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Conditions for Congruence Pre-Algebra Day 1 Warmup: (Use Questions 1 and 2 to review vocabulary) 1. Draw an acute angle. How do you know it is an acute angle? 2. Draw an obtuse angle. How do you know it is an obtuse angle? Now let them work on some for themselves. 3. For each of the following pictures, determine if it is: ‘bigger’ ‘smaller’ or ‘equal’ to 90°. Be prepared to defend your answer. (Note: #7 has 3 angles, and #8 has 2.) Display OT1: Name that Angle Closure: let students discuss answers. Remind them of the vocabulary – acute, obtuse, right. Activity: Classifying triangles Put students in groups of 2-3, and give each group a bag with a set of cut-out triangles in them, a protractor and a centimeter ruler. (The questions are duplicated on OT2 for your convenience.) 1. Which of the shapes are obtuse triangles? 2. Which of the shapes are right triangles? 3. Which of the shapes are acute triangles? 4. Which of the shapes are equilateral triangles? 5. Which of the shapes are isosceles triangles? 6. Which of the shapes are scalene triangles? 7. Which of the shapes are EXACTLY the same triangle? (These are called congruent.) How do you know that they are the same? 8. Why can’t these triangles be congruent? a) C and H b) E and I c) G and I d) A and G 9. How will you know when two triangles are congruent? Activity: “Moves”, “Flips”, “Spins” and congruence Repeat this part of the activity twice -- using shapes B and E the first time, and shapes F and J the second time. (The questions are duplicated on OT3 for your convenience.) 1. Place the two triangles directly on top of each other. (Call this the HOME position.) How can that help to convince you that they are congruent? 2. Move the top triangle over to the right and up until it no longer touches the bottom one. a. Sketch a drawing of the two triangles. b. Are the two triangles congruent? 3. Take the triangle you moved and flip it over along one of its sides. a) Sketch a drawing of the two triangles now. b) Are the two triangles congruent? 4. Now take that same triangle and spin it around one of its corners just a little bit. a) Sketch a drawing of the two triangles now.. b) Are the two triangles congruent? 5. If you start with two congruent triangles, a) Will moving one affect whether they are congruent? (translation) b) Will flipping one affect whether they are congruent? (reflection) c) Will spinning one affect whether they are congruent? (rotation) Activity: Pair Up Equal Parts The given triangles are congruent; students need to name vertex points, angles and sides that correspond. Use OT4: Pair Up Equal Parts Homework: Congruence I Name that Angle ACUTE, RIGHT, or OBTUSE #1. #2. #3. #4. #5. #6. #7. #8. A G H D E B I J C F Classifying triangles 1. Which of the shapes are obtuse triangles? 2. Which of the shapes are right triangles? 3. Which of the shapes are acute triangles? 4. Which of the shapes are equilateral triangles? 5. Which of the shapes are isosceles triangles? 6. Which of the shapes are scalene triangles? 7. Which of the shapes are EXACTLY the same triangle? (These are called congruent.) How do you know that they are the same? 8. Why can’t these triangles be congruent? a) C and H b) E and I c) G and I d) A and G 9. How will you know when two triangles are congruent? “Moves”, “Flips”, “Spins” and Congruence 1. Place the triangles directly on top of each other. How can that help to convince you that they are congruent? 2. Move the top triangle over to the right and up until it no longer touches the bottom one. a) Sketch a drawing of the two triangles. b) Are the two triangles congruent? 3. Take the triangle you moved, and flip it over along one of its sides. a) Sketch a drawing of the two triangles now. b) Are the two triangles congruent? 4. Now take that same triangle and spin it around one of its corners just a little bit. a) Sketch a drawing of the two triangles now. b) Are the two triangles congruent? 5.If you start with two congruent triangles: a) does moving one affect whether they are congruent? (translation) b) does flipping one affect whether they are congruent? (reflection) c) does spinning one affect whether they are congruent? (rotation) Pair Up Equal Parts Directions: The two triangles are known to be congruent. Determine the vertex points, angles and sides that match up. A C B D A E C D B A D C B Congruence I 1. 2. Hint: What do you know about the angles in a triangle? 3. Conditions for Congruence Pre-Algebra Day 2 Warmup -- Use OT1: Congruence and Area 1. A rectangle is formed by putting two congruent squares adjacent to each other. Then a triangle is formed by drawing a line from one corner to the opposite side, and then to the other corner as shown. Explain why the triangle must have an area which is one half the area of the rectangle. Provide a hint by drawing an auxiliary line to form an altitude for the triangle. 2. “If two triangles have the same area, then they are congruent.” Is this a true statement? If it is, provide an argument for why it must be true. If it is not, then find an example to support your answer. Activity: Conditions for Congruence Purpose: Students use paper strips, brads and cut-out angle measures to investigate the sufficient (SSS, SAS, ASA, AAS) and insufficient (SSA and AAA) conditions for proving triangles congruent. Materials (per group): 6 paper strips of different lengths (out of 3 possible) made from computer paper edges 6 angle cut-outs (out of 3 possible) made from transparency acetate 6 brads Introduction – Use OT2: Building Congruent Triangles Use this to reinforce the instructional outcome of yesterday’s lesson – that when two triangles are congruent, the corresponding parts match. Notice that it is stated as part of a definition, and that it is technically the converse statement: when all corresponding parts match, the two triangles are congruent. Activity Directions – Use OT3: “Conditions for Congruence” Activity Two triangles are defined to be congruent if the three pairs of corresponding angles and the three pairs of corresponding sides are congruent. However, there are several ways to construct congruent triangles using fewer than the six parts. As a group, try to use the least amount of information to create a pair of congruent triangles with the material given. Keep marked drawings of your findings and what information was needed. Also keep track of methods which do not produce congruent triangles for a class discussion. Clarifying Directions – Use OT4: Sample student solution Use this if you see students need more specific directions about how to record the examples that they find. Class Discussion – Use OT5: Discussion Questions • Is it possible to build congruent triangles when only 1 piece of information is identified? (one side or one angle) Explain. • Is it possible to build congruent triangles when only 2 pieces of information are identified? (two sides or two angles or one side and one angle) Explain. • Is it possible to build congruent triangles when only 3 pieces of information are identified? o For what kinds of situations did this work? (SSS, SAS, ASA, AAS) o For what kinds of situations did this not work? (SSA, AAA) • Look at this drawing: how does it help convince us that SSS is enough information to conclude that two triangles are congruent? S2 S3 (Note: Figure provided on OT6) S1 • Look at this triangle that is only partly finished. All that is left to do is put the other side on it (the angles will be determined then.) S2 If another triangle started out the same way (same two lengths, same angle), would IT be A1 congruent to this triangle when the third side S1 was put on IT? Explain. (Note: Figure provided on OT6) • Look at this triangle that is only partly finished. All that is left to do is extend the two remaining arms until a triangle is formed. A1 A2 If another triangle started out the same way S1 (same two angles, same length), would IT be congruent to this triangle when the third side was put on IT? Explain. (Note: Figure provided on OT6) • You may have found that two angles and a side which is not between them is enough information to build congruent triangles (AAS). 5 Suppose you have a situation like the one shown 30° 40° where the side has length 5, and the angles have measures 30° and 40°. (Note: Figure provided on OT7) o What will the third angle have to be? Why? o Once you know that angle, this AAS situation should look like one of the other ones we just discussed. Will it be like SSS, SAS, or ASA? How come? • You probably found that knowing two sides and an angle which is not between them is not enough information to build congruent triangles. Here is one way to explain that: S1 S2 A1 (Note: Figure provided on OT7) You can extend the part of angle A1 in the direction that it must go, but you don’t know how far to go. You can swing the side S2 around, since you don’t know another angle. What does the picture suggest the problem is with a SSA approach to showing two triangles are congruent? • Why is it that specifying all three angles can never provide enough information to show that two triangles are congruent? (Note: Figure provided on OT7) Homework: Congruence II Congruence and Area 1. A rectangle is formed by putting two congruent squares adjacent to each other. Then a triangle is formed by drawing a line from one vertex to the opposite side, then to the other vertex. Explain why the triangle must have an area which is one-half the area of the rectangle. 2. “If two triangles have the same area, then they are congruent.” Is this a true statement? • If it is, provide an argument for why it must be true. • If it is not, then find an example to support your answer. Building Congruent Triangles Two triangles are congruent if all 3 pairs of corresponding angles and all 3 pairs of corresponding sides are congruent. A1 A1 A2 A2 A3 A3 A1 A1 A2 A2 A3 A3 A1 A1 A2 A2 A3 A3 S1 S1 S1 S2 S2 S2 S3 S3 S3 S3 S3 S3 S1 S1 S1 S2 S2 S2 “Conditions for Congruence” Activity There are several ways to construct congruent triangles using fewer than the six parts. As a group, try to use the least amount of information to create a pair of congruent triangles with the material given. Keep marked drawings of your findings and what information was needed. Also keep track of methods which do not produce congruent triangles for a class discussion. Sample Student Solution Using 2 pairs of congruent paper strips and 1 pair of congruent angles, we created a pair of congruent triangles. We call this method “Side-Angle-Side” Triangle Congruence. Discussion Questions • Is it possible to build congruent triangles when only 1 piece of information is identified? (one side or one angle) Explain. • Is it possible to build congruent triangles when only 2 pieces of information are identified? (two sides or two angles or one side and one angle) Explain. • Is it possible to build congruent triangles when only 3 pieces of information are identified? o For what kinds of situations did this work? o For what kinds of situations did this not work? S2 S3 S1 S2 A1 S1 A1 A2 S1 5 30° 40° S1 S2 A1 A3 A2 A1 Congruence II Conditions for Congruence Pre-Algebra Day 3 Warmup -- Use OT1: Congruence Notation and Vocabulary When we know that two triangles are congruent, it is important to identify corresponding parts – that is, the ones that match up with each other. • We say: “Triangle ABC is congruent to triangle TUV” • We write: ∆ABC ≅ ∆TUV • That means the following parts are corresponding: AB ≅ TU AC ≅ TV BC ≅ UV ∠ABC ≅ ∠TUV ∠ACB ≅ ∠TVU ∠BAC ≅ ∠UTV • Notice the markings that indicate which sides (or angles) match up Now, answer the following questions: Use OT2: Congruence Relations 1) Suppose ∆CAT ∆DOG. Find 3 pairs of sides and 3 pairs of angles that are also congruent. 2) Suppose that ∆ABC ∆XYZ, m A = 110°, and m C = 35°. Find four pairs of congruent angles. Note: Be sure to point out the notation in Question #2 about “the measure of an angle” Activity: Conditions for Congruence (cont.) If necessary, finish the follow-up discussion from yesterday’s class. Activity: Hey, Cableman! • Form student groups; provide students with rulers and a copy of the handout. • Let students read the directions, and solve the problem using measurement methods. • When students have determined the solution reasonably well, display the OT3: Cableman’s Boss Ask students to check their solution using this method and explain how that drawing can help solve the problem faster, and more accurately. Then ask them to identify the geometric ideas that are being used in the drawing. Homework: Congruence III Congruence Notation and Vocabulary You are told that ∆ABC ≅ ∆TUV U V T B A C Congruence Relations 1) Suppose ∆CAT ≅ ∆DOG. Find 3 pairs of sides and 3 pairs of angles that are also congruent. 2) Suppose that ∆ABC ≅ ∆XYZ, m∠A = 110°, and m∠C = 35°. Find four pairs of congruent angles. Hey, Cableman! Two houses are to be hooked up to cable TV by running service lines from the houses to the same point where they will both connect to the main cable. The main cable runs underground along the edge of the street in front of the houses. By measuring, determine where the connection should be made to use the least amount of cable altogether for the two service lines. House A House B Main Cable Cableman’s Boss House A House B P M Congruence III Directions: All of the following drawings show two congruent triangles. In each case, identify which argument can be used to conclude that they are congruent: SSS (side, side, side) SAS (two sides and the angle in between them) ASA (two angles and the side in between them) Also, write the statement which says that the two triangles are congruent using mathematical notation. 1. 2. C Y Z A B X A 3. 4. C B D 5. 6.