1 Unit 4 lgebraic and Geometric Proof Math 2 Spring 2017 1
Table of ontents Introduction to lgebraic and Geometric Proof... 3 Properties of Equality for Real Numbers... 3 lgebra Proofs... 4 Introduction to Lines & ngle Proofs... 6 Types of ngle Pairs... 7 Practice and losure... 8 Parallel Lines and Transversals with lgebra... 10 Practice and losure... 12 dding and Subtracting Line Segments and ngles... 15 Practice and losure... 19 Formal Flow Proof... 22 Flow Proofs ontinued... 25 Practice and losure... 28 Math 2 Unit 4 Review Sheet... 29 2 2
3 Introduction to lgebraic and Geometric Proof Properties of Equality for Real Numbers Using the word bank, write each property next to its corresponding definition. Word ank: Transitive Property Reflexive Property Symmetric Property istributive Property Substitution Property (Simplifying Property) ddition/subtraction Properties Multiplication/ivision Properties If two things are equal, then you can add/subtract the same thing on both sides of the equal sign (e.g., if a = b, then a + c = b + c and a c = b c). If two things are equal, then you can multiply/divide the same thing on both sides of the equal sign (e.g., if a = b, then a c= b c and a b c c. If a = b, then a may be replaced by b in any equation or expression. a (b + c) = ab + ac. If a = b and b = c, then a = c. Everything is equal to itself (e.g., 3 = 3). If two things are equal, you can write that equality either way (e.g. if a = b, then b = a). 3
lgebra Proofs 4 Proof a set of statements (and reasons) that lead to a logical conclusion. lgebraic Proof an algebra equation that is solved using the two-column proof format. Ex. 1) Given: 2x + 3= 15 Prove: x = 6 Statements Reasons. 1. 1. 2. 2. 3. 3. 4. 4. 5. 5. Ex 2) Given: 2 5x 3 = 8+ 3x Prove: x = 2 Statements Reasons. 1. 1. 2. 2. 3. 3. 4. 4. 5. 5. 6. 6. 7. 7. 8. 8. 4
Ex.3. Given: 7x + 16 = 3x + 48 5 Prove: x = 8 Statements Reasons. 1. 1. 2. 2. 3. 3. 4. 4. 5. 5. 6. 6. 7. 7. 2x 13 Ex.4. Given: = 9 3 Prove: x = 7 Statements Reasons. 1. 1. 2. 2. 3. 3. 4. 4. 5. 5. 6. 6. 7. 7. 5
Introduction to Lines & ngle Proofs 6 Linear Pair ngles ngles that are and. Vertical ngles Two formed by a pair of intersecting lines. Vertical ngles onjecture - Vertical angles are. Intersecting Lines onjecture - ongruent ngles Supplementary ngles Lines and Transversals Interior Exterior Transversal Interior ngles Exterior ngles 6
Types of ngle Pairs 7 orresponding ngles Same-Side Interior ngles aka ( ) Same-Side Exterior ngles aka ( ) lternate Interior ngles lternate Exterior ngles 7
Practice and losure 8 Using the diagram to the right, identify all pairs of angles which match the given angle pair name. 1. Same-Side Interior t 2. lternate Exterior 3. lternate Interior 4. Same-Side Exterior 5. orresponding 6. Vertical 1 5 2 6 3 7 4 8 m l 7. Linear Pair Name each angle pair and the transversal used. (Note: o not use congruent or supplementary as angle pair names.) 15. 6 and 14 r t ngle pair name: Transversal: 1 2 3 4 5 6 7 8 l 16. 4 and 7 ngle pair name: Transversal: 9 10 11 12 13 14 15 16 m 17. 3 and 10 ngle pair name: Transversal: 8
For problems 8 14, write the Special Name for each angle pair. (Note: o not use congruent or supplementary as angle pair names.) 9 8. 1 and 9 9. 3 and 10 10. 7 and 13 11. 6 and 16 12. 11 and 14 13. 2 and 3 14. 7 and 8 Name each angle pair and the transversal used. (Note: o not use congruent or supplementary as angle pair names.) 18. 11 and 14 ngle pair name: Transversal: 19. 13 and 16 ngle pair name: Transversal: 20. 3 and 8 ngle pair name: Transversal: 21. 2 and 4 ngle pair name: Transversal: 9
Parallel Lines and Transversals with lgebra 10 Steps To Solve for x: 1. etermine the types of angle pair or pairs a. orresponding, tl. Int., lt. Ext., Same-Side Interior, Vertical, Linear Pair 2. Set-up the problem using the appropriate algebraic relationship a. orresponding, tl. Int., lt. Ext.,, Vertical are congruent b. Same-Side Interior, Linear Pair are supplementary (add to 180 ) 3. Solve for x 4. heck your answer Example 1: Solve for x Example 2: Example 3: 10
Example 4: 11 Example 5: You Try: Solve the following example for x 11
Practice and losure 12 For problems 1 7, identify the special name for each angle pair. 1. 1 and 4 r t 2. 13 and 10 3. 5 and 13 1 2 3 4 5 6 7 8 l 4. 12 and 16 5. 11 and 14 6. 2 and 7 9 10 11 12 13 14 15 16 m 7. 7 and 8 For problems 8 13, the figure shows l m. Find the measures of each angle and list the angle pair name. Treat each problem independently. 8. If m 1 = 120, find m 5 = pair name: t 9. If m 6 = 72, find m 4 = pair name: 10. If m 2 = 64, find m 8 = pair name: 1 2 3 4 l 11. If m 4 = 112, find m 5 = pair name: 12. If m 2 = 82, find m 7 = pair name: 13. If m 2 = 80, find m 5 = pair name: 5 6 7 8 m 12
For problems 14 15, the figures show p q. 14. m 1 = 3x 15 and m 2 = 2x + 7, find x and m 1. t 13 x = 1 p m 1 = 2 q 15. m 3 = 7x 12 and m 4 = 12x + 2, find x and m 4. t x = m 4= 3 p 4 q For problems 13 14, find the values of x, y and z in each figure. 16. 17. (y+12) (3z + 18) x 72 3y x (y-18) z 13
lass work For problems 1 7, the figure at the right shows p q, m 1 = 78 and m 2 = 47. Find the measures of the following angles. 14 1. m 1 = 78 o 2. m 2 = 47 o p q 3. m 3 = 4. m 4 = 5. m 5 = 3 1 9 8 2 6 7 6. m 6 = 4 5 7. m 7 = 8. m 8 = 9. m 9 = For problems 10 12, find the values of x and y in each figure. 10. 11. (y+8) (6x-14) (3x+5) 5x 9y 12. (8x + 40) 7y 6x (3y 10) 14
dding and Subtracting Line Segments and ngles Everybody knows you can add and subtract numbers: 7 + 3 = 10 makes perfect sense. However, adding and subtracting objects is different. It is nonsense to say that an apple + banana = banapple Line segments are somewhere in between. In general, you can t add or subtract just any two random line segments and get another segment. ut sometimes it makes sense. Your job is to understand when. IMPORTNT: 1) only makes sense when,, and are collinear and is between and. In other words, to add segments, they must be collinear and the second one must start where the first one ends. 15 nonsense nonsense nonsense 2) and only make sense when,, and are collinear and is between and. In other words, to subtract segments, the one being subtracted must be part of the one being subtracted from and they must share an endpoint. nonsense nonsense nonsense nonsense 6. ased on the diagram at right, tell if each of the following is True or False. Remember the difference between and. P a. + = P b. P c. + = d. 4 5 e. = f. g. P P = h. P P 2 3 1 7. In the diagram at right, FLG. For each of the following, either fill in the appropriate line segment or write nonsense. 15 a. L G b. FL LP c. F LG d. FL G e. FL LG f. FL L G g. FP FL h. F L i. F L j. FP FL. k. FG FL l. FG L F L G P
16 1. Use the diagram at right to answer the following. a. How many angles in the diagram have their vertex at? b. How many angles in the diagram have their vertex at? c. What angle (number) is named? 6 1 5 2 3 4 d. Name two adjacent angles in the diagram. e. re and adjacent? f. Give three alternate names for 4. g. Explain why we should not refer to in the diagram. (Yes, you may lose points for sloppy notation on quizzes and tests.) h. Name one acute angle on the diagram. R i. Name one obtuse angle on the diagram. j. Which angle on the diagram appears to be closest to a right angle? P Q S 2. In the diagram at right, which angle has a larger measure, PQ or RS? N 3. In the diagram at right, NOP, OR OQ, and m POQ = 40. Find m NOR. 4. The measures of two supplementary angles are in the ratio 5:7. Find the measure of the smaller angle. O R P Q 5. The measure of the complement of an angle is 18 less than twice the measure of the angle. What is the numerical measure of the angle? 6. If ET bisects EG, m ET = x 2 and m GET = 5x + 14, find the numerical measure of EG. 7. If OY bisects OT, m OY = 3x + 8 and m OT= 8x 2, find the numerical measure of TOY. 16
RE: : Numbers can always be added and subtracted. It makes no sense to add or subtract people. Line segments can sometimes be added or. ngles are like segments. They can sometimes be added and subtracted. Remember, represents an actual angle (a geometric object); m is a number that represents the degree measure of. 1) dding two angles only makes sense if they are adjacent: they share a vertex and one side but have no interior points in common (one is not inside the other). P P P P + P = P P + P = nonsense P + P = nonsense P + P = nonsense 2) Subtracting two angles only makes sense if they share a vertex and one side and the second side of the smaller angle is on the interior of the larger angle (the smaller angle is part of the larger angle). P P P P = P P P = nonsense P P = nonsense P P = P P P = nonsense 8. ased on the diagram at right, tell if each of the following is True or False. Remember the difference between and m. a. m + m = m b. + = 55 c. m + m = m d. + = e. m m = m f. = g. m m = m 40 15 70 h. = P 17 9. Use the diagram at right to fill in an appropriate angle for each of the following or write nonsense. a. NG + LG = b. SEG + EL = c. NS + NSE = d. LGS EGS = e. NSE ESG = f. LG LE = g. LGS + EGS = h. LSN LE = 17 N S E L G
You Try: In problems #1-5, for each given, state a valid conclusion and a reason based on the definitions we have covered. (Note: some of these have more than one correct answer.) 1. Given: onclusion: Reason: 18 2. Given: X is the midpoint of PQ. onclusion: Reason:.. P X Q. 3. Given: bisects. onclusion: Reason: 4. Given: bisects at E. onclusion: Reason: E 5. Given: onclusion: Reason: 6. In the diagram at right, bisects, m = 66 2x and m = 3x 24. Find the numerical value (a number, not just an algebraic expression) of m. 66 3x 18
Practice and losure 19 1. If the angles of have the following measures: m = 3x + 2, m = 5x 3, m = 6x 1, list the sides of from Longest to Shortest. For problems 14 22, find the values of the given variables in each of the figures below. 14] 15] 16] 108 53 40 x 142 33 x x x 30 17] 18] 19] 26 x x x 19 x 105 42 73 44 x 20] 21] 22] x 2y 8x+4 72 5x-32 7x-8 4x+19 42 6y 19
For #1-4, name the postulate that justifies the conclusion. 20 1. Given: FT T, T RT onclusion: FT RT Reason: F T R 2. Given: (iagram at right) onclusion: m E = m 4 + m 2 + m 5 4 2 5 E Reason: 1 3 3. Given: (iagram at right) onclusion: T T Reason: F T R 4. Given: m 1 + m 2 = 180, m 2 = m 3 (iagram at right) onclusion: m 1 + m 3 = 180 1 2 Reason: 3 5. Given: m 1 + m 2 = 180; m 3 = m 1. onclusion: Reason: 6. Given: Q bisects U. 1 2 U 3 onclusion: Q Reason: 7. Given: m O = 90. Statement: m O = m OX + m XO onclusion: X Reason: O 20
lass work If two line segments are added or subtracted, the result is another line segment. (See diagram below.) Ex: a. b. c. nothing (why?) d. nothing (why?) e. nothing (why?) f. nothing (why?) g. E nothing (why?) If two angles are added or subtracted, the result is another angle. (Same diagram.) Ex: a. FE + E = F b. F + F = nothing (why?) c. E FE = F d. F F = nothing (why?). F 21 E 1. Use the diagram at right to answer the following: a. P P b. S S. P c. S R d. Q Q. Q e. Q f. S. g. SR h. R R. S R 2. Use the same diagram to answer the following: a. + =. P b. QR + QR =. Q c. RQ + RSQ =. d. Q QP =. e. QS Q =. S R f. Q PQ =. 3. If M is the midpoint of Y, M = x + 8 and Y = 3x 2, find the numerical length of Y. 4. If HOT is the perpendicular bisector of OG, HO = 2x + 1, OT = 3x 2, O = 4x 5, and OG = 2x + 3, find the numerical length of HOT. 21
Formal Flow Proof 22 Today we will begin formal flow proofs. Just like a graphic organizer, flow proofs can be used to organize your work into a picture that is easy to read and understand. 1) Given: WT SU ; W U; VW OU Prove: OSU VTW O T W S U V 2) Given: S; U ; SU Prove: MSU M S U 22
23 3) Given: 1 2; LO OE Prove: LOV EOV 1 O 2 L V E 4) Given: 1 2; 3 4 E Prove: EU USE 3 1 S 2 4 U 5) Given: ; MT TH M Prove: MT HT T 23 H
6) Given: R RW ; E is the midpoint of W Prove: RE REW R 24 E W 7) Given: S; is the midpoint of S U Prove: U SK K S 24
Flow Proofs ontinued 25 8) Given: O ; ; is a right angle; ; O is a right angle Prove: OG G O 9) Given: JU TN ; JS TI ; J T J T Prove: ; U N U S I N 25
26 10) Given: J S; JN NS Prove: O J N O S 11) Given: JN NS ; bisects Prove: 26
27 12) Given: ; O U Prove: 1 2 1 2 O U 13) Given: ; O U H Prove: 1 2 1 O S M 2 27 E
Practice and losure 1) Given: OS SU ; ; US ; U Prove: O O 28 U S 2) Given: T R; is the midpoint of SE Prove: RE ST S T R E 3) Given: O bisects GOL; GO OL Prove: G L O L V E 28
Math 2 Unit 4 Review Sheet 29 Solve the given proof. Show all possible steps. 1. Given: 1 2 Prove: Δ Δ 1 2 2. Given: G is the midpoint of FH EF LH EG LG Prove: E L E I F G H 3. Given: bisects Prove: 1 2 29
30 4. Given: G is the midpoint of FI F I Prove: EF IH F G H E I 5. Given: ; E EF; ; E Prove: F E F For problems 6 12, identify the special name for each angle pair. 6. 1 and 9 r 7. 3 and 10 8. 7 and 13 9. 6 and 16 10. 11 and 14 11. 2 and 3 1 2 3 4 12. 7 and 8 t 5 6 7 8 l 9 10 11 12 13 14 15 16 m 30
31 For problems 13 17, identify the special name for each angle pair. 13. If m 1 = 120, find m 5 = t 14. If m 6 = 72, find m 4 = 15. If m 4 = 112, find m 5 = 16. If m 2 = 82, find m 7 = 17. If m 2 = 80, find m 5 = 1 2 3 4 5 6 7 8 m l 18. Solve for x 19. Solve for x t t 3x 15 p 7x 12 p 2x + 7 q 12x + 2 q 20. 21. 3y + 12 y 65 y 2x + 40 3x + 5 x =, y = x =, y = 31