Chapters 1 & 2 Basics of Geometry & Reasoning/Proof

Transcription

1 1 st Semester Chapters 1 & 2 Basics of Geometry & Reasoning/Proof Name: Teacher: Mrs. Gerardot or Mrs. Brown Period: Gerardot and Brown 1

2 1.2 Points Lines and Planes HW: 1.2 worksheet Point UNDEFINED Terms of Geometry Line Plane *** YOU MUST BE BLE TO IDENTIFY ND LIST THESE *** Ex 1 Classify each of the following objects as one of the 3 undefined terms of geometry? a.) e.) h.) Point Line Plane b.) Point Line Plane Point Line Plane Point Line Plane f.) i.) c.) Point Line Plane Point Line Plane Point Line Plane g.) j.) d.) Point Line Plane Point Line Plane Point Line Plane Gerardot and Brown 2

3 Definition Example Counterexample Collinear Points that lie on the same line. Between When three points lie on a line, you can say that one of them is between the other two. Coplanar Points that lie on the same plane. How would it be represented geometrically? Point Definition Example Naming/Notation Undefined! Plane Undefined! C B Line Undefined! B Segment Part of a line that consists of two points (endpoints) and all the points on the line between the endpoints. B Ray Opposite Rays Part of a line that consists of a point (initial point) and all the points on the line that extend in one direction. Two rays that share a common endpoint and form a line. C B B *Must start with the initial point! *Need TWO rays. Each must start with the common endpoint. Gerardot and Brown 3

4 Ex 2 Classify and name each of the following objects T Object What is it? Name it geometrically. X Name it TWO ways R U P Which TWO objects are shown? Name BOTH objects. B U G Name it TWO ways W E Name it TWO ways Z P Ex 3 Things that make you go hmmm. 1. Think about a LINE What is the fewest number of points needed to make a line? 4. What object is formed when a distinct line and a plane intersect? Point, Line, or Plane (circle your answer) Give a real-life example How many points are on a line? 2. Think about a PLNE What is the fewest number of points needed to make a plane? How many points are in a plane? 5. What object is formed when two distinct planes intersect? Point, Line, or Plane (circle your answer) Give a real-life example 3. What object is formed when two lines intersect? Point, Line, or Plane (circle your answer) Give a real-life example Gerardot and Brown 4

5 You Try Ex 4 Draw the following objects. a.) Opposite Rays PR and PQ b.) Ray with an endpoint Q in the direction of K c.) ME Ex 5 Use the diagram to name the figures. C Ex 6 nswer the following questions about the picture. B F C N D D E a.) Three non-collinear points a.) Name all points that are NOT collinear to F and C b.) Opposite rays b.) Name all the points that are NOT in c.) One line segment plane N d.) Three collinear points c.) What is the intersection of plane DE and plane N e.) Two rays that are not opposite rays Gerardot and Brown 5

6 Pop Quiz Will you pass? Refer to the diagram and decide if the statement is true or false. a.) Line E G lies in plane R. b.) Line B lies in plane R. c.) D lies in plane R. E F D R d.) e.) H lies in plane R C and E are coplanar. Q C G f.) h.) C and E are collinear. Line B intersects plane R at F. H B Gerardot and Brown 6

7 Chapter 1 Vocabulary HW Day 1: Notecards & Memorize #1-7 HW Day 2: Finish Notecards & TEST NEXT CLSS Front of Notecard # Theorem, Word, Given/Quest or Property 1. What are the 3 undefined term 2. Pythagorean Theorem geometry? Pythagorean Theorem Back of Notecard tion Draw a picture Statement ms of Write the PYTHGOREN THEOREM: Reason When would you use this formula? 3. Distance Formula 4. Midpoint Formula Distance Formula Midpoint Formula Write the DISTNCE FORMUL: Write the MIDPOINT FORMUL: 5. Segment ddition Postulate Z is between U and L 6. Segment Bisector B bisects RP at T 7. Midpoint I is the midpoint of RP 8. Right ngle HT is a right angle m HT = ngle ddition Postulate Y is in the interior of PUR Gerardot and Brown 7

8 # Theorem, Word, or Property 10. ngle Bisector Front of Notecard Given/Question Draw a picture Statement B bisects HT Back of Notecard Reason 11. Complementary ngles and B complementary m + m B = Supplementary ngles Q and Z supplementary m Q + m Z = Linear Pair (definition) Linear Pair Postulate 7 and 9 are a linear pair 15. Vertical ngles (definition) Thm and 4 are vertical angles 17. Perpendicular Lines ME NO at Gerardot and Brown 8

9 1.3 Segments and their Measures HW: 1.3 worksheet Review Do you remember? SIMPLIFYING RDICLS It s helpful to know your perfect squares 13 2 = 12 2 = 11 2 = 10 2 = 9 2 = 8 2 = 7 2 = 6 2 = 5 2 = 4 2 = 3 2 = 2 2 = ( 2 7 ) 2 lways your radicals and leave answers as fractions. NO DECIMLS UNLESS THE PROBLEM STRTS WITH DECIML! You must also always write a theorem/postulate/definition to support your initial equation for a problem. Identify and name the geometric figures using proper notation. X Y D X What is the object? What is the object? What is the object? What is the object? E What is its name? What is its name? What is its name? What is its name? Gerardot and Brown 9

10 Ruler Postulate, Segment ddition Postulate, Distance Formula, Pythagorean Theorem pply what you have learned so far. For each statement, draw a picture and write a geometric statement and reason. a.) is between B and C. b.) X is between Y and Z. Geo Stmt Reason Geo Stmt Reason Ex 1 N is between M and P; MN = 3x + 2; NP = 18; and MP = 5x. Find MN.. Use the given to draw and label a picture: Fill in the information below. Then use it to find your answer. IF N is between M and P Geo Stmt: Reason: lg Stmt: Ex 2 J(9, 4) and K(-5, 12). Find JK. What are they asking you to find? Ex 3 5 Find LG. L What can you use to solve this problem? W 10 G Why? Gerardot and Brown 10

11 You Try Ex 4 U is between T and B; TU = 2x; UB = 3x + 1; and TB = 21. Find TU. Draw and label a picture: Fill in the information below. Then use it to find your answer. IF U is between T and B Geo Stmt: Reason: lg Stmt: Ex 5 (10, -1) and B(-3, 6). Find B. Ex 6 Find B. 6 C 7 B Gerardot and Brown 11

12 1.4 ngles and Their Measures HW: 1.4 worksheet Review Do you remember? 1. CD 2. B What is the object? Draw a picture: What is the object? NOTE: Unless there is a number given use 3 letters to name a angle!!!! 8. Name the angle where the arrow is pointing. T S Q R E 9. Name the sides in JKL. Draw a picture: K 3. GH What is the object? J L Draw a picture: 4. Points X, Q and R are collinear Draw a picture: 5. CB and CD are opposite rays Draw a picture: 10. Name two angles adjacent to RES. T S Q R E Use the diagram below to answer questions D 6. Name the angle where the arrow is pointing. U 2 B 3 C 11. List all the possible names for BD. R O K C 7. Name the angle where the arrow is pointing. 12. Name the vertex of 3. B C 13. What are the sides of 1? Gerardot and Brown 12

13 ngle, Sides of an angle, Vertex of an angle, Interior of an angle, Exterior of an angle, cute angle, Right angle, Obtuse angle, Straight angle, djacent angle, ngle ddition Postulate, pply what you have learned. For each statement, draw a picture and write a geometric statement and reason. a.) P is in the interior of DOG. b.) X is in the interior of CT. Geo Stmt: Geo Stmt: Reason: Reason: Ex 2 D lies in the interior of BC. Th Find m BC. he m BD = (2x + 7), m CBD = (3x 5) and m BC =(7x 8). a. Draw and label a picture. b. Fill in the information below. Then use it to find your answer. IF D lies in the interior of BC Geo Stmt: Reason: lg Stmt: Ex 2 P lies in the interior of BIT. The Find m PIT. e m BIT = (25x 1), m BIP = (10x + 7) and m PIT = (5x + 2). a. Draw and label a picture. b. Fill in the information below. Then use it to find your answer. IF P lies in the interior of PIT Geo Stmt: Reason: lg Stmt: Gerardot and Brown 13

14 Review Do you remember? 1. Suppose m T=m S, m T = (12n 6), and m S = (4n + 18). Is T acute, obtuse, or straight? 2. Suppose is between B and C. B = (x +2), BC = 3x 4, and C = 44. Find BC. 3. Find C if (-11, 7) and C(-9, 3) Solve each equation. x = 3 2 y 6. x + 30 = 2x = 4 1 Gerardot and Brown 14

15 1.5 Segment and ngle Bisectors HW: 1.5 worksheet ngle Bisector, Midpoint, Segment Bisector, Midpoint Formula Ex 3 Find the midpoint ofbd, with endpoints Ex 4 Find the endpoint ofbd, if B(0, -7) and D(-2, -1). B(-3, -1) and the midpoint is (3, -4) What can you use to solve this problem? What can you use to solve this problem? Draw and label a picture: Draw and label a picture: Ex 5 C is the midpoint ofd. If C = and CD = 16-3x, find x. 5x 8 Ex 6 BD bisects BC, m BD = (5x + 5) and m BC = (4x + 100). Find x. Draw and label a picture. Draw and label a picture. Complete the stmts & reasons, then solve for x. IF C is the midpoint ofd Complete the stmts & reasons, then solve for x. IF BD bisects BC Geo Stmt: Reason: Geo Stmt: Reason: lg Stmt: IF D is in the interior of BC Geo Stmt: Reason: lg Stmt: Gerardot and Brown 15

16 You Try: 1. Find the endpoint ofrp, if R(-1, 7) 2. Find the midpoint of WB, with endpoints and the midpoint is (2, 4). W(4, 5) and B(-3, -8). Draw and label a picture: Draw and label a picture: 3. U is the midpoint of XY. If XY = 16x 6 4. QS bisects RQP, m RQS = (2x + 10) and UY = 4x + 9, find x. and m SQP = (3x 18). Find x. Draw and label a picture. Draw and label a picture. IF U is the midpoint of XY IF QS bisects RQP Geo Stmt: Reason: Geo Stmt: Reason: IF U is between X and Y lg Stmt: Geo Stmt: Reason: lg Stmt: Gerardot and Brown 16

17 1.6 ngle Pair Relationships HW: 1.6 worksheet Skills You ll Need Complementary angles, Supplementary angles, Linear pair, Vertical angles, Linear Pair Postulate, Thm 2.6 djacent (DJ), Linear Pair (LP), Vertical ngles (V) Supplementary (Supp), Complementary (Comp), Congruent (Cong) Ex 1 Determine the type of angle pair given (if any) and its relationship (if it can be determined.) Circle all that apply Type of ngle Pair Relationship a.) 4 and 5 dj LP V None Congruent Supp Comp b.) 1 and 4 dj LP V None Congruent Supp Comp c.) 2 and 3 dj LP V None Congruent Supp Comp d.) 3 and 5 dj LP V None Congruent Supp Comp e.) 3 and 4 dj LP V None Congruent Supp Comp f.) 1 and 2 dj LP V None Congruent Supp Comp Can t Determine Can t Determine Can t Determine Can t Determine Can t Determine Can t Determine Can you figure it out? If the measure of an angle is x, then what is the measure of its complement? HINT FOR SET UP: + = Draw an example of complementary angles. If the measure of an angle is y, then what is the measure of its supplement? HINT FOR SET UP: + = Draw an examplee of supplementary angles. Gerardot and Brown 17

18 Ex 1 and B are complementary. m = (5x - 28) and m B = (2x - 1). Find both angles. What can you use to solve this problem? Write and solve an algebraic expression. Make sure you find everything the problem is looking for!!! Ex 2 The measure of an angle is 20 more than four times its supplement. Find both angles. Ex 3 The measure of an angle is 6 less than five times its complement. Find both angles. What can you use to solve this problem? Write and solve an algebraic expression. Make sure you find everything the problem is looking for!!! Ex 4 Given m 1 = ( 9x 4) and m 2 = ( 4x 11), find 3 m. What kind of special angle pair are 1 and 2? What do you know about that type of angle pair relationship? Fill in the information below. Solve for x. Use value of x to find m 3. Geo Stmt: Reason: lg Stmt: Gerardot and Brown 18

19 Pop Quiz - Will you pass? Use the figure to answer the questions a.) re 1 and 2 a linear pair? a.) re 1 and 5 a linear pair? b.) re 1 and 3 vertical angles? b.) re 1 and 2 a linear pair? c.) re 1 and 4 a linear pair? c.) re 1 and 4 vertical angles? d.) re 2 and 4 vertical angles? d.) re 3 and 5 vertical angles? Solve for variable in each diagram. B (8x) E (3x +59) D C F I G (5y -50) J (4y -10) H 5. Use the diagram for problem #4 to answer the questions below a.) Name the sides of the angle FJG b.) Name a pair of opposite rays c.) Name 3 non collinear points d.) Name a vertical angle to FJI e.) Name an adjacent angle to FJG Gerardot and Brown 19

20 Writing Reasons HW: Writing Reasons Wkst # Front of Notecard Problem/Given If B ET Then B = ET If m W = m Q What happened? Back of Notecard Statement Reason No statement! What is the RESON? No statement! What is the RESON? Then W Q If 10x + 7 = 2x 3 Then 8x + 7 = 3 If 100x = 500 Then x = 5 If 3x + 7 = 5( x + 1) No statement! What is the RESON? No statement! What is the RESON? No statement! What is the RESON? If x + y Then 3x + 7 = 5x + 5 Then x + y = x + y If 10x + 7 = 20 Then 20 = 10x + 7 If x = 5 and x + y = 4 No statement! What is the RESON? No statement! What is the RESON? No statement! What is the RESON? 26. Then 5 + y = 4 If x + 15 = y and 7x + 50 = y No statement! What is the RESON? Then x + 15 = 7x + 50 If 10x = 2x + x 5 Then 10x = 3x 5 G & O are each right s B is supp to E E is supp to P H is comp to E R is comp to E No statement! What is the RESON? Gerardot and Brown 20

21 Writing Reasons in Proofs HW: Complete the blanks of the sequences below. Use the Reason Bank to help with the reasons. 1. Statement Reason XY bisects ZXC Given ZXY YXC m ZXY = m YXC 2. Statement Reason Z is the mdpt of B Given POSSIBLE RESONS Segment ddition Def n Midpoint Def n Segment Bisector ngle ddition Def n ngle Bisector Def n Congruence (seg or angles) Z BZ Defn of (segments) 3. Hint: Label the picture at the picture to the right based on you given information. Use your LOOK BOOK! Statement XY bisects CD at N Reason Given Defn of Segment Bisector Defn of Midpoint Defn of (segments) Things to remember a.) b.) c.) d.) e.) The 1 st step is to state the given information. Label the given information in the picture ND label additional information as you go If you can label something in the picture, WRITE IT down as a step. Read through the problem and think about why it makes sense. You must justify everything in a proof using a theorem, definition, or postulate. You must have these memorized. Gerardot and Brown 21

22 1. Given: 5 ( 3x 1) = 9x Prove: x = 6 Statement a. 5 ( 3 1) = 9x + 2 Reason (Think what happened? or why? ) x a. b. 15 x 5 = 9x + 2 b. c. 6 x 5 = 2 c. d. 6 x = 7 d. e. 7 x = e. 6 RESON BNK Given dd/sub Property Mult/Divide Property Distribute Property Reflexive Property Symmetric Property Transitive Property Substitution Property Simplify 2. Given: ST = RN, IT = RU Prove: SI = UN S I T R U N Statement Reason (Think what happened? or why? ) a. ST = RN, IT = RU a. b. SI + IT = ST b. c. RU + UN = RN c. d. SI + IT = RN d. e. SI + IT = RU + UN e. f. SI + RU = RU + UN f. g. SI = UN g. RESON BNK Given Segment ddition Postulate Defn of Segment Bisector Defn of Midpoint Defn of Defn of Right ngles Defn of Perpendicular Lines ngle ddition Postulate Defn of ngle Bisector Defn of Complementary ngles Defn of Supplementary ngles LP Supp ngles V ngles ngles Supp to Same ngles ngles Comp to Same ngles dd/sub Property Mult/Divide Property Distribute Property Reflexive Property Symmetric Property Transitive Property Substitution Property Simplify Gerardot and Brown 22

23 Given Segment ddition Postulate Defn of Segment Bisector Defn of Midpoint Defn of Defn of Right ngles Defn of Perpendicular Lines RESON BNK ngle ddition Postulate Defn of ngle Bisector Defn of Complementary ngles Defn of Supplementary ngles LP Supp ngles V ngles ngles Supp to Same ngles ngles Comp to Same ngles dd/sub Property Mult/Divide Property Distribute Property Reflexive Property Symmetric Property Transitive Property Substitution Property Simplify 3. Given: m 1 = m 3 m 2 = m 4 Prove: DEF BC F E 4 3 D 1 2 B C Statement Reason (Think what happened? or why? ) a. m 1 = m 3 a. b. m 2 = m 4 b. c. m 1 + m 2 = m BC c. d. m 3 + m 4 = m DEF d. e. m 3 + m 2 = m BC e. f. m 3 + m 4 = m BC f. g. m DEF = m BC g. h. DEF BC h. Gerardot and Brown 23

24 Pop Quiz/Review Do you remember? 1. Given: is between B and C. Picture: Statement: Reason: 2. Find DC. D(5, -2) and C(-8, 4) 3. Find the other endpoint of XY, if the midpoint is M(-3, 5) and X(4, 8). Use the given property to complete the statement. 4. ddition/subtraction property of Equality: If B = 5, then 10 + B =. 5. Symmetric property of equality: If m DCF = m MJC, then. 6. Transitive property of equality: If YZ = JK + UR and YZ = 36, then. Gerardot and Brown 24

25 2.1 Conditional Statements HW: 2.1 worksheet Review - Do you remember? Match the statement with the property on the right. Properties may be used more than once. 1. If JK = PQ and PQ = ST, then JK = ST.. Reflexive property 2. If m S = 30, then 5 + m S = 35. B. Substitution property 3. If 2x + x = 7, then 3x = 7. C. Transitive property 4. Given B, then B = B D. Symmetric property 5. If x = 4 and y = x + 5, then y = E. Multiplication/Division property 6. If m K = 45, then 3( m K) = 135. F. ddition/subtraction property 7. If m P = m Q, then m Q = m P G. Simplify 8. If CD PM and PM RV, then. CD RV 9. Given 10. If DS, then R DB, then DS DS. DB R. Type of Reasoning Example Explanation Josh knows that Brand X computers cost less than Brand Y computers. ll other brands that Josh knows of cost less than Brand X. Josh reasons that Brand Y costs more than all other brands. Inductive Deductive ndrea knows that Robin is a sophomore and Todd is a junior. ll other juniors that ndrea knows are older than Robin. ndrea reasons that Todd is older than Robin. Josh knows that Brand X computers cost less than Brand Y computers. He also know that Brand Y computers cost less than Brand Z. Josh reasons that Brand X cost less than Brand Z. ndrea knows that Todd is older than Chan. She also knows that Chan is older than Robin. ndrea reasons that Todd is older than Robin. Gerardot and Brown 25

26 Proof Types 2 column Flow Paragraph Indirect VOCBULRY In this lesson you will study a type of logical statement called a conditional statement. conditional statement has two parts, a & When the statement is written in if-then form, the if part contains the hypothesis and the then part contains the conclusion. Note: Conditional statements are necessarily true! Underline the hypothesis and circle the conclusion. Rewrite the statement in if-then form. Ex 1 ll mammals breathe oxygen. Ex 2 Two points are collinear if they lie on the same line. Ex 3 number divisible by 9 is also divisible by 3. Gerardot and Brown 26

27 Ex 4 When we have a conditional statement we can rewrite it in several different forms. We will use symbolic notation to help us write the different forms. Given the following statement 30 angle is acute. Conditional Contrapositive Two statements are said to be Equivalent Statements when they are they are either true OR false. Is the conditional true or false? Is the contrapositive true or false? re the conditional and the contrapositive equivalent statements? Inverse Converse Is our inverse true or false? Is our converse true or false? re the inverse and the converse equivalent statements? Gerardot and Brown 27

28 Ex 5 Conditional statements can be either or. For a conditional statement to be TRUE, it must be true for possible case. To prove that a conditional statement is FLSE, you only need to show counterexample. counterexample is an example that satisfies the but not the. Prove that each conditional statement is FLSE by finding a counterexample. a.) If a number is odd, b.) If the car has a full tank of gas, then it is divisible by 3. then the engine will start. You Try: Write the inverse, converse, and contrapositive of these statements: Given the following statement If you do not eat meat, then you are a vegan. Conditional Contrapositive Inverse Converse Gerardot and Brown 28

29 Complete each blank with the word point, line or plane. Through any two points there exists exactly one. line contains at least two. If two lines intersect, then their intersection is exactly one. Through any three non-collinear points there exists exactly one. plane contains at least three non-collinear. If two points lie in a plane, then the line containing them lies in the. If two planes intersect, then their intersection is a. *** Use page 73 of your textbook to check your answers. *** Pop Quiz/Review Do you remember? Use the diagram to solve for the missing angle measures If m 1 = 37, then m 3 =. Why? 2. If m 2 = 48, then m 1 =. Why? 3. If m 4 = (x + 17), then m 2 =. Why? Use the given property to complete the statement. 4. Reflexive property of equality: F =. 5. Multiplication/Division property of equality: If m C = 30, then ( m C 6. Substitution property of equality: If MN = 3 + XY and XY = 5(MN), then ) = ( ) Gerardot and Brown 29

30 2.2 Definitions & Biconditional Statements HW: 2.2 Worksheet Biconditional Statements Contain the phrase if and only if which is abbreviated. Can be read forward also called the ( ) Can be read backward also called the ( ) Symbolically we write a the biconditional statement as ( ) Ex 1 Write the statement forward and backward Given: right angle is an angle whose measure equals 90. Forwards ( ) Backwards ( ) Ex 2 nswer the questions about the given statement. Given: I will go to Bub s if and only if it is Saturday. a.) b.) Is the statement a biconditional? How do you know? Rewrite the statement as its conditional and its converse Conditional ( ) Converse ( ) Gerardot and Brown 30

31 For a biconditional statement to be TRUE the CONDITIONL and CONVERSE must be statements. Ex 2 Given: The definition for Perpendicular Lines: Lines that intersect to form 4 right angles. Conditional Converse If two lines are perpendicular, then they form right angles. Biconditional Conditional true? Converse true? Biconditional true? Ex 3 Given: The theorem for Vertical ngles: Vertical angles are congruent. Conditional Converse If two angles are vertical, then they are congruent. Biconditional Conditional true? Converse true? Biconditional true? Gerardot and Brown 31

32 SUMMRY In general If the CONDITIONL is true, the is also true. If the CONVERSE is true, the is also true. IF a BICONDITIONL STTEMENT is true then ND are true (by definition of a true biconditional) IF a BICONDITIONL STTEMENT is TRUE then RE LL TRUE!!! Note: LL can be written as TRUE biconditional statements, but ONLY theorems can be written as true biconditionals. Turn the page for the Pop Quiz/Review Gerardot and Brown 32

33 Pop Quiz/Review Do you remember? 1. B lies on the interior of XYZ. Picture: Statement: Reason: 2. Name the equivalent statement for each: a.) Conditional Converse? Inverse? Contrapositive? b.) Converse Conditional? Inverse? Contrapositive? 3. Rewrite in if-then form. ll monkeys have tails. 4. Write a counterexample. If a number is divisible by 2, then it is divisible by Write the following for the given statement Given: Complementary angles are two angles whose sum is 90. Conditional Converse Gerardot and Brown 33

34 Writing Statements HW: Writing Statements Wkst 1. Given (If/known from a previous step,) B C Statement (for THIS problem) Then Statement ( GENERL Form) Reason Why? Defn Midpoint 2. B C Segment ddition 3. B D C Defn Segment Bisector 4. E X ngle ddition 2 B 1 C X 2 1 B 1 3 C Defn ngle Bisector Definition of Vertical ngles 7. 5 and 6 are vertical angles Vertical ngles Congruent Definition of Linear Pair 9. 5 and 6 are a linear pair Linear Pair Postulate (or Linear Pairs Supp ngles) Defn Perpendicular Lines Gerardot and Brown 34

35 11. Given (If/known from a previous step,) 2 1 Statement (for THIS problem) Then Statement ( GENERL Form) Reason Why? ll Right ngles Congruent 12. m 1 = 90 Defn Right ngle is a right angle Defn Right ngle and 2 are supplementary Defn Supp. ngles 15. m 1 + m 2 = 180 Defn Supp. ngles and 2 are complementary Defn Comp. ngles 17. m 1 + m 2 = 90 Defn Comp. ngles & 3 are supplementary and 3 & 2 are supplementary 4 & 5 are complementary and 4 & 7 are complementary Congruent Supplements ( 's supp to the same are ) Congruent Complements ( 's comp to the same are ) 20. B C Defn Midpoint (You need to say something more than what is marked in the picture.) 21. D Defn ngle Bisector B (You need to say something more than what is marked in the picture.) Gerardot and Brown 35

36 Writing Statements Until now we have been writing reasons to support each step in our problems. Now we will look at the problems from the other direction. The reason will be given, and we must fill in the appropriate statements. It is very important to understand what each statement should look like in its general form so that you can more easily fill in the reason specific to that problem. Given: BD bisects BC D Prove: BD C 1 B 2 C Statement Reason a.) a.) Given b.) b.) Defn ngle Bisector c.) c.) Defn Congruence d.) d.) Defn of LP e.) e.) LP Supp ngles f.) f.) Defn Supp ngles g.) g.) Substitution h.) h.) Simplify i.) i.) Mulitplication/Division j.) j.) Defn Right ngle k.) k.) Defn Perpendicular Lines Gerardot and Brown 36

37 Pop Quiz/Review Do you remember? 1. Given: BD bisects BC. Picture: Statement: Reason: 2. Write the inverse, converse, and contrapositive of the statement: Given: If you do not have a sibling, then you are not a twin Conditional Contrapositive If you do not have a sibling, then you are not a twin Inverse Converse 3. Write the following Given: Two angles are supplementary if and only if their sum is 180. Conditional Converse Gerardot and Brown 37

38 Pop Quiz/Review Do you remember? 1. Given: (5, -7) and B(-10, 2). Find the following B Slope of B Midpoint of B 2. Given: and T are complementary and the measure of T is three times the measure of. Find the measures of and T. 3. If a biconditional is true, which of the following statements must be true? (circle all that apply) Conditional Inverse Converse Contrapositive 4. Write the statements and answer the questions below. Conditional Converse If two planes intersect, then they contain the same line. Biconditional Conditional true? Converse true? Biconditional true? Gerardot and Brown 38

39 Great job!! You finished Chapter 1 and Chapter 2! Gerardot and Brown 39