIVEN information! 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 Follow a five-step procedure to draw logical conclusions AND the “then...” part matches the CONCLUSION being justified! CAUTION! Be sure not to reverse that order!!!

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

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

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

THEOREM #4 After studying this SECTION, you should be able to... If angles are supplementary to the same angle, then they are congruent 22 Chapter 2, Section 4: “Congruent Supplements and Complements” 2.4 Prove angles congruent by means of four new theorems 120 ⁰ 1 G 2 ∡ 1 is supplementary to ∡ G ∡ 2 is also supplementary to ∡ G What can we conclude about ∡1 and ∡2? 60 ⁰ 60 ⁰ =

After studying this SECTION, you should be able to... If angles are supplementary to congruent angles, then they are congruent 23 Chapter 2, Section 4: “Congruent Supplements and Complements” 2.4 Prove angles congruent by means of four new theorems G ∡ G is supplementary to ∡ E ∡ O is supplementary to ∡ M What can we conclude about ∡G and ∡M? ∡E ≅ ∡O M E O 50 ⁰ 50 ⁰ 130 ⁰ 130 ⁰ THEOREM #5

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

Complete a Proof! After studying this SECTION, you should be able to... Given: PROVE: Chapter 2, Section 4: “Congruent Supplements and Complements” ∡1 is comp to ∡4 25 R S V ∡2 is comp to ∡3 RT bisects ∡SRV TR bisects ∡STV T ? ? 4) ∡3 ≅ ∡4 1) ∡1 is comp to ∡4 2) ∡2 is comp to ∡3 3) RT bisects ∡SRV 1) Given 2) Given 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 ≅ StatementsReasons 6) TR bisects ∡STV 6) If an ∡ is div into 2 ≅ ∡s, then it was bisected by a ray!

2.5 Apply the addition properties of segments and angles Related Vocabulary After studying this SECTION, you should be able to... Chapter 2, Section 5: “Addition and Subtraction Properties” Apply the subtraction properties of segments and angles AC = BD, because AB + BC = BC + CD,  If two segments have the same measure, they are congruent! (7) +(7)(3) =(3) + (Commutative Property of Addition!) 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.

2.5 Apply the addition properties of segments and angles Related Vocabulary After studying this SECTION, you should be able to... Chapter 2, Section 5: “Addition and Subtraction Properties” Apply the subtraction properties of segments and angles If two angles have the same measure, they are congruent! (Commutative Property of Addition!) m ∡ ABC = 50 ⁰ m ∡ DBE = 50 ⁰ 50 + ∡ CBD = ∡ CBD + 50 m ∡ ABC + m ∡ CBD = m ∡ CBD + m ∡ DBE m ∡ ABD = m ∡ CBE, so 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.

 ABC   DBE 2.5 Apply the addition properties of segments and angles Related Vocabulary After studying this SECTION, you should be able to... Chapter 2, Section 5: “Addition and Subtraction Properties” Apply the subtraction properties of segments and angles If two angles have the same measure, they are congruent! m ∡ ABD = 80 ⁰ m ∡ CBE = 80 ⁰ 80 - ∡ CBD = ∡ CBD 80 - m ∡ ABD - m ∡ CBD = m ∡ CBE - m ∡ CBD m ∡ ABC = m ∡ DBE, so 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.

Chapter 2, Section 5: “Addition and Subtraction Properties” After studying this SECTION, you should be able to 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 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

Chapter 2, Section 5: “Addition and Subtraction Properties” After studying this SECTION, you should be able to 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.

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 Apply the addition properties of segments and angles 2.5 Apply the subtraction properties of segments and angles

2.5 Apply the addition properties of segments and angles After studying this SECTION, you should be able to... Chapter 2, Section 5: “Addition and Subtraction Properties” ∡ 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.

 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: 5. If two segments have the same measure then they are congruent Addition of Segments 4. PR = QS 3. Additive Property of Equality 3. PQ + QR = RS + QR 2. If two segments are congruent, then they have the same measure 2. PQ = RS 1. Given 1. ReasonsStatements PQ  RS PR  QS

Given: Conclusion:StatementsReasons1. 2. How to use this theorem in a proof: 2. If a segment is subtracted from congruent segments, then the resulting segments are congruent. (Subtraction) 1. Given ??

I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. ABCDEFGH

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

 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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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