Basic Triangle Congruence Lesson Plan Developed by CSSMA Staff Drafted August 2015 Prescribed Learning Outcomes: Introduce students to the concept of triangle congruence and teach them about the congruency conditions of SSS and SAS. This short lesson is to be delivered through a series of interactive, hands-on activities. Materials needed : Straws, tape, protractor, ruler, scissors, and construction paper or cardstock Warmup Q: What s a triangle made of? - Possible answer: 3 sides and 3 angles Teacher: Let s play a game. Cut out a triangle and give me its side lengths. Without looking at your triangle, I claim that I can make the same triangle myself. ( Teacher uses straws of the given lengths and constructs a triangle, then show that they are equal by laying it on top of their triangle) After students are convinced, do the same except ask them for two sides and the angle between these two sides (students should use the protractor to determine this angle). A note on technicality: Q: What does it mean for two things to be equal? - Possible answers: - Numbers: 3 = 3, always the same. This is the logic that we use to solve equations. - Geometrically: - same shape
- same size - Bonus: can it be rotated? Reflected? - yes! In geometry, we use the word congruent to mean equal. SSS (Teacher hands out sets of straws of equal lengths, say 2 straws each of 4, 5, and 7 cm, ideally each set is of a different color. Asks the student, can you make two different triangles from each set? After students have had a chance to try, make sure they are convinced that such is impossible. Ask them to measure the angles of each triangle. What do they find?) Definition of Congruence is congruent to Two geometric objects are congruent, i.e. equal, if, after rotations and translations, they physically occupy the same space. - Intuitive definition: For triangles (and other flat objects), we can cut them out and physically stack them on top of each other. If they fit exactly, they are congruent. Triangles are congruent when all corresponding sides and interior angles are congruent. Q: What is the relationship between the areas of congruent triangles? If we know all three sides and all three angles of two triangles, we can figure out if they are congruent or not. As the previous exercise shows, however, just knowing three sides is sufficient---the triangle is set in place if we fix all three of its side lengths. - Intuitively, once you draw the boundary, the inside is fixed. - This is called SSS Congruence, where S stands for side SAS Congruence
1. Take 2 of the straws, place them on a piece of paper, and form a 60 degree angle between them. 2. Trace the third side on paper, then try rotating and flipping the two straws while keeping the angle constant and trace the third side. Measure the length each time. 3. Do the third side always have the same length? What does this tell you? What about the other angles? 4. Are the triangles always congruent? 5. Is this true in general, if we change the numbers? This is called SAS congruence, where A stands for angle. Note that the A is between the two S s, so we need to know two sides and the angle between them. Give students the handout titled Thought Questions & Classroom Activities. Teaching ends here, and now, allow students to think about the questions in the Thought Questions & Classroom Activities handout for the remaining duration of class. Allow students to discuss problems with each other as collaborative learning can be very effective! If students do not finish the problems on this handout by the end of class, assign the rest as homework. At the beginning of the next class, discuss some of the problems on the Thought Questions & Classroom Activities handout. The problems on this handout can be quite theoretical, so we actually do not expect every student to be able to provide a fully justified answer to every problem. The goal of this handout is to encourage students to think more abstractly about the concept of congruent triangles, so when you discuss these problems, use physical models to help you illustrate the concepts. The next logical step after teaching SSS and SAS is to move onto ASA, AAS, and HS, the other methods of checking whether two triangles are congruent or not. The Food for Thought handout is a good jumping off point for that. Use it during the next class period as a starting point of discussion for other relevant concepts in congruent triangle! An answer key has been included at the end of this lesson plan to offer some of the answers that we believe are reasonable.
Thought Questions & Classroom Activities Name: 1. In class, we discussed conditions for two triangles to be congruent. Now, let s look at squares: what must be true for two squares to be congruent? 2. Play the warm-up game without straws: in other words, after you are given the three side lengths, try to directly cut out such a triangle. Is it easy to do so? 3. In SAS, we stressed that the angle has to be between the two sides. Is this necessary? As in, if you know two sides of a triangle, a and b, and an angle A between sides b and c (the unknown side), can you make two different triangles? - It may help to draw side b and angle A on paper, with a ray extending in the direction of the third side coming from angle A, then pivot a straw of length a and see where the third side lies. 4. You learned about SSS and SAS today, both of which are methods of showing two triangles as being congruent. It turns out there are more methods of doing so, but they all require us knowing 3 facts about the triangle s sides or angles. Why do you think this is so? 5. Suppose I have two triangles whose respective angles are equal. Do you know if they are congruent? How do you know?
Food for Thought Name: 1. What do you think are other ways of showing that two triangles are congruent to each other? 2. For two triangles to be congruent, they must share a side of equal length. Is the reverse statement true: in other words, are two triangles that share one side of equal length always congruent? 3. We know that SSS shows two triangles to be congruent. If you have two triangles such that at least one side length differ (so SSS is not satisfied), are those two triangles necessarily not congruent? 4. Create two congruent triangles, A and B. Now make a triangle C with the same side lengths as triangle B. What s the relationship between A and C? Why? 5. Sammy tells you that he has two triangles with the following measurements: Side a Side b Side c Angle A Angle B Angle C Triangle 1 5 7 9 80 70 30 Triangle 2 5 7 10 85 65 30 If he uses SSS, from problem 3, the two triangles are not congruent; however, if he uses SAS with sides a, b, and angle C (the angle opposite side C), he get that they are congruent. What s going on here?
Thought Questions & Classroom Activities Suggested Answers In class, we discussed conditions for two triangles to be congruent. Now, let s look at squares: what must be true for two squares to be congruent? They have the same side length. They have the same area/perimeter. (Not correct!) They have the same angles. Play the warm-up game without straws: in other words, after you are given the three side lengths, try to directly cut out such a triangle. Is it easy to do so? With only a ruler and a pencil, it s hard to construct such a triangle. The easiest way to do it also requires a protractor. First, draw a line segment with the same length as one of the sides of the original triangle. Then, determine the angles of the other two sides emerging from the ends of this line segment on the original triangle. Finally, replicate those angles with the correct side lengths on the new line segment to make the new triangle. This can be very inaccurate and tedious to do. This shows that just because something is doable or true doesn t mean we can easily do it or show it! In SAS, we stressed that the angle has to be between the two sides. Is this necessary? As in, if you know two sides of a triangle, a and b, and an angle A between sides b and c (the unknown side), can you make two different triangles? Yes! As shown in the figure below, you can have AB=PQ (side), BC=QR (side), and angle A equal to angle P. Yet, these two triangles are obviously different! The reason that this happens can be explained through noticing that once angle A and AB is fixed, there are two points on the ray AC that are of a particular distance away from B.
This shows that the placement of the angle matters in determining whether triangles are congruent! You learned about SSS and SAS today, both of which are methods of showing two triangles as being congruent. It turns out there are more methods of doing so, but they all require us knowing 3 facts about the triangle s sides or angles. Why do you think this is so? To fully describe any particular triangle, we generally need to know its three side lengths because it has three sides. If we were given less information, there would be no way of determining the lengths of all three sides. Hence, when determining whether another triangle is exactly the same (i.e. congruent) to the given triangle, we need to know enough about the new triangle to fully describe it. So we need to know three facts about it. Suppose I have two triangles whose respective angles are equal. Do you know if they are congruent? How do you know? They are not necessarily congruent! As seen in the figure below, angle A is equal to angle P, angle B is equal to angle Q, and angle C is equal to angle R, yet the two triangles are clearly different in size. Equal respective angles can show that two triangles are similar to each other, but since there s nothing known about side lengths involved, the two triangles could have dramatically different sizes!
Food for Thought Suggested Answers What do you think are other ways of showing that two triangles are congruent to each other? ASA: two corresponding angles with the side between sandwiched between these two angles all equal in both triangles. AAS: two corresponding angles with any corresponding side between the two triangles equal. In a right-angled triangle, HS: hypotenuse and any other corresponding side between the two triangles equal. For two triangles to be congruent, they must share a side of equal length. Is the reverse statement true: in other words, are two triangles that share one side of equal length always congruent? False! For example, draw a line on a piece of paper so that its two endpoints form two of the vertices of the triangle. Now, choose a random point in the plane to be the third vertex of the triangle. Clearly, there are many choices for this third point that would make the resulting triangles very different! We know that SSS shows two triangles to be congruent. If you have two triangles such that at least one side length differ (so SSS is not satisfied), are those two triangles necessarily not congruent? This is a bit of a trick question, but the definition of congruent triangles implies that SSS is satisfied in all cases that two triangles are congruent. Hence, if SSS is not satisfied, the two triangles cannot be congruent. This should be pretty obvious from intuition Create two congruent triangles, A and B. Now make a triangle C with the same side lengths as triangle B. What s the relationship between A and C? Why? By SSS, triangle C is congruent to triangle B. Since triangle B is congruent to triangle A, it makes sense that congruency, as a measure of equality in geometry, is transitive. Since in algebraic equality, A=B and B=C implies A=C, we would also expect triangle A to be congruent to triangle C.
Sammy tells you that he has two triangles with the following measurements: Side a Side b Side c Angle A Angle B Angle C Triangle 1 5 7 9 80 70 30 Triangle 2 5 7 10 85 65 30 If he uses SSS, from problem 3, the two triangles are not congruent; however, if he uses SAS with sides a, b, and angle C (the angle opposite side C), he get that they are congruent. What s going on here? Trick question! These two triangles actually don t exist!!! Try drawing them out and you ll see that if you have a triangle with side lengths 5, 7, and 9, the angles won t be 80, 70, and 30 degrees. Similarly, if you have a triangle with side lengths 5, 7, and 10, the angles won t be 85, 65, and 30 degrees. Hence, SSS and SAS doesn t even apply as these shapes are nonexistent! Moral of the story: in this problem, we gave you too many conditions for the triangle that ultimately turned out to be self contradictory. A triangle needs 3 key defining conditions, usually its three side lengths. More information must coincide with these three already given conditions.