Chapter 1: Shapes and Transformations

– Justifications and Proofs

Measuring, describing, and transforming: these are three major skills in geometry that you have been developing. In this chapter, you will focus on comparing; you will explore ways to determine if two figures have the same shape (called similar) and if they have the same size (congruent).

Making logical and convincing arguments that support specific ideas about the shape you are studying is another important skill. In this chapter you will learn how you can document facts to support a conclusion in a flowchart and two-column proof.

In this chapter, you will learn:  how to support a mathematical statement using flowcharts and two-column proofs  what a converse of a conditional statement is and how to recognize whether or not the converse is true  how to disprove a statement with a counterexample  about the special relationships between shapes that are similar or congruent  how to determine if triangles are similar and/or congruent 44

Justification and Proofs

pg.1 CHAPTER 4 INDEX The following is a list of vocabulary used in this chapter. Make sure that you are familiar with all of these words and know what they mean. Section Page # Vocabulary Meaning Proof Method to show something is true Reasons Justification for each statement 4.1 3-6 Reflexive Property Anything is equal to itself. a = a Transitive Property If a=b and b=c, then a = c Dilation Stretch a shape, keeping congruent angles and side ratios 4.2 7-9 Scale Factor How much something is increased or decreased Symbols Equal (=) Approximately (), Similar (~) and Congruent () Proportion Setting two ratios equal, cross multiply 4.3 10-13 Similarity Statement Order of letters shows what angles are equal AA~ Two triangles are similar if two corresponding angles are = 4.4 14-15 SSS~ Two triangles are similar if all 3 sides have the same ratio Flow Chart Way to organize a proof 4.5 16-18 // lines, name congruent angles, look for reflexive of sides Extra Information touching, triangle sum add to 180, etc Two triangles are similar if two sides have the same ratio and SAS~ 4.6 19-22 the angle in between is equal ASS and SSA There is no ASS or ASS backwards in geometry! Congruent Triangles If triangles are similar with side ratio of 1 4.7 23-26 Proof Congruent ∆s SSS, SAS, HL, ASA, AAS 4.8 27-29 Triangle Proofs Each statement should have one reason CPCTC If triangles are congruent, their parts are congruent 4.9 30-32 Proof by contradiction Prove something isn't true with a counterexample

pg.2 4.1 – What Is A Proof? ______Types of Proof

Whenever you buy a new product that needs to be put together, you are given a set of directions. The directions are written in a specific order that must be followed closely in order to get the desired finished product. Sometimes they clarify their directions by explaining why you are completing each step. This is the same idea we use in geometry in proofs.

4.1 – ORDERING STATEMENTS When you write a proof, the statements must be in a specific order, building off of each other. You can't just jump to the end without breaking down each part. To illustrate this, with your group explain how to make a peanut butter and jelly sandwich. Work with your team to include all steps to make sure the sandwich will be made correctly.

4.2 – STATEMENTS AND REASONS When you write a proof in geometry, each statement you make must have a reason to support it. This helps people understand why each statement was listed. This can be done in a flowchart proof or a two-column proof. Examine the two types below. Notice where the statements and reasons are. Also, notice how the statements are in a specific order.

4x + 1 = 5 Statements Reasons given 4x + 1 = 5 Given

4x = 4 Subtraction 4x = 4 subtraction x = 1 Division

x = 1 division

pg.3 4.3 – REASONS The reasons for certain statements come from definitions, properties, postulates, and theorems. Below are some commonly used reasons. Name Property of Equality Addition Property If a = b, then Subtraction Property If a = b, then Multiplication Property If a = b, then Division Property If a = b, then Distribution Property If a(x + b), then Substitution Property If a = b, then Reflexive Property Transitive Property If a = b and b = c, then

a. Write the reason for each statement Statement Reason If 4(x + 7), then 4x +28 BD@ BD If 2x + 5 = 9, then 2x = 4 If X  Y and Y  Z, then X  Z If x – 7 = 2, then x = 9 If 4x = 12, then x = 3 If a = 20, then 5a = 5(20)

c. write a reason for each statement. Statements Reasons 1. 5(x – 3) = 4(x + 2) 1. ______

2. 5x – 15 = 4x + 8 2. ______

3. x – 15 = 8 3. ______

4. x = 23 4. ______

4.4 – REASONS pg.4 Complete the proof by writing a reason for each step. a. GIVEN: M is the midpoint of AB , AM = 6in PROVE: AB = 12in

Statements Reasons

1. M is the midpoint of AB 1. ______2. AM = MB 2. ______3. AM = 6in 3. ______4. AM + MB = AB 4. ______5. AM + AM = AB 5. ______6. 6 + 6 = AB 6. ______7. AB = 12in 7. ______

 b. GIVEN: BD bisects ABC, mABD = 20° PROVE: ABC = 40°

Statements Reasons  1. BD bisects ABC 1. 2. ABD  DBC 2. 3. mABD = 20° 3. 4. ABC = ABD + DBC 4. 5. ABC = ABD + DBC 5. 6. ABC = 20° + 20° 6. 7. ABC = 40° 7.

s s c. GIVEN: AB intersectsCD PROVE: 蠤COA BOD Statements Reasons s s 1. AB intersectsCD 1. 2. COA + AOD = 180 2. 3. AOD + BOD = 180 3. 4. COA + AOD = AOD + BOD 4. 5. COA = BOD 5. 6. 蠤COA BOD 6

4.5 – STATEMENTS AND REASONS pg.5 Complete each proof with statements and reasons. a. Given: g//h, 1 is supplementary to 4. Prove: p//r

Statements Reasons

1. g//h 1.

2. 1 = 2 2.

3. 2 + 3 = 180 3.

4. 4. substitution

5. 1 is supplementary to 4 5.

7. 1 + 3 = 1 + 4 7.

8. 8. subtraction

b. Given: MATH is a parallelogram, MN@ AT Prove: 1  2

Statements Reasons

2. MH@ AT 2.

3. MN@ AT 3.

5. 1 = 2 5.

6. 1  2 6.

4.2 – What Do These Shapes Have in Common?______pg.6 Similarity

This year you organized shapes into groups based on their size, angles, sides, and other characteristics. You identified shapes using their characteristics and investigated relationships between different kinds of shapes, so that now you can tell if two shapes are both parallelograms or trapezoids, for example. But what makes two figures look alike?

Today you will be introduced to a new transformation that enlarges a figure while maintaining its shape, called a dilation. After creating new enlarged shapes, you and your team will explore the interesting relationships that exist between figures that have the same shape.

4.11 – WARM-UP STRETCH Before computers and copy machines existed, it sometimes took hours to enlarge documents or to shrink text on items such as jewelry. A pantograph device (like the one at right) was often used to duplicate written documents and artistic drawings. You will now employ the same geometric principles by using rubber bands to draw enlarged copies of a design. Your teacher will show you how to do this. a. What do you notice about the angles of the original and the dilation?

b. What do you notice about the sides of the original and the dilation?

4.12 – DILATION In problem 4.11, you created designs that were similar, meaning that they have the same shape. But how can you determine if two figures are similar? What do similar shapes have in common? To find out, your team will need to create similar shapes that you can measure and compare. a. Get the resource page from your teacher. Find quadrilateral ABCD on the graph. Dilate (stretch) the quadrilateral from the origin by a factor of 2, 3, 4, or 5 to form A' B ' C ' D '. Each team member should pick a different enlargement factor. You may want to imagine that your rubber band chain is stretched from the origin so that the knot traces the perimeter of the original figure. For example, if your job is to stretch ABCD by a factor of 3, then A' would be located as shown in diagram #2 at right. b. Carefully cut out your enlarged shape and compare it to your teammates' shapes. How are the four shapes different? How are they the same? As you investigate, make sure you record what qualities make the shapes different and what qualities make the shapes the same. Then complete the conditional statement.

If a shape is similar, then its ______are congruent and its sides are ______,

4.13 – ZOOM FACTOR AND SCALE FACTOR

pg.7 In the previous problem, you learned that you can create similar shapes by multiplying each side length by the same number. This number is called the zoom factor or the scale factor. You may have used a zoom factor when using a copy machine. For example, if you set the zoom factor on a copier to 50%, the machine shrinks the image in half (that is, multiplies by 0.5) but keeps the shape the same. In this course, the zoom factor and the scale factor will be used to describe the ratio of the new figure to the original.

What zoom factor was used to enlarge the puppy shown at right?

4.14 – CASEY'S "C" Casey decided to enlarge her favorite letter: C, of course! Your team is going to help her out. Have each member of your team choos a different zoom factor below. Then on the grid, enlarge (or reduce) the block "C" below by your zoom factor.

(1) 3 (2) 2

1 (3) 1 (4) 2

Look at the different "C's" that were created with your team. What happened with each zoom factor? When did the shape stay the same size? When did it grow? When did it shrink?

4.15 – MULTIPLY VS. ADD Can you create a similar shape if you add the same number to each side? Examine this again with the rectangle at right. Multiply each side by 3. Then add 3 to each side of the rectangle. When does this shape appear to be similar to the original?

4.16 – SYMBOLS

pg.8 Talk about the differences of each of the symbols below. What do you think each one is written the way it is? How are they alike? How are they different? a. Equal to (=) b. Approximately () c. Similar (~) d. Congruent ()

4.17 – PICTURES Draw an example of a shape that is similar and congruent to the following.

4.18 – CONGRUENT VS. EQUAL When you are talking about numbers that are the same, we can say they are equal. However, when it is a shape, we call it congruent, because it isn't exactly the same. Use this idea to complete the statements below. A AB @ ______AB = ______

蠤A ______m� A ______45 B C 8 in

4.3 – How Can I Use Equivalent Ratios?______pg.9 Applications and Notation

Now that you have a good understanding of how to determine similarity, you are going to use proportions to find missing parts of similar shapes.

4.19 – EQUAL RATIOS OF SIMILARITY Casey wants to learn more about her enlarged "C's". a. Since the zoom factor multiplies each side of the original shape, then the ratio of the widths must equal the ratio of the lengths.

Casey decided to show these ratios in the diagram at right. Verify that her ratios are equal by reducing each one.

b. When looking at Casey's work, her brother wrote a different 8 24 pair of ratios. He wrote and . Are his ratios equal? And 6 18 how can he show his work on his diagram. Add arrows to show what sides Casey's brother compared.

c. She decided to create an enlarged "C" for the door of her bedroom. To fit, it needs to be 20 units tall. If x is the width of this "C", write and solve an proportion to find out how wide the "C" on Casey's door must be. Be ready to share your equation and solution with the class.

4.20 – PROPORTIONS Use your observations about ratios between similar figures to answer the following: a. Are the triangles similar? How do you know?

b. If the pentagons at right are similar, what are the values of x and y?

pg.10 4.21 – PROPORTIONS Your team may have used a proportion equation to solve for the previous problem. It is important that parts be labeled to help you follow your work. The same measures need to match to make sure you will get the right answer.

Likewise, when working with geometric shapes such as the similar triangles below, it is easier to explain which sides you are comparing by using notation that everyone understands. a. One possible proportion equation you can write for these AC DF triangles is = . Write at least three more proportional AB DE equations based on the similar triangles above.

b. Jeb noticed that RA@ R D and RC@ R F. But what about RB and R E ? Do these angles have the same measure? Or is there not enough information?

4.22 – WRITING SIMILARITY STATEMENTS The two triangles are similar. We use the similar symbol (~) to state the shapes that are similar. When we name the letters, the parts of the triangle must match up. Notice how 蠤A Z. Those letters must match when we write our statement. a. What other angles should match up? b. Complete the similarity statement for the triangles.

DABC ~ ______c. Examine the triangles below. Which of the following statements are correctly written and which are not? Hint: more than one statement is correct.

a. DDOG~ D CAT

b. DDOG~ D CTA

c. DOGD~ D ATC

d. DDGO~ D CAT

pg.11 4.23 – READING SIMILARITY STATEMENTS Read the similarity statements below. Determine which angles must be equal. Then determine which sides match up.

a. DABC~ D DEF b. DGEO~ D FUN Angles Sides Angles Sides

A = ______AB matches with _____ G = ______GE matches with _____

B = ______BC matches with _____ E = ______EO matches with _____

C = ______AC matches with _____ O = ______GO matches with _____

4.24 – PROPORTION PRACTICE Find the value of the variable in each pair of similar figures below. Make sure you match the correct sides together. a. ABCD ~ JKLM b. DNOP~ D XYZ

c. DGHI~ D PQR d. DABC~ D XYZ

pg.12 4.25 – NESTING TRIANGLES Rhonda was given the diagram and told that the two triangles are similar. a. Rhonda knows that to be similar, all corresponding angles must be equal. Are all three sets of angles equal? How can you tell?

b. Rhonda decides to redraw the shape as two seperate triangles, as shown. Write a proporitonal equation using the corresponding sides, and solve. How long is AB? How long is AC?

pg.13 4.4 – What Information Do I Need?______Conditions for Triangle Similarity

Now that you know what similar shapes have in common, you are ready to turn to a related question: How much information do I need to know that two triangles are similar? As you work through today's lesson, remember that similar polygons have corresponding angles that are equal and corresponding sides that are proportional.

4.26 – ARE THEY SIMILAR? Erica thinks the triangles below might be similar. However, she knows not to trust the way figures look in a diagram, so she asks for your help. a. If two shapes are similar, what must be true about their angles? Their sides?

b. Measure the angles and sides of Erica's Triangles and help her decide if the triangles are similar or not.

4.27 – HOW MUCH IS ENOUGH? Jessica is tired of measuring all the angles and sides to determine if two triangles are similar. "There must be an easier way," she thinks. "What if I know that all of the side lengths have a common ratio? Does that mean that the triangles are similar?" a. Before experimenting, make a prediction. Do you think that the triangles have to be similar if the pairs of corresponding sides share a common ratio?

b. Experiment with Jessica's idea. To do this, use either a manipulative or dynamic geometry tool to test triangles with proportional side lengths. Can you create two triangles that are not similar?

c. Jessica then asks, "Is it enough to know that each pair of corresponding angles are congruent? Does that mean the triangles are similar?" Again, make a prediction, test this claim use either a manipulative or dynamic geometry tool to test triangles with equal corresponding angles. Can you create two triangles with the same angle measures that are not similar?

pg.14 4.28 – WHAT'S YOUR ANGLE? Scott is looking at the set of shapes at right. He thinks that DEFG~ D HIJ but he is not sure that the shapes are drawn to scale. a. Are the corresponding angles equal? Convince Scott that these triangles are similar by finding the missing angles.

b. How many pairs of angles need to be congruent to be sure that triangles are similar? c. Use this information to determine if DABX~ D RQX . Do you see any other angles that are equal?

4.29 – ARE THEY SIMILAR? Based on your conclusions, decide if each pair of triangles below are similar. Explain your reasoning by stating which angles are equal OR which sides have the same ratio. Then determine if you are using Angle- Angle similarity or Side-Side-Side similarity. a. b.

pg.15 4.5 – How Can I Organize My Information?______Creating a Longer Flowchart

In Lesson 4.4, you developed the AA~ and SSS~ conjectures to help confirm that triangles are similar. Today you will continue working with similarity and will learn how to use flowcharts to organize your reasoning.

4.30 – FLOWCHARTS Examine the triangles at right. a. Are these triangles similar? Use full sentences to explain your reasoning.

b. Julio decided to use a diagram (called a flowchart) to explain his reasoning. Compare your explanation to Julio's flowchart. Did Julio use the same reasoning you used? given given c. What appears to go in the bubbles of a flowchart? What goes outside the bubbles?

4.31 – WRITING FLOWCHARTS Besides showing your reasoning, a flowchart can be used to organize your work as you determine whether or not triangles are similar. a. Are these triangles similar? Why?

b. What facts must you know to use the triangle similarity conjecture you chose? Julio started to list the facts in a flowchart at right. Complete the third oval.

given given ______c. Once you have the needed facts in place, you can conclude that you have similar triangles. Add to your flowchart by making an oval and filling in your conclusion. d. Finally, draw arrows to show the flow of the facts that lead to your conclusion and record the similarity conjecture you used, following Julio's example.

pg.16 4.32 –FLOWCHART ARROWS Lindsay was solving a math problem and drew the flowchart: A  R B  Q given given a. Draw and label two triangles that could represent Lindsay's problem. What question did the problem ask her? How can you tell?

∆ABC ~ ∆RQS AA~ b. Lindsay's teammate was working on the same problem and made a mistake in his flowchart. How is this flowchart different from A  R B  Q Lindsay's? Why is this the wrong way to explain the reasoning in given given Lindsay's problem?

∆ABC ~ ∆RQS AA~

4.33 – HOW MANY OVALS? Ramon is examining the triangles at right. He suspects they may be similar by SSS~. a. Why is SSS~ the best conjecture to test for these triangles? b. Set up ovals for the facts you need to know to show that the triangles are similar. Complete any necessary calculations and fill in the ovals. c. Are the triangles similar? If so, complete your flowchart, using an appropriate similarity statement. If not, explain how you know.

4.34 – START FROM SCRATCH Now examine the triangles at right. a. Are these triangles similar? Justify your conclusion using a flowchart.

b. What is mR C ? How do you know?

pg.17 4.35 – EXTRA INFORMATION NEEDED Determine if you can prove the triangles are similar by AA~ or SSS~. What information are you given? Do you need to find more information to determine if the triangles are similar? What extra information do you need in order to prove these are congruent? Find the missing information and complete the flowchart that uses the given information to state what else we know about the triangles to prove them similar.

RT//QU ∆HIG is a right triangle given U given

given given

∆SRT ~ ∆SQU AA~

GI GH T  Q = 1 = 1 JI JH given given given

∆GHI ~ ∆JHI ∆SRT ~ ∆QRP SSS~ AA~

XW//ZV given

∆ABE ~ ∆DBC AA~

∆XWY ~ ∆ZVY AA~

pg.18 4.6 – What Information Do I Need?______More Conditions for Triangle Similarity

So far, you have worked with two methods for determining that triangles are similar: AA~ and SSS~. Are these the only ways to determine if two triangles are similar? Today you will investigate similar triangles and complete your list of triangle similarity conjectures.

4.36 – ANOTHER WAY Richard's team is using SSS~ Conjecture to show that two triangles are similar. "This is too much work," Richard says. "When we're using the AA~ Conjecture, we only need to look at two angles. Let's just calculate the ratios for two pairs of corresponding sides to determine that triangles are similar."

Is SS~ a vailid similarity conjecture for triangles? That is, if two pairs of corresponding side lengths share a common ratio, must the triangles be similar?

In this problem you will investigate this question using a manipulative or a dynamic geometry tool a. Richard has a triangle with side lengths 5cm and 10cm. If your triangle has two sides that share a common ratio with Richards, does 5 10 your triangle have to be similar to his? b. Kirk asks, "What if the angles between the two sides have the same measure? Would that be enough to know the triangles are similar?" 5 10 c. Kirk calls this the "SAS~ Conjecture," placing the "A" between the two "S"s because the angle is between the two sides. He knows it works for Richard's triangle, but does it work on all other triangles?

4.37 – SSA~ or ASS~ Cori's team put "SSA~" on their list of possible triangle similarity conjectures. Investigae if this is a valid similarity conjecture. If a triangle has two sides sharing a common ratio with Richard's, and has the same angle "outside" these sides, must it be similar?

5 10 5 10

pg.19 4.38 – ANYTHING ELSE? What other triangle similarity conjectures involving sides and angles might there be? List the names of every other possible triangle similarity conjecture you can think of that invovles sides and angles.

4.39 – AAS~ or SAA~ Betsy's team came up with a similarity conjecture they call "AAS~," but Betsy thinks they should cross it off their list. Betsy says, "This similarity conjecture has extra, unnecessary information There is no point in having it on our list." a. What is Betsy talking about? Why does the AAS~ method contain more information than you need?

b. Go through your list of possible triangle similarity conjectures, crossing off all the invalid ones and all the ones that contain unnecessary information. c. How many valid triangle similarity conjectures are there? List them.

4.40 – FLOWCHARTS Lynn wants to show that the triangles are similar. a. What similarity conjecture should Lynn use? b. Make a flowchart showing that these triangles are similar.

pg.20 4.41 – USING SIMILARITY Examine the triangles. a. Are these triangles similar? If so, make a flowchart justifying their similarity. Hint: It might help to draw the triangles seperately first.

25 60

b. Charles has DDCG~ D ECF as the conclusion of his flowchart. Lisa has DCDG~ D CEF as her conclusion. Who is correct? Why?

c. Find all the missing side lengths and all the missing angle measures in the two triangles.

4.42 – FLOWCHARTS Write a flowchart to prove the following triangles are similar. a. b.

pg.21 4.43 – CONCLUSIONS Describe how to show the triangles are similar using the reasons listed below. If ______angles are

______, then the triangles are similar by

AA~. AA~

If ______

sides are ______, then SSS~ the triangles are similar by SSS~.

If ______sides are

______and the angle

SAS~ ______them is

______, then the triangles are

similar by SAS~.

pg.22 4.7 – How Can I Use Equivalent Ratios?______Triangle Similarity and Congruence

By looking at side ratios and at angles, you are now able to determine whether two figures are similar. But how can you tell if two shapes are the same shape and the same size? In this lesson you will examine properties that guarantee that shapes are exact replicas of one another.

4.44 – MORE THAN SIMILAR A E Examine the triangles. 3 D 4 5 5 a. Are these triangles similar? How do you know? Use a flowchart to organize your explanation. 4 B C 3 F

b. Cameron says, "These triangles aren't just similar – they're congruent!" Is Cameron correct? What special value in your flowchart indicates that the triangles are congruent? c. Write a conjecture (in "If...,then..." form) that explains how you know when two shapes are similar.

"If two shapes are ______with a side ratio of ______, the the two shapes are ______." d. Cameron wanted to write a statement to convey that these two triangles are congruent. He started with "∆CAB...", but then got stuck because he did not know the symbol for congruence. Now that you know the symbol for congruence, complete Cameron's statement for him.

BC e. In Don's congruence flowchart for the above problem, one of the ovals said "= 1". In Phil's flowchart, DE one of the ovals said, "BC = DE". Discuss with your team whether these ovals say the same thing. Can equality statements like Phil's always be used in congruence flowcharts?

4.45 – ANOTHER WAY OF PROOF Stephanie is tired of drawing flowcharts because the bubbles can be messy. She A E decides to organize her proof in columns instead. 3 D 4 5 5 Compare your proof in the previous problems with the one below. How do they 4 alike? How are they different? B C 3 F Statements Reasons 1. BC = DE 1. given 2. AB = FD 2. given 3. AC = FE 3. given 4. ∆ABC  ∆FDE 4. SSS

pg.23 4.46 – A QUICKER WAY Examine the triangles at right.

a. Are these triangles similar? Explain your reasoning.

b. Are the triangles congruent? Explain your reasoning.

c. Derek wants to find general shortcuts that can help determine if triangles are congruent. To help, he draws the diagram at right to show the relationship between the triangles in part (a). If two triangles have the relationship shown in the diagram, do they have to be congruent? How do you know?

d. Complete the conjecture below based on this relationship. What is a good abbreviation for this shortcut?

"If two triangles have two pairs of equal ______and the angles between them are ______,

then the triangles are ______.

4.47 – ARE THEY CONGRUENT OR JUST SIMILAR? Determine if the triangles are similar, conngruent, or neither. Justify your answers.

SIMILAR CONGRUENT NEITHER SIMILAR CONGRUENT NEITHER SIMILAR CONGRUENT NEITHER

Reason:______Reason:______Reason:______

SIMILAR CONGRUENT NEITHER SIMILAR CONGRUENT NEITHER SIMILAR CONGRUENT NEITHER

Reason:______Reason:______Reason:______pg.24 4.48 – ARE THERE OTHERS? Derek wonders, "What other types of information can determine that two triangles are congruent?"

Your Task: Examine the pairs of triangles below to decide what other types of informaiton force triangles to be congruent. Notice that since no measurements are given in the diagrams, you are considering the general cases of each type of pairing. For each pair of triangles below that you can prove are congruent, come up with a shortcut name to use for now on.

4.49 – STATE THE CONJECTURE Use your triangle congruence conjectures to state why the following triangles are congruent.

pg.25 4.50 – ARE THEY CONGRUENT? Use your triangle congruence conjectures to determine if the following pairs of triangles must be congruent. Note: the diagrams are not necessarily drawn to scale.

4.51 – CONGRUENCE STATEMENTS Suppose you are working on a problem involving the two triangles ∆UVW and ∆XYZ, and you know that ∆UVW  ∆XYZ. What can you conclude about the sides and angles of ∆UVW and ∆XYZ? Complete each equation below invoving side lengths and angle measures. Drawing a picture might help.

蠤U_____, 蠤 V _____, 蠤 W _____ UV@_____, VW @ _____, UW @ _____

pg.26 4.8 – What Information Do I Need?______Conditions for Triangle Congruence

Now that you have shortcuts for establishing triangle congruence, how can you organize information in a flowchart to show that triangles are congruent? Consider this as you work today with your team.

4.52 – HIDDEN STATEMENTS Sometimes a proof needs a statement that is not given. For example, on Problem 6.14, you needed to state the vertical angles are congruent. Find each hidden fact in the pictures below.

4.53 – STATE YOUR REASON For each picture below, determine if the triangles are similar, congruent, or neither. If they are, write a similarity or congruence statement. Then state your reason. Be sure to mark any hidden facts in the picture. If they are not, explain why not.

pg.27 4.54 – TRIANGLE CONGRUENCE Complete the following proofs. A a. Given: AC bisects DB , AD@ AB Prove: ∆ACD  ∆ACB Statements Reasons

b. Given: DB^ AC, CA bisects DAB Prove: ∆ADC  ∆ACB A Statements Reasons

c. Given: AB DC , D = B

Prove: ∆ABC  ∆CDA A B Statements Reasons

pg.28 4.55 – PROVE TRIANGLES CONGRUENT Complete the proofs below. A a. Given: AC bisects 蠤 DAB , DA BA Prove: ∆ACD  ∆ACB

Statements Reasons

b. Given: ABCD is a parallelogram Prove: ACD  ACB

Statements Reasons

c. Given: C is the midpoint of AD , ED // AB Prove: DEDC @ D BAC Statements Reasons

pg.29 4.9 – What Is Left To Prove? ______Parts of Congruent Triangles

Now that you know how to prove triangles are congruent, how can we prove more about their individual parts?

4.56 – CONGRUENT TRIANGLES When you have proven that two triangles are congruent, what can you say about their corresponding parts?

a. Examine the two triangles at right and the proof below. What is the given? What are you trying to prove? Statements Reasons A 1. ABCD is a kite 1. Given D B 2. AD = AB 2. Definition of Kite 3. DC = BC 3. Definition of Kite 4. AC = AC 4. 5. ∆ABC  ∆ADC 5. C 6. D  B 6.

b. Complete the missing reasons for #4 and #5 above.

c. Notice there is no reason given for Statement #6. Why do you know those angles will be congruent based on this proof?

c. This reason is called "Corresponding Parts of Congruent Triangles Are Congruent." It can be shortened to CPCTC. Or you can write an arrow diagram to show the meaning by stating:  ∆   parts. Complete the reason for the proof above.

4.57 – DIAGONALS OF A RECTANGLE Use the proof below to show that the diagonals of a rectangle are congruent.

Given: ABCD is a rectangle Prove: AC = BD

pg.30 4.58 – DIAGONALS OF A RHOMBUS What can congruent triangles tell us about the diagonals of and angles of a rhombus? Prove that the diagonals of a rhombus bisect the angles.

Given: ABCD is a rhombus Prove: ABD = CBD

4.59 – DIAGONALS OF A RHOMBUS Prove that if one pair of opposite sides are congruent and paralell, the shape is a parallelogram.

Given: AB@ CD, AB // CD Prove: ABCD is a parallelogram A B

pg.31 4.60 – PROOF BY CONTRADICTION Sometimes you cannot prove something directly and need to prove it by disproving other ideas. Come up with a way to disprove the following claims.

a. The product of an odd number and an even number is always odd.

b. A number minus another number will always be smaller.

c. A quadrilateral with perpendicular diagonals is a kite.

d. All quadrilaterals with two pairs of congruent sides is a parallelogram.

e. Interior angles of a pentagon are always 108