Name: Date: School: Facilitator: 7.03 Proof Look at the Explore! Parallel Lines and Triangles interactive. Use this interactive to answer the questions below. http://www.geogebratube.org/student/m5160 1. How are segments PR and TS related? 2. If PR and ST were extended as lines, name two transversals of these lines. and 3. What type of angles are PQR and SQT? How are they related? 4. Look at transversal PS and segments PR and ST only. How could you describe the relationship between angles RPQ and TSQ? 5. Look at transversal RT and PR and ST only. How could you describe the relationship between angles QRP and QTS? 6. List the congruent angle pairs in this diagram. and and and 7. Complete the similarity statement: By the Similarity Theorem, 8. Based on this interactive, will two triangles created by the intersection of two transversals between parallel lines always be similar? Why or why not? Write your answer using complete sentences and support your answer using theorems and explanations from this activity. Name: Date: School: Facilitator: 7.03 Proving Triangles Similar Part 1. 1. Look at the triangles and decide who has chosen the triangle that is similar to EFG? and why? John says triangle A and Jill says B Answer: A B C 2. Decide whether ABC and DEF are similar, not similar, or cannot be determined for the given information. Include your reason. Answer: 3. The triangles shown are similar. List all the pairs of congruent angles and write the statement of proportionality. / = / = / Part 2: Determine whether the triangles are similar. If they are, a. List the congruent angles (if applicable) b. Set up a statement of proportionality for known sides (in the form AB/DE = BC/EF) (if applicable) c. Write a similarity statement in the form ABC DEF. In your similarity statement, include the theorem you would use to prove your answer. 1. Are the triangles similar? a. If yes, which angles are congruent? and , and , and b. What is the statement of proportionality? / = / = / Complete the similarity statement: By the Similarity Theorem, 2. Are the triangles similar? a. If yes, which angles are congruent? and , and , and b. What is the statement of proportionality? / = / = / Complete the similarity statement: By the Similarity Theorem, 3. Are the triangles similar? a. If yes, which angles are congruent? and , and , and b. What is the statement of proportionality? / = / = / Complete the similarity statement: By the Similarity Theorem, Name: Date: School: Facilitator: 7.04 Problem Solving With Proportions Find the geometric mean of the two numbers. 1. 8 and 12 geometric mean: 2. 8.5 and 12.4 geometric mean: Use the diagram and the given information to find the unknown length. 3. LJ/JN = MK/KP, find JN JN = 4. JK/KL = JM/MN, find JN JN = 5. Note that the two triangles are similar. Find the measure of x = Hint: Careful, the image flips 12f 4in x 6in Name: Date: School: Facilitator: 7.05 Scale Drawings Part A: Determine the type of dilation (enlargement, reduction, or isometric) given the scale factor. http://www.geogebratube.org/student/m21446 A. enlargement B. reduction C. isometric 1. 3 5 2. 8 3 3. 7 7 4. 3 Part B: Use the ACCESS - Dilation of Lines and Segments Geogebra activity to answer the questions below. Line AB is dilated with center of dilation at point C. Adjust the scale factor with the slider. Adjust the scale factor to a value greater than 1. 1. What can you say about the position of line A'B' compared to line AB and point C? (Is it farther from point C than line AB, between AB and point C, or on the other side of point C?) 2. Use A and B to adjust the slope of line AB. How does the slope of A'B' change? Adjust the scale factor to a value between 0 and 1. 3. What can you say about the position of line A'B' compared to line AB and point C? 4. Use A and B to adjust the slope of line AB. How does the slope of A'B' change? Adjust the scale factor to a value between 0 and -1. 5. What can you say about the position of line A'B' compared to line AB and point C? 6. Use A and B to adjust the slope of line AB. How does the slope of A'B' change? Adjust the scale factor to a value between less than -1. 7. What can you say about the position of line A'B' compared to line AB and point C? 8. Use A and B to adjust the slope of line AB. How does the slope of A'B' change? 9. Adjust the scale factor to 0. What happens to A'B'? 10. Adjust the scale factor to 1. What happens to A'B'? 11. Adjust the scale factor to a number other than 1. Based on this and all of the experiments you did earlier, how can you describe the relationship between AB and A'B'? (Hint: look at their slopes.) 12. Move point A so that it is on C. This is now a line passing through the center of dilation. What happens to A'B'? Check the scale factor to make sure this holds for all values. Hide line AB and Show segment AB. 1. Change the scale factor for AB. What do you notice about the relationship of the ration A'B'/AB and the scale factor? 2. Complete the statements. Use the slider to test them if necessary. a. If you dilate a segment AB to form a new segment A'B', the ratio / will be equal to the . b. If you dilate a segment AB to form a new segment A'B' by a scale factor greater than 1, segment X'Y' will be than segment XY. c. If you dilate a segment AB to form a new segment A'B' by a scale factor less than 1 but greater than zero, segment X'Y' will be than segment XY. d. If you dilate a segment AB by a scale factor of -1, the result would be the same as if you had . (Hint - what other tranformation would create this image?) Name: Date: School: Facilitator: 7.03 Proving Triangles Similar Part 1. 1. Look at the triangles and decide who has chosen the triangle that is similar to EFG? and why? John says triangle A and Jill says B Answer: Both, All the three triangles have all their three angles equal, 30, 60 and 90 degrees. A B C 2. Decide whether ABC and DEF are similar, not similar, or cannot be determined for the given information. Include your reason. Answer: Similar, ABC DEF, BCA DFE, BAC FDE 3. The triangles shown are similar. List all the pairs of congruent angles and write the statement of proportionality. PQR STM QRP TMS RPQ MST PQ/ST = QR/TM = RP/MS Part 2: Determine whether the triangles are similar. If they are, a. List the congruent angles (if applicable) b. Set up a statement of proportionality for known sides (in the form AB/DE = BC/EF) (if applicable) c. Write a similarity statement in the form ABC DEF. In your similarity statement, include the theorem you would use to prove your answer. 1. Are the triangles similar? Yes a. If yes, which angles are congruent? PQR and SQT,QRP and QTS, RPQ and TSQ b. What is the statement of proportionality? PQ/QS = QR/QT = PR/TS Complete the similarity statement: By the AAA Similarity Theorem, PQR SQT, 2. Are the triangles similar? Yes a. If yes, which angles are congruent? ABC and DEF,ACB and DFE, BAC and EDF b. What is the statement of proportionality? AB/DE = BC/EF = AC/DF Complete the similarity statement: By the SAS Similarity Theorem, ABC DEF 3. Are the triangles similar? Yes a. If yes, which angles are congruent? MNO and FDE,NOM and DEF, OMN and EFD b. What is the statement of proportionality? MN/FD = NO/DE = OM/EF Complete the similarity statement: By the SSS Similarity Theorem, MN - FD, NO - DE, OM - EF Name: Date: School: Facilitator: 7.04 Problem Solving With Proportions Find the geometric mean of the two numbers. 1. 8 and 12 geometric mean: 10 2. 8.5 and 12.4 geometric mean: 10.45 Use the diagram and the given information to find the unknown length. 3. LJ/JN = MK/KP, find JN JN = 24/7 4. JK/KL = JM/MN, find JN JN = 9.6 5. Note that the two triangles are similar. Find the measure of x= 8 Hint: Careful, the image flips 12f 4in x 6in Name: Date: School: Facilitator: 7.03 Proof Look at the Explore! Parallel Lines and Triangles interactive. Use this interactive to answer the questions below. http://www.geogebratube.org/student/m5160 1. How are segments PR and TS related? Are parallel lines 2. If PR and ST were extended as lines, name two transversals of these lines. PS and RT 3. What type of angles are PQR and SQT? How are they related? Are vertical angles 4. Look at transversal PS and segments PR and ST only. How could you describe the relationship between angles RPQ and TSQ? Alternate angles 5. Look at transversal RT and PR and ST only. How could you describe the relationship between angles QRP and QTS? Alternate angles 6. List the congruent angle pairs in this diagram. RPQ and TSQ PQR and SQT QRP and QTS 7. Complete the similarity statement: By the AAA Similarity Theorem, PQR SQT 8. Based on this interactive, will two triangles created by the intersection of two transversals between parallel lines always be similar? Why or why not? Write your answer using complete sentences and support your answer using theorems and explanations from this activity. They would be similar. The angles made by the transversals will always be equal and so will the triangles. All opposite angles, alternate and adjacent angles will be equal. Name: Date: School: Facilitator: 7.05 Scale Drawings Part A: Determine the type of dilation (enlargement, reduction, or isometric) given the scale factor. http://www.geogebratube.org/student/m21446 A. enlargement B. reduction C. isometric reduction 1. 3 5 enlargment 2. isometry 3. 8 3 7 7 enlargment 4. 3 Part B: Use the ACCESS - Dilation of Lines and Segments Geogebra activity to answer the questions below. Line AB is dilated with center of dilation at point C. Adjust the scale factor with the slider. Adjust the scale factor to a value greater than 1. 1. What can you say about the position of line A'B' compared to line AB and point C? (Is it farther from point C than line AB, between AB and point C, or on the other side of point C?) farther from point C than line AB 2. Use A and B to adjust the slope of line AB. How does the slope of A'B' change? Equally to that of AB Adjust the scale factor to a value between 0 and 1. 3. What can you say about the position of line A'B' compared to line AB and point C? A'B' is between AB and point C 4. Use A and B to adjust the slope of line AB. How does the slope of A'B' change? Equally to that of AB Adjust the scale factor to a value between 0 and -1. 5. What can you say about the position of line A'B' compared to line AB and point C? on the other side of point C 6. Use A and B to adjust the slope of line AB. How does the slope of A'B' change? Equally to that of AB Adjust the scale factor to a value between less than -1. 7. What can you say about the position of line A'B' compared to line AB and point C? on the other side of point C and further away from point C than AB 8. Use A and B to adjust the slope of line AB. How does the slope of A'B' change? Equally to that of AB 9. Adjust the scale factor to 0. What happens to A'B'? Becomes equal to point C 10. Adjust the scale factor to 1. What happens to A'B'? Becomes equal to line AB 11. Adjust the scale factor to a number other than 1. Based on this and all of the experiments you did earlier, how can you describe the relationship between AB and A'B'? (Hint: look at their slopes.) AB and A'B' are parallel lines 12. Move point A so that it is on C. This is now a line passing through the center of dilation. What happens to A'B'? Check the scale factor to make sure this holds for all values. Passes through center of dilation and moves along line AB Hide line AB and Show segment AB. 1. Change the scale factor for AB. What do you notice about the relationship of the ration A'B'/AB and the scale factor? Equal in magnitude 2. Complete the statements. Use the slider to test them if necessary. a. If you dilate a segment AB to form a new segment A'B', the ratio A'B'/AB will be equal to the Scale factor. b. If you dilate a segment AB to form a new segment A'B' by a scale factor greater than 1, segment X'Y' will be bigger than segment XY. c. If you dilate a segment AB to form a new segment A'B' by a scale factor less than 1 but greater than zero, segment X'Y' will be Smaller than segment XY. d. If you dilate a segment AB by a scale factor of -1, the result would be the same as if you had reflection. (Hint what other tranformation would create this image?)