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Mathematics 20 Module 3 Lesson 16 Mathematics 20 The Geometry of Triangles 87 Lesson 16 Mathematics 20 88 Assignment 16 The Geometry of Triangles Introduction Geometry includes the study of the properties of shapes. The study of geometry in this course is mainly about triangles. Another basic concept in geometry is the study of arranging concepts in a logical order through definitions of properties and assumptions. This lesson will start with this concept by introducing the Congruence Postulates. Through deductive reasoning, you will also be introduced to the idea of two column proofs. The study of geometry has been happening for years. Such professionals as architects, engineers and drafters use the concepts of geometry every day. This lesson will also show you how to construct congruent triangles, both informally and formally. Irrational Number Math Gordon wants to place a 4-meter pole completely inside a box with dimensions 3 m by 3 m by 1.5 m. Can this be done? Mathematics 20 89 Assignment 16 Mathematics 20 90 Assignment 16 Objectives After completing this lesson you will be able to • informally and formally construct congruent angles and congruent triangles. • determine the properties of congruent triangles. • identify and state the corresponding parts of congruent triangles. • determine whether triangles are congruent by SSS, SAS, ASA, AAS, HL or LL properties. • prove that two triangles are congruent by supplying the statements and reasons in a guided deductive proof. Mathematics 20 91 Assignment 16 Mathematics 20 92 Assignment 16 16.1 Congruent Triangles A short review of triangles is necessary to begin this lesson. A triangle can be classified by the relationship between the sides of the triangle or the angles of the triangle. Classification by Sides Equilateral Triangle Isosceles Triangle Scalene Triangle An equilateral triangle has three congruent sides. An isosceles triangle has two congruent sides. A scalene triangle has no congruent sides. Classification by Angles Acute Triangle Equiangular Triangle An acute triangle has three acute angles. (Each angle is less than 90 ) An equiangular triangle is an acute triangle with all three angles congruent. Right Triangle A right triangle has exactly one right angle. Obtuse Triangle An obtuse triangle has exactly one obtuse angle. (greater than 90 0 and less than 180 0 ). Mathematics 20 93 Assignment 16 Two figures that have the exact same size and shape are congruent. Activity 16.11 In each of the following puzzles, determine which two are congruent. If necessary, trace the figures and slide them over one another. The figures may be flipped or turned in order to be a match. The symbol for congruence is and is read "is congruent to". If two segments have the same measure or length, they are congruent segments. AB = 3 cm CD = 3 cm AB = CD therefore AB CD . Notation: Notice that AB represents a number but AB represents the set of points which makes the line. It is incorrect to write AB CD or AB CD Congruent segments are segments which have the same measure. Mathematics 20 94 Assignment 16 If two angles have the same measure, they are congruent angles. m RST = 20 m XYZ = 20 RST XYZ Congruent angles are angles that have the same measure. You will notice that in order to place triangle ABC on triangle XYZ, you must line up vertices A and X. This means that these two vertices are corresponding vertices. The angles at these vertices also correspond. Three pairs of sides and three pairs of angles correspond. A X B Y C Z AB XY BC Y Z AC X Z The same can be shown by the way the triangles are labelled. The corresponding vertices are arranged in the same order. ABC XYZ Mathematics 20 95 Assignment 16 Congruent Triangles If ABC is congruent to JKL , then the corresponding angles and the corresponding sides of the two triangles are congruent. The congruent corresponding parts are marked by a single line, double lines and triple lines. Corresponding angles are: A J • B K • C L • • • • Corresponding sides are: AB JK BC KL AC JL Also note that in the same way, congruence can be established in polygons. Two polygons with the same number of sides have corresponding sides and angles. This correspondence is determined by the way that the two polygons are labelled. pentagon EFGHI pentagon MNOPQ . This is true if the corresponding angles and the corresponding sides are congruent. Mathematics 20 96 Assignment 16 Activity 16.12 On each of the coordinate planes plot the required points to make the indicated triangles congruent. 1. 2. ABC LMN 1 ABC LMN DEF XYZ 3. 4. DOG CAT 2 EFG OPQ DOG CAT Mathematics 20 97 Assignment 16 Example 1 Name all of the congruent parts of the following two triangles. Solution: Angles: • • Z H A N • Q B Sides: • ZA HN AQ NB • • ZQ HB What can you say about the two triangles? Mathematics 20 98 Assignment 16 This shows that the definition of congruence is reversible. If two triangles are congruent, then the pairs of corresponding angles and sides are congruent. If the pairs of corresponding angles and sides of a triangle are congruent, Example 2 then the triangles are congruent. If FGH TUV , draw a diagram and mark the diagram to show the congruent parts. Solution: Example 3 From the following corresponding congruent parts, state the triangles that are congruent. a) b) AB DE , BC EJ , CA JD , A D, B E, C J WB RF , BY FP , YW PR , W R, B F, Y P Solution: a) b) ABC DEJ WBY RFP Mathematics 20 99 Assignment 16 Exercise 16.1 1. In the diagram, UVW PQR . State which sides are congruent. 2. If RUN CAB , state which sides are congruent. 3. Correctly write the symbols which state that the two triangles are congruent. 4. a. b. Is ABC DEF the same as BCA EFD ? Justify your answer. Is ABC DEF the same as ABC FED ? Justify your answer. Mathematics 20 100 Assignment 16 5. For each diagram correctly write the symbols which state that certain triangles are congruent. In some cases there are more than one set of congruent triangles. a. c. b. d. e) Mathematics 20 101 Assignment 16 16.2 Exploring Congruent Triangles Sometimes it is helpful to describe the parts of a triangle in terms of their relative positions. • • • • AB AB A A is opposite C . is included between A and B . is opposite BC . is included between AB and AC . Besides the properties or basic terms and definitions associated with geometry, there are also certain postulates and theorems that are used. A postulate is a statement that is accepted as being true without proof. A theorem is a statement that must be proved before it is accepted as being true. It is not always necessary to have all three sides and all three angles congruent to say that two triangles are congruent. The minimum requirements for the proving that two triangles are congruent are stated in the congruence postulates. Side-Side-Side (SSS) Congruence Postulate If three sides of one triangle are congruent, respectively, to three sides of a second triangle, then the two triangles are congruent. Mathematics 20 102 Assignment 16 Side-Angle-Side (SAS) Congruence Postulate If two sides and the included angle of one triangle are congruent, respectively, to two sides and the included angle of a second triangle, then the two triangles are congruent. Angle-Side-Angle (ASA) Congruence Postulate If two angles and the included side of one triangle are congruent, respectively, to two angles and the included side of a second triangle, then the two triangles are congruent. Angle-Angle-Side (AAS) Congruence Postulate If two angles and the non-included side of one triangle are congruent to two angles and the non-included side of a second triangle, then the two triangles are congruent. Mathematics 20 103 Assignment 16 Example 1 For each of the pairs of triangles, state the congruence postulate, if any, that will prove the triangles congruent. Solution: a) b) ASA Postulate - The congruent side is included between the congruent angles. SAS Postulate - Two sides and the included angle of one triangle are congruent, respectively, to the two sides and the included angle of a second triangle. SSS Postulate - Two sides and the common side are congruent. None - There is not an AAA Postulate. c) d) Activity 16.2 Why is there not an AAA Postulate? Use a protractor and ruler for the following constructions. Triangle One • • • • • Draw a line 4 cm long. Label the line RS. Construct an angle of measure 900 at point R. Construct an angle of measure 300 at point S. The point where the two lines join will be called T. You now have RST with angles measuring 30, 60 and 90 degree. Mathematics 20 104 Assignment 16 Triangle Two • • • • • Draw a line 6 cm long. Label the line GH. Construct an angle of measure 900 at point G. Construct an angle of measure 300 at point H. The point where the two lines join will be called I. You now have GHI with angles measuring 30, 60 and 90 degree. What can you say about the angles in these two triangles? What can you say about the two triangles? Is RST GHI ? (Remember the definition of congruence triangles) Example 2 Each pair of triangles below has two corresponding sides or angles marked as congruent. Indicate the additional information that is needed to enable you to apply the specified congruence postulates. a) For ASA b) For SAS c) For SAS d) For SSS e) For AAS f) For ASA Mathematics 20 105 Assignment 16 Solution: a) b) c) d) e) f) M R NO ST C F AB DE K Z T Y An angle is needed on the other side of the included side. A side is needed on the other side of the included angle. The angle has to be included between the two sides. A third side is necessary. The side is not included between the two angles. The side is included between the two angles. Equivalence relations play a role in understanding the properties of congruences. The following chart will show you the properties of reflexivity, symmetry and transitivity. This course will utilize the properties of segment congruence and angle congruence. Property Number Equality Reflexive a=a Symmetric Transitive Mathematics 20 Segment Congruence Angle Congruence AB AB ABC ABC If a = b, then b = a If AB CD , then CD AB If ABC XYZ then XYZ ABC If a = b and b = c, then a=c If AB CD and CD EF , then AB EF If ABC XYZ and XYZ RST then ABC RST 106 Assignment 16 Here are some examples where the reflexive property can be used. Statement Reason AD AD Reflexive Property of Congruence X X Reflexive Property of Congruence CD CD Reflexive Property of Congruence Exercises 16.2 1. Are the triangles congruent? Justify your answers by stating the congruence postulate that applies. a. b. Mathematics 20 107 Assignment 16 c. d. 2. If the triangles are congruent, state the postulate which justifies your answer. If the triangles are not congruent, state the additional information needed to use a certain postulate to make the triangles congruent. a. ABD, CBD b. AEB, DEC c. AEC, DEB d. ABC, DCB Mathematics 20 108 Assignment 16 3. Complete the statements for the given diagram. ADC BDC because of the -------- Postulate. It is given that two pairs of sides are ---------, and for the third side, CD CD because of the --------Property of congruence of segments. 4. If ABC DEF and DEF GHI , why is AB GH ? 16.3 Proving Triangles Congruent Our minds are often called upon to make decisions. Sometimes these decisions require a great deal of thought and certain steps must be followed to reach the outcome. The thought process in these situations is very important. Mathematics 20 109 Assignment 16 Activity 16.3 Mr. Boyes, Mr. Banadyga, Mr. Crow and Mr. Petersen are gentlemen living in rural Saskatchewan. They live in Meadow Lake, Kamsack, Estevan and Ponteix. Use the table and clues to determine where each of the gentlemen live. a) Mr. Boyes is a business partner of the men living in southern Saskatchewan. b) Mr. Banadyga is a cousin of the men living in Kamsack and Estevan. c) Mr. Crow and Mr. Petersen just visited northern Saskatchewan. d) Mr. Petersen does not live in a city. Mr. Boyes Mr. Banadyga Mr. Crow Mr. Petersen Meadow Lake Kamsack Ponteix Estevan This has been an example of logical thinking. The most important thing to remember is that a certain thought process must occur in able for you to solve a problem. This is the same with proofs in Geometry. Many of the steps in the process of solving geometric proofs use deductive reasoning. Deductive reasoning is reasoning that begins with a preliminary statement. Other statements can be logically developed from this statement through the process of deduction. Mathematics 20 110 Assignment 16 Deductive reasoning is often in the form "If .... then" where the "if" part is the preliminary statement and the "then" parts are the logical statements that follow from the preliminary statement. • • • If it rains out, then the class will not being going on their field trip. If a person drives a car in Saskatchewan, then he must wear a seat belt. If a triangle has two congruent sides, then it is isosceles. Example 1 Express each of the following statements in the "If ... then" form. a) b) c) d) When a b is a positive number, b is less than a. A square with a side length of 2 cm has a perimeter of 8 cm. Any two right angles are congruent. Two circles which have congruent radii have equal areas. Solution: a) b) c) d) If a b is a positive number, then b is less than a. If a square has a side length of 2 cm, then it has a perimeter of 8 cm. If two angles are right angles, then they are congruent. If two circles have congruent radii, then they have equal areas. In this form of deductive reasoning, the "if" statement is the hypothesis, and the "then statement is the conclusion. If Then Hypothesis Conclusion When the hypothesis and the conclusion of a statement are reversed, the converse statement is produced. This reversal does not always result in a correct statement. Example 2 Take the converse of the statements in Example 2, and determine whether the converse is true or false. a) b) c) d) Mathematics 20 If a b is a positive number, then b is less than a. If a square has a side length of 2 cm, then it has a perimeter of 8 cm. If two angles are right angles, then they are congruent. If two circles have congruent radii, then they have equal areas. 111 Assignment 16 Solution: a) If b is less than a, then a b is a positive number. True b) If a square has a perimeter of 8 cm, then it has a side length of 2 cm. True c) If two angles are congruent, then they are right angles. False - Two congruent angles are not necessarily right angles. d) If two circles have equal areas, then they have congruent radii. True Problem solving steps and your ability to use deductive reasoning will help you to prove that two triangles are congruent. The problem solving steps are the same ones that you have been using throughout this course. • Read the problem. • Develop a plan. • Carry out the plan. • Look back. Any time that you are asked to prove that two triangles are congruent, you will be given information about the two triangles. You will need to take this information and develop a way to use the information in order to prove the two triangles congruent. You have learned four ways to prove that two triangles are congruent: • SSS Postulate • SAS Postulate • ASA Postulate • AAS Postulate In each of these cases, you need three congruences of either angles or sides to prove that the triangles are congruent. It is important to organize the steps that are necessary in proofs. In this course the majority of proofs will be done using two columns. • • The first column will include the statements and show the logical steps that will result in the proof of the triangles being congruent. The second column will list the reasons why each statement is true. Mathematics 20 112 Assignment 16 Example 3 ABC and KLM Given: AB KL , A K, B L ABC KLM Prove: Solution: Read the problem. Given: ABC and KLM AB KL , A K, B L Prove: ABC KLM Develop a plan. Mark the congruent parts on the two triangles. AB KL (Side) A K (Angle) B L (Angle) Determine the congruence postulate that can be applied. ASA Postulate Carry out the plan. 1. 2. 3. 4. Statement Reason AB KL A K B L ABC KLM 1. 2. 3. 4. Given Given Given ASA Postulate Look Back The final statement of the proof will always be what was originally asked to be proven. In this example, the last step was that the triangles were in fact congruent, and this was determined by using the ASA Postulate. Mathematics 20 113 Assignment 16 Example 4 Given: ABC AB AC , BD CD Prove: BAD CAD Solution: Read the problem. Given: ABC AB AC , BD CD Prove: BAD CAD Develop a plan. Mark the congruent parts on the two triangles. AB AC BD CD Mark the common side between the two triangles. AD AD Determine the congruence postulate that can be applied. SSS Postulate Carry out the plan. 1. 2. 3. 4. Statement Reason AB AC BD CD AD AD BAD CAD 1. 2. 3. 4. Given Given Reflexivity property SSS Postulate Look back. The final statement showed that the triangles were proven congruent by the SSS Postulate. Many times, a common angle or side can be the third pair of corresponding parts when proving triangles congruent. Mathematics 20 114 Assignment 16 Exercise 16.3 1. Write the converse of each statement and then state whether it is true or not. a. b. c. d. e. f. g. If two angles are right angles, then they are congruent. If two segments have the same measure, then they are congruent. For real numbers, if a = b , then b = a . If two angles are vertically opposite angles, then they are congruent. If two triangles are congruent, then the three corresponding pairs of angles of the triangles are congruent. If a triangle is isosceles, then it is equilateral. Given two supplementary angles, if they are right angles, then they are congruent. Write a two column proof for each of the following problems. Include GIVEN and PROVE statements. 2. If segments AD and CB bisect each other, prove that ABE DCE . 3. Prove that ADC BDC Mathematics 20 115 Assignment 16 4. Prove that ABD ACD . 5. Prove that ABC AED . 16.4 Constructing Congruent Angles and Triangles There are many different ways of constructing angles and triangles informally. • • • • paper folding mira tracing ruler and protractor Mathematics 20 116 Assignment 16 Activity 16.41 Given: ABC Construct: DEF such that ABC DEF by tracing. • • • • Cut out the triangle. Trace the triangle on another piece of paper. Label the new triangle. List the congruent sides. Activity 16.42 Given: MNO Construct: XYZ such that MNO XYZ by paper folding. • • • • • Choose one arm of the angle. Fold over the paper using this arm as the edge. Trace over top of the other arm. Label the new angle. List the angle so that it corresponds with the congruence. There are also formal methods of proof that are more exact. Informal methods may be easier, but are sometimes not as accurate. The formal method of proof that is used is a straight edge and compass. This course will review the construction of congruent angles from Mathematics 10, and then explore constructing congruent triangles by utilizing the knowledge that has been developed from the congruence postulates. Mathematics 20 117 Assignment 16 Constructing Congruent Angles Example 1 Construct an angle congruent to the following angle A using a straightedge and compass. Solution Step 1 Step 2 Use a straightedge to draw a ray and label its endpoint D. With the point of the compass at A, draw an arc through both arms, labelling points B and C, of the original angle. Using the same compass setting or distance, and with the point of the compass on D, draw an arc crossing the new ray. Label this point E. Mathematics 20 118 Assignment 16 Step 3 Step 4 Step 5 Set your compass to the length of BC . With the point of the compass on E and with the distance BC , draw an arc through the arc containing E. Label the intersection of these two arcs F. With the straightedge, draw the ray that has D as its endpoint and passes through point F. FDE is now congruent to CAB . Constructing Congruent Triangles Constructing a congruent angle or triangle means that the new angle or triangle has the exact same measure as the original one. Example 1 Given: Segments measuring 3 cm, 4 cm and 5 cm Construct: ABC with side lengths measuring these segments. Mathematics 20 119 Assignment 16 Solution: Step 1 Use a straight edge to draw a ray and label its endpoint A. Step 2 Set the compass to the length of the 5 cm segment. With the point of the compass on point A, draw an arc with a length of 5 cm. Label this point B. Step 3 Set the compass to the length of the 4 cm segment. With the point of the compass on point A, draw an arc with a length of 4 cm above AB . Step 4 Set the compass to the length of the segment which is 3 cm. With the point of the compass on point B, draw an arc with a length of 3 cm above AB . Mathematics 20 120 Assignment 16 Label the point C where these two arcs intersect. Join AC and BC . Step 5 ABC is a triangle with side lengths of 3 cm, 4 cm and 5 cm. We can construct a congruent triangle that has sides of these lengths because the SSS Congruence Postulate, which states all triangles with congruent sides will be congruent. Example 2 Given: RST Construct: JKL such that RST JKL using the SAS Postulate. Solution: Read the problem. The measures of two sides and the included angle of RST must be congruent to two sides and the included angle of JKL . Develop a plan. (Here is one plan of three) • • • Construct a segment congruent to RS . Construct a segment congruent to ST . Construct the included angle congruent to RST . Mathematics 20 121 Assignment 16 Carry out the plan. Construct the angle first. Step 1 Use a straight edge to draw a ray and label the endpoint K. Step 2 Construct an angle congruent to RST with the vertex at K. Follow the instructions from "Constructing Congruent Triangles". Step 3 With the compass, measure the length of RS . With the point of the compass at K, draw an arc with the length of RS on the arm of the angle. Label the point J where the arc crosses the arm. J K Step 4 L With the compass, measure the length of ST . With the point of the compass at K, draw an arc with the length of ST on the ray. Label this point L. J K Mathematics 20 L 122 Assignment 16 Step 5 Join JL . J You now have RST JKL using the SAS Postulate. K L Exercise 16.4 1. Given A and B construct, with a compass and straight edge, both angles adjacent to each other to form angle C. With a protractor find the measure of each angle and the measure of the constructed combined adjacent angles (angle C). Write a general statement about your observations. 2. Given A and B construct an angle which is the supplement of the sum of the measures of angles A and B. Use the same angles as those given in Question 1. 3. Given two angles and the included side construct the DEF . State the postulate which assures that there is only one such triangle. Use the same angles as those given in Question 1. Mathematics 20 123 Assignment 16 4. Given two angles and a non included side construct a triangle. State the postulate which assures that there is only one such triangle. Hint: The measures of the angles of a triangle must add up to 180 . First construct the supplement of the sum of the two given triangles. Mathematics 20 124 Assignment 16 Conclusion In the introduction you were given a Irrational Number Math problem. The solution to this problem is: Yes 2 2 2 3 + 3 + 1. 5 = 22 .25 4 .7 The 4 m pole will fit diagonally. Note: A summary of acceptable reasons for proofs can be found in the Appendix on page 381. Summary The following is a list of concepts that you have learned in this lesson: • Congruent segments are segments which have the same measure. • Congruent angles are angles which have the same measure. • Two triangles are congruent if the corresponding angles and sides of the two triangles are congruent. • The Congruence Postulates are: Side-Side-Side (SSS) Congruence Postulate If three sides of one triangle are congruent, respectively, to three sides of a second triangle, then the two triangles are congruent. Side-Angle-Side (SAS) Congruence Postulate If two sides and the included angle of one triangle are congruent, respectively, to two sides and the included angle of a second triangle, then the two triangles are congruent. Angle-Side-Angle (ASA) Congruence Postulate If two angles and the included side of one triangle are congruent, respectively, to two angles and the included side of a second triangle, then the two triangles are congruent. Angle-Angle-Side (AAS) Congruence Postulate If two angles and the non-included side of one triangle are congruent, respectively, to two angles and the non-included side of a second triangle, then the two triangles are congruent. Mathematics 20 125 Assignment 16 • Equivalence relations have the properties of reflexivity, symmetry and transitivity. • Deductive reasoning is reasoning that begins with a preliminary statement. Other statements can be logically developed from this statement through the process of deduction. Deductive reasoning is often in the form "if ... then". • Triangles can be constructed through both informal and formal methods. The formal method of construction is done with a straight edge and compass. • The Congruence Postulates can also be used in the construction of congruent triangles Mathematics 20 126 Assignment 16 Answers to Exercises Exercise 16.1 1. UV PQ VW QR UW PR 2. RU CA UN AB RN CB 3. ABC RTS 4. a. b. 5. Mathematics 20 Yes. In both cases the same pairs of sides and angles are congruent. No. The same pairs of sides and vertices are not congruent. For example, in the first case A D , but in the second case A F . a. ADC BDC b. AEB CED c. ABD ABE BAC DAE d. ADE BCE ADB BCA AEB BEA e. BDF BDC ABE BFC BAC ACE ACD CAB EAD CEF CEB ACD CFB CAB 127 Assignment 16 Exercise 16.2 1. a. b. c. d. Yes, No, Yes, Yes, 2. a. b. Yes, SSS, Yes, SAS, congruent. Yes, SSS, Yes, ASA, c. d. SAS The angle is not included between the two sides. AAS ASA 3. SSS, congruent, 4. AB DE DE GH AB GH BD BD because it is a common side. BEA CED because vertically opposite angles are AC DB because BC is common to AC and DB . BC BC because it is a common side. reflexive since ABC DEF . since DEF GHI . by transitive property of congruent segments. Exercise 16.3 1. a. b. c. d. e. f. g. 2. If two angles are congruent, then they are right angles. False If two segments are congruent, then they have the same measure, True For real numbers, if b = a, then a = b . True If two angles are congruent, then they are vertically opposite. False If three corresponding pairs of angles of two triangles are congruent, then the triangles are congruent. False If a triangle is equilateral, then it is isosceles. True Given two supplementary angles, if they are congruent, then they are right triangles. True Given: AD and CB bisect each other. Prove: ABE DCE Proof: Mathematics 20 1. Statement AEB DEC 2. 3. CE BE , AE DE ABE DCE Reason 1. Vertically opposite angles are congruent. 2. The segments bisect each other. 3. SAS Postulate 128 Assignment 16 3. Given: AD BD ADC and BDC Prove: ADC BDC Proof: 4. 1. 2. 3. Statement AD BD ADC BDC CD CD 4. ADC BDC Given: AB AC BD DC Prove: ABD ACD Proof: 5. 1. 2. 3. Statement AB AC BD DC AD AD 4. ABD ACD Given: ABC AED BC ED Prove: ABD AED Proof: Mathematics 20 are right angles. 1. 2. 3. Statement ABC AED BC ED BAC EAD 4. ABC AED Reason 1. Given 2. Right angles are congruent. 3. Reflexive Property of Congruence 4. SAS Postulate Reason 1. Given 2. Given 3. Reflexive Property of Congruence 4. SSS Postulate Reason 1. Given 2. Given 3. Vertically opposite angles are congruent 4. AAS Postulate 129 Assignment 16 Exercise 16.4 1. with compass and straight edge measuring with protractor m A 32 m B 52 m A m B 84 The constructed angle (combined adjacent angles) and the sum of angle A and angle B are both 84°. 2. Mathematics 20 130 Assignment 16 3. Steps 1. Draw DG 2. At D construct DE such that DE AB . 3. Construct angles at D and E such that D A and E B . 4. Extend the arms of the angles to intersect at F. DEF is constructed using the ASA congruence postulate. 4. Mathematics 20 131 Assignment 16 With the information given we could use the AAS congruence postulate but, we are not given enough information to confidently construct the triangle. We must first find the third angle ( C ) and use the ASA congruence postulate using A and C with included side AC . See question 3 for detailed steps for ASA. Mathematics 20 132 Assignment 16 Mathematics 20 Module 3 Assignment 16 Mathematics 20 133 Assignment 16 Mathematics 20 134 Assignment 16 Optional insert: Assignment #16 frontal sheet here. Mathematics 20 135 Assignment 16 Mathematics 20 136 Assignment 16 Assignment 16 Values (40) A. Multiple Choice: Select the best answer for each of the following and place a () beside it. Your calculations will not be evaluated and need not be shown. 1. The statement that the given two triangles are congruent is ***. ____ ____ ____ ____ 2. a. b. c. d. RT RT RS ST XY XZ YZ ZX a. b. c. d. AB IH BC HI AB GH ABC GHI If ABC DFE and DEF GHI , then ***. ____ ____ ____ ____ Mathematics 20 DEF EFD FED FDE If ABC FED and DEF GHI , then ***. ____ ____ ____ ____ 4. ABC ABC ABC ABC If RST XYZ, then ***. ____ ____ ____ ____ 3. a. b. c. d. a. b. c. d. ABC ABC ABC ABC HIG HGI IHG GIH 137 Assignment 16 5. 6. 7. In ABC ***. ____ a. ____ ____ b. c. ____ d. For the given triangles ***. ____ a. ____ b. ____ ____ c. d. a. b. c. d. m F 5 4 m D 7 0 m E 5 6 m F 5 6 For ABC to be congruent to DFE it is sufficient to have ***. ____ ____ ____ ____ Mathematics 20 ABC DEF by SSS ABC DEF by SAS ABC DEF by ASA the triangles need not be congruent If m A = 56 , m B = 70 , and ABC DEF, then ***. ____ ____ ____ ____ 8. A is included between B and C. A is opposite B and C. A is included between AB and AC A is opposite BA a. b. c. d. AB BC AB E DF FD DE C 138 Assignment 16 9. In the statement, "if two circles are concentric, they have the same center", the phrase "...they have the same center" is the ***. ____ ____ ____ ____ 10. 11. ____ a. ____ b. ____ c. ____ d. If one angle is acute, then the other angle is obtuse in a linear pair If the angles form a linear pair, then they are acute and obtuse If one angle is acute and the other angle is obtuse, they form a linear pair Given a linear pair, if one angle is obtuse, the other is acute In the theorem "If one angle of a linear pair is acute, then the other is obtuse” the Given statement is ***. a. b. c. d. A, B form a linear pair A, B form a linear pair, A is acute one angle is acute one angle is obtuse The postulate that can be used to prove that the triangles are congruent is ***. ____ ____ ____ ____ Mathematics 20 congruence hypothesis conclusion converse The converse of the statement "If one angle of a linear pair is acute, then the other angle is obtuse." is "***". ____ ____ ____ ____ 12. a. b. c. d. a. b. c. d. ASA SAS SSA SSS 139 Assignment 16 13. The postulate that can be used to prove that the triangles are congruent is ***. ____ ____ ____ ____ 14. a. b. c. d. SSS SAS ASA AAS Given that ACB DEF , the length of EF is ***. ____ ____ ____ ____ Mathematics 20 ASA SSS SAS AAS The postulate that can be used to prove that the triangles are congruent is ***. ____ ____ ____ ____ 15. a. b. c. d. a. b. c. d. 6 2 72 170 5 34 140 Assignment 16 16. The least common multiple of 4 x , 8 x 2 , and 12 x 3 is ***. ____ ____ ____ ____ 17. 18. 19. Mathematics 20 a. b. c. d. 4x 24 x 3 384 x 6 48 x 3 The simplified form of ____ a. ____ b. ____ c. ____ d. 1 x 1 2x is ***. 1 x 1 2x 1 + 2x x 1 1 2 x 1 2x + 1 x 1 The solution to 6 x 2 + 11 x + 3 = 0 is ***. ____ a. ____ b. ____ c. ____ d. 0, 3 2 , 3 3 3 1 , 2 3 3 1 , 2 3 The solution to 49 x 2 81 = 0 is ***. ____ a. ____ b. ____ c. ____ d. 7 9 9 7 81 49 49 , 81 141 Assignment 16 20. The solution to 2 x( x 5) = x 2 8 x 1 is ***. ____ ____ ____ ____ Mathematics 20 a. b. c. d. x= x= x= x= 1 5 0, 5 1 142 Assignment 16 Part B can be answered in the space provided. You also have the option to do the remaining questions in this assignment on separate lined paper. If you choose this option, please complete all of the questions on the separate paper. Evaluation of your solution to each problem will be based on the following. B. • A correct mathematical method for solving the problem is shown. • The final answer is accurate and a check of the answer is shown where asked for by the question. • The solution is written in a style that is clear, logical, well organized, uses proper terms, and states a conclusion. Complete the two column proof for each of the following problems by completing a, b, c, and d. (8) 1. Prove that the given isosceles triangles are congruent. Given: a. Isosceles ABC and DEF with C __________ AB __________ BC __________ DE __________ Prove: b. ___________________________________________________ Proof 1. 2. 3. 4. 5. Mathematics 20 Statement B C , E F C F B E BC EF ABC DEF Reason 1. c. _____________________ 2. Given 3. d. _____________________ 4. Given 5. ASA Postulate of Congruence 143 Assignment 16 (8) 2. If ABC is an isosceles triangle with base BC , and D is any point on the bisector of A in the interior of ABC , prove that ADB ADC . A C B Proof: 3. Statement Reason 1. AB AC 1. a. _____________________ 2. BAD CAD 2. b. _____________________ 3. AD AD 3. c. _____________________ 4. ADB ADC 4. d. _____________________ Suppose that AB and CD are perpendicular bisectors of each other at point E. Prove that CEB DEA . a. Given: Proof Mathematics 20 AB AC ADB ADC Prove: D (16) Isosceles ABC B C Given: a _____________________________________________ Statement Reason 1. CE DE 1. b. _____________________ 2. c. __________________ 2. Definition of segment bisector. 3. AED BEC 3. d. _____________________ 4. CEB DEA 4. SAS Postulate 144 Assignment 16 Prove that CEB DEB . b. Given: Proof (8) 4. a _____________________________________________ Statement Reason 1. CE DE 1. a. _____________________ 2. CEB DEB 2. b. _____________________ 3. EB EB 3. c. _____________________ 4. CEB DEB 4. d. _____________________ Prove that a diagonal of a rectangle divides the rectangle into two congruent triangles. A D Given: Rectangle ABCD and diagonal AC Prove: a. B Proof 1. b. Statement Reason _____________________ 1. Opposite sides of a rectangle are congruent. _____________________ Mathematics 20 _______________________ C 2. AC AC 2. c. _________________________ 3. ABC CDA 3. d. _________________________ 145 Assignment 16 (10) C. 1. Complete the statements in the proof of this theorem. The HL (hypotenuse leg) theorem for right triangles. If one pair of corresponding sides are congruent and the hypotenuse of one triangle is congruent to the hypotenuse of the other triangle, the right triangles are congruent. Given: In right triangles ABC and DEF , AC DF and BC EF . Prove: ____________________________ Proof: Statement Mathematics 20 Reason 1. AC DF 1. ________________ 2. BC EF 2. ________________ 3. AB 2 BC 2 AC 2 3. ________________ 4. _______________ 4. Pythagorean Theorem 5. _______________ 5. Subtraction 6. DE 2 DF 2 EF 2 6. _________________ 7. DE 2 AC 2 BC 2 7. _________________ 8. AB 2 DE 2 8. _________________ 9. ________________ 9. Square root of both sides 10. AB DE 10. _________________ 11. _________________ 11. SSS postulate 146 Assignment 16 Write a two column proof for each of the following problems. Include a diagram and the given and prove statements. (5) 2. Prove the following theorem. The LL (leg leg) theorem for right triangles. If the corresponding legs of two right triangles are congruent, then the triangles are congruent. D A C B E Given: Right triangles ABC and DEF AB DE , BC EF Prove: ABC DEF F Proof Statement Mathematics 20 Reason 147 Assignment 16 (5) 3. Make use of the HL or LL theorems to prove the following. The perpendicular from the vertex to the base of an isosceles triangle divides the triangle into two congruent triangles. 100 Mathematics 20 148 Assignment 16