, N NO FE. 7. Write a conjecture about two triangles with congruent angles.
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1 Part 1: Is AAA a Similarity Shortcut? In this activity you will explore the following question: If three angles of one triangle are congruent to three corresponding angles of another triangle, are the triangles similar? 1. Draw a triangle, by copying the angles above, and label it MNO. 2. Construct a second triangle Δ DEF with M D, N F and O E. 3. Measure all the angles of the two triangles to verify that corresponding angles are congruent. 4. Look at the definition of similar triangles. You know the angles are congruent. What else do you need to know to determine if the triangles are similar? 5. Measure all the lengths of the sides of the two triangles. Use the calculator to compute the ratios of lengths of corresponding sides. Are the sides proportional? What can you say about the triangles? This is Trial 1. Complete Trial 1 for the table below. MN DF MO DE NO FE Are Sides Proportional? Are Triangles Similar? Trial 1 6. Do you think the sides will always be proportional if the angles of two triangles are congruent? Write a conjecture (prediction) about the sides of two triangles with congruent angles. 7. Write a conjecture about two triangles with congruent angles. 8. Go back and look at your three angles. Do we really need three angles to come to this conclusion? Explain why or why not. 3
2 Part 2: Is SSS a Similarity Shortcut? In this activity you will explore the following question: If the sides of two triangles are proportional, are the triangles similar? 1. Draw a triangle, by copying the side lengths above, and label it Δ ABC. 2. Choose a scale factor (ratio) like 1 2, 3 4, 1 3, 7 3, or Apply the scale factor you chose to Δ ABC. 4. Measure the lengths of the sides of Δ ABC and A' B' C'. Compute the ratios of the lengths of corresponding sides. Confirm (check) that the sides are proportional. 5. Measure the angles of the two triangles. What do you notice? 6. Do you have enough information to determine that Δ ABC and A' B' C' are similar? Explain. 7. Write a conjecture about two triangles with proportional sides. 4
3 Part 3: SAS Similarity In this activity you will explore the following question: If two sides of two triangles are proportional, and the included angles are congruent, are the triangles similar? Use paper, a ruler and protractor for this activity. 1. Copy the angle and label it D. 2. Use the two segments to construct the sides of the triangle and then use a straightedge to connect Points E and F to finish the triangle. 3. Now draw another angle congruent to D. Choose a scale factor and apply the same scale factor to DE and DF to create DE and DF. Use these to create a second triangle. EF DE DF 4. Compute, and. Are these pairs of sides proportional? E' F' DE' DF' 5. Measure the other angles of the triangles. Are the triangles similar? 6. Write a conjecture about two triangles with two pairs of proportional sides and congruent included angles. 5
4 Part 4: ASA Similarity In this activity you will explore the following question: If two angles of two triangles are congruent, and the included sides are proportional, are the triangles similar? Use paper, a ruler and protractor for this activity. 1. Copy the angles and label them G and H. 2. Use the segment to construct the side of the triangle between the angles and then use a straightedge to extend the sides of the angles until they intersect to finish the triangle. Lable the point of intersection Point J. 3. Now draw two more angles congruent to G and H and apply a scale factor to GH to start another triangle. Extend the sides of the new angles until they intersect. Use these to create a second triangle. Label the new triangle with vertices G, H and J. GH HJ GJ 4. Compute, and. Are these pairs of sides proportional? G' H ' H ' J ' G' J ' 5. Measure the other angles of the triangles. Are the triangles similar? 6. Write a conjecture about two triangles with two pairs of congruent angles and a proportional pair of included sides. 6
5 Part 5: SSA Similarity In this activity you will explore the following question: If two sides of two triangles are proportional, and the non-included angles are congruent, are the triangles similar? Use paper, a ruler and protractor for this activity. 1. Copy the angle and label it J. 2. Use the two segments to construct the sides of the triangle. Point L will be where KL intersects the side of J. 3. Now draw another angle congruent to J and apply the same scale factor to KL and KJ to create K J and K L. Use these to create a second triangle. KJ KL JL 4. Compute, and. Are these pairs of sides proportional? K' J ' K' L' J ' L' 5. Measure the other angles of the triangles. Are the triangles similar? 6. Write a conjecture about two triangles with two pairs of proportional sides and congruent non-included angles. 7
6 Part 6: AAS Similarity In this activity you will explore the following question: If two angles of two triangles are congruent, and the included sides are proportional, are the triangles similar? Use paper, a ruler and protractor for this activity. 1. Copy angle Q. 2. Use the segment to construct the side of the triangle that is not between the angles. 3. Copy P to finish the triangle. 4. Now copy angle Q again and apply a scale factor to QR to start another triangle. Now add a Copy P to finish the triangle. Use these to create a second triangle. Label the new triangle with vertices Q, R and P. 5. Compute QP, Q' P' PR and P' R' Q' R' QR. Are these pairs of sides proportional? 6. Measure the other angle of the triangles. Are the triangles similar? 7. Write a conjecture about two triangles with two pairs of congruent angles and a proportional pair of non-included sides. 8
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