L9 Congruent Triangles 9a Determining Congruence. How Do We Compare?

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1 How Do We Compare? Using patty paper, compare the sides and angles of the following triangle pairs. Record what is the same for each pair and what is different. 1. What is common? What is different? Is there a rigid motion that shows they are congruent? 2. What is common? What is different? Is there a rigid motion that shows they are congruent? Page 1

2 3. What is common? What is different? Is there a rigid motion that shows they are congruent? 4. Based upon your results, what are some conjectures about when triangles are congruent? Page 2

3 Grandma s Garden Boxes Grandma has been watching the garden channel again. She fell in love with some triangular shaped raised garden boxes. She asked Uncle Bobby to build them, but Uncle Bobby did not want to measure all three sides and all three angles of all the triangles. While he was complaining to you, you mentioned that you heard a rumor that in order to be sure that two triangles are congruent, you only need to measure three pieces of information. You just couldn t remember what three pieces. Let s investigate in the next activity. When you are finished with the activity, write a note to Uncle Bobby explaining what three measurements he would need to make sure the triangles are congruent. Note to Uncle Bobby: Page 3

4 WHAT DOES IT TAKE TO BE THE SAME? Scenario 1: Side-Side-Side (SSS) In this scenario you will explore if having three sides of one triangle congruent to three sides of another triangle guarantees that the two triangles are congruent. 1. Draw a scalene triangle on a sheet of tissue paper. 2. Using three other pieces of tissue paper, trace each of the sides of the triangle onto a separate piece of paper. Mark the ends of each segment to make them easier to see. 3. Slide the three pieces together to make a triangle and copy the new triangle onto another piece of tissue paper. 4. Is your new triangle congruent to the original? Explain why or why not. 5. Can you rearrange the pieces to create a new triangle that is not congruent to the original? Explain why the two triangles must be congruent, or why not. 6. Conjecture: If three sides of one triangle are congruent to three sides of another triangle, then.(#thm) Page 4

5 Scenario 2: Angle-Angle-Angle (AAA) In this scenario you will explore if having three angles of one triangle congruent to three angles of another triangle guarantees that the two triangles are congruent. 1. Draw a scalene triangle on a sheet of tissue paper. 2. Using three other pieces of tissue paper, trace each of the angles of the triangle onto a separate piece of paper. Extend the rays of the angles. 3. Slide the three pieces together to make a new triangle and copy the new triangle onto another piece of tissue paper. Recall that a ray has no end, hence you will only be using a portion of each ray as a side. 4. Is your new triangle congruent to the original? Explain why or why not. 5. Can you rearrange the pieces to create a new triangle that is not congruent to the original? Explain why the two triangles must be congruent, or why not. 6. Conjecture: If three angles of one triangle are congruent to three angles of another triangle, then Page 5

6 Scenario 3: Side-Side-Angle (SSA) In this scenario you will explore if having two concurrent sides and the angle adjacent to the second side of one triangle congruent to two concurrent sides and the angle adjacent to the second side of another triangle guarantees that the two triangles are congruent. 1. Draw a scalene triangle on a sheet of tissue paper. 2. Using three other pieces of tissue paper, trace two concurrent sides and the angle adjacent to the second side (i.e. opposite the first side) of one triangle onto a separate piece of paper. Mark the ends of each segment to make them easier to see, making sure you keep track of which side was the first side, and extend the rays of the angles. 3. Slide the three pieces together to make a new triangle, making sure the angle is still opposite the first side, and copy the new triangle onto another piece of tissue paper. Recall that a ray has no end, hence you will only be using a portion of each ray as a side. 4. Is your new triangle congruent to the original? Explain why or why not. 5. Can you rearrange the pieces to create a new triangle that is not congruent to the original, where the angle is still opposite the first side? Explain why the two triangles must be congruent, or why not. 6. Conjecture: If two concurrent sides and the angle adjacent to the second side of one triangle are congruent to two concurrent sides and the angle adjacent to the second side of another triangle, then Page 6

7 Scenario 4: Side-Angle-Side (SAS) In this scenario you will explore if having two sides and the angle between them of one triangle congruent to two sides and the angle between them of another triangle guarantees that the two triangles are congruent. 1. Draw a scalene triangle on a sheet of tissue paper. 2. Using three other pieces of tissue paper, trace two sides and the angle between them of the triangle onto a separate piece of paper. Mark the ends of each segment to make them easier to see and extend the rays of the angles. Recall that a ray has no end, hence you will only be using a portion of each ray as a side. 3. Slide the three pieces together to make a new triangle, making sure the angle is still between the two sides, and copy the new triangle onto another piece of tissue paper. Recall that a ray has no end, hence you will only be using a portion of each ray as a side. 4. Is your new triangle congruent to the original? Explain why or why not. 5. Can you rearrange the pieces to create a new triangle that is not congruent to the original, where the angle is still between the two sides? Explain why the two triangles must be congruent, or why not. 6. Conjecture: If two sides and the angle between them of one triangle are congruent to two sides and the angle between them of another triangle, then. (#THM) Page 7

8 Scenario 5: Side-Angle-Angle (SAA) In this scenario you will explore if having two angles and the side not between them of one triangle congruent to two angles and one of the sides not between them, of another triangle guarantees that the two triangles are congruent. 1. Draw a scalene triangle on a sheet of tissue paper. 2. Using three other pieces of tissue paper, trace two angles and one of the sides not between them of the triangle onto a separate piece of paper. Mark the ends of each segment to make them easier to see and extend the rays of the angles. 3. Slide the three pieces together to make a new triangle, making sure the side is still not between the two angles, and copy the new triangle onto another piece of tissue paper. Recall that a ray has no end, hence you will only be using a portion of each ray as a side. 4. Is your new triangle congruent to the original? Explain why or why not. 5. Can you rearrange the pieces to create a new triangle that is not congruent to the original, making sure the side is still not between the two angles? Explain why the two triangles must be congruent, or why not. 6. Conjecture: If two angles and the side not between them of one triangle are congruent to two angles and the side not between them of another triangle, then.(#thm) Page 8

9 Scenario 6: Angle-Side-Angle (ASA) In this scenario you will explore if having two angles and the side between them of one triangle congruent to two angles and the side between them of another triangle guarantees that the two triangles are congruent. 1. Draw a scalene triangle on a sheet of tissue paper. 2. Using three other pieces of tissue paper, trace two angles and the side between them of the triangle onto a separate piece of paper. Mark the ends of each segment to make them easier to see and extend the rays of the angles. 3. Slide the three pieces together to make a new triangle, making sure the side is still between the two angles, and copy the new triangle onto another piece of tissue paper. Recall that a ray has no end, hence you will only be using a portion of each ray as a side. 4. Is your new triangle congruent to the original? Explain why or why not. 5. Can you rearrange the pieces to create a new triangle that is not congruent to the original, making sure the side is still between the two angles? Explain why the two triangles must be congruent, or why not. 6. Conjecture: If two angles and the side between them of one triangle are congruent to two angles and the side between them of another triangle, then.(#thm) Page 9

10 Summary Complete the following. Use this sheet as a summary for your class and homework. 1. List the four Congruence Theorems here. Write the acronym and then describe what that acronym means. Be specific and clear when describing an angle or side. 2. By definition, congruent triangles have and. 3. Thus, we can say Corresponding parts of triangles are. (#THM) We use this statement very often in geometry. When we use it, we use an acronym, CPCTC. Congruence Statement If ABC is congruent to DEF, then we write ABC DEF. So, if ABC DEF then complete the following: E C D AB EF AC Now go back to Grandma s Garden Boxes and write your note to Uncle Bobby. Page 10

11 Are We Identical Twins? Which of the following pairs of triangles are congruent? Explain which criteria for triangle congruence you used to determine your answer C B P O S D I T W C M N A T N 3. R 4. B P T P L H I O Page 11

12 R V X M S Y L U T Z N P R K I S B K J M P N C Page 12

13 9b Proving Congruent Triangles Now we will use SSS, SAS, ASA, AAS and CPCTC to prove statements involving congruent triangles. 1. Given: A is the midpoint of CE A is the midpoint of BD Prove: ΔBCA ΔDEA What transformation could take ΔBCA onto ΔDEA? Statement Reason Page 13

14 9b Proving Congruent Triangles 2. Given: GH! JI G I Prove: GJ IH What transformation could take ΔJGH onto ΔHIJ? Statement Reason Page 14