2-3. Copy the diagrams below on graph paper. Then draw the result when each indicated transformation is performed.
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1 2-1. Below, ΔPQR was reflected across line l to form ΔP Q R. Copy the triangle and its reflection on graph paper. How far away is each triangle from the line of reflection? Connect points P and P Q and Q, R and R, what do you notice about the segments? 2-2. Copy the diagrams below on graph paper. Reflect the triangle across the x-axis. Reflect the triangle across the y-axis. Label the reflected points using prime notation, ie. A B C a. b. c. B A C
2 2-3. Copy the diagrams below on graph paper. Then draw the result when each indicated transformation is performed. a. Reflect Figure A across line l. b. Reflect Figure C across line m. c. Reflect A across line l. d. Reflect D across line m While playing a game of pool, Montana Mike needed to hit the last remaining ball into pocket A, as shown in the diagram below. However, to show off, he decided to make the ball first hit one of the rails of the table. Copy the pool table and determine where Mike could bounce the ball off one rail so that it will land in pocket A. Draw the other two reflections of pocket A and the path the ball will follow to go in pocket A. Aim shot at reflection of A
3 2-5. Copy ΔABC and lines n and p(shown below)on graph paper. What happens when ABC is reflected across line n to form ΔA B C and then ΔA B C is reflected across line p to form ΔA B C? First visualize the reflections and then test your idea of the result by drawing both reflections. Examine your result from part (a). Compare the original triangle ΔABC with the final result, ΔA B C. What single motion would change ΔABC to ΔA B C? 2-6. Describe the translation. That is, how many units to the right and how many units down does the translation move the triangle? a. On graph paper, plot ΔEFG with coordinates E(4, 2), F(1, 7), and G(2, 0). Find the coordinates of ΔE F G if ΔE F G is translated the same way as ΔABC was in part (a). b. For the translated triangle in part (b), draw a line segment connecting each vertex to its translated image. What do you notice these line segments? What does this tell you about how a translation moves each point of the graph? 2-7. Lourdes has created the following challenge for you: She has given you three of the four points necessary to determine a rectangle on a graph. She wants you to find the points that complete each of the rectangles below. a. (3, 7), (5, 7), (5, 3) c. ( 1, 3), ( 1, 2), (9, 2) b. ( 5, 5), (1, 4), (4, 2) d. Find the area of rectangle (a) and (c).
4 2-8.Copy each figure below on graph paper. a. Reflect each shape across the x-axis. Name the coordinates of the vertices. b. Translate each shape so that A is at (2, 6). Name the coordinates of the vertices Plot ΔABC on graph paper if A (6, 3), B (2, 1), and C (5, 7). a. ΔABC is translated left 6 units and down 3 units to become ΔA B C. Name the coordinates of A, B, and C. b. This time ΔABC is reflected across the y-axis to form ΔA B C. Name the coordinates of B. c. If ΔABC is translated to form ΔA B C, where A has coordinates of (2, 3), describe the translation and graph ΔA B C On graph paper, draw the quadrilateral with vertices ( 1, 3), (4, 3), ( 1, 2), and (4, 2). What kind of quadrilateral is this? Translate the quadrilateral 3 units to the left and 2 units up. What are the new coordinates of the vertices? On graph paper, draw the quadrilateral with vertices (1, 3), (4, 3), (1, 2), and (4, 2). Reflect the quadrilateral across the vertical line x = 2. What are the new coordinates of the vertices?
5 2-12. Copy each shape on graph paper and rotate about the given point. Use tracing paper if needed. a. 180 b. 180º c d. 90º e. 90º f. 180º Copy ΔABC below on graph paper. a. Rotate ΔABC 90 counter-clockwise ( ) about the origin to create ΔA B C. Name the coordinates of ΔA B C. b. Rotate ΔABC 180 clockwise ( ) about the origin to create ΔA B C. Name the coordinates of ΔA B C.
6 2-14. Copy the diagrams below on graph paper. Then find the result when each indicated transformation is performed. a. Rotate 90 clockwise about P. b. Rotate 180 about point Q. c. Rotate 90 counterclockwise about P. d. Rotate C 180 about point Q What if you have the original figure and its image after a sequence of transformations? Examine ΔABC and ΔA'B'C' in the graph below. Describe at least two different ways to move ΔABC onto ΔA'B'C'. Use complete sentences in your description.
7 2-16. Examine the triangles below. a. Are these triangles congruent? Explain how you know. b. Luis wanted to write a statement to convey that these two triangles are congruent. He started with ΔCAB, but then got stuck because he did not know the symbol for congruence. Now that you know the symbol for congruence, complete Luis s statement for him Consider square MNPQ with diagonals intersecting at R, as shown below. a. How many triangles are there in this diagram? (Hint: There are more than 4) b. On your paper, draw and label 2 different pairs of congruent triangles in the square. c. Write as many triangle congruence statements as you can that involve triangles in this diagram Copy the triangles below on to your paper. a. If two triangles have the relationships shown in the diagram, do they have to be congruent? How do you know? b. Write a triangle congruence statement that involves triangles above.
8 2-18 c. Suppose you are working on a problem involving the two triangles ΔUVW and ΔXYZ and you know that ΔUVW ΔXYZ. What can you conclude about the sides and angles of ΔUVW and ΔXYZ? Write down every congruence statement involving sides or angles that must be true. (ie, U X ) Copy ΔABC below on graph paper. a. Rotate ΔABC 90 counter-clockwise about the origin to create ΔA B C. Name the coordinates of ΔA B C. b. Rotate ΔABC 180 clockwise about the origin to create ΔA B C. Name the coordinates of ΔA B C. c. Rotate ΔABC 90 clockwise about the origin to create ΔA ' B C. Name the coordinates of ΔA ' B C The diagrams below are not drawn to scale. Copy each pair of triangles: List the congruent sides and angles in each pair of triangles below. If you find congruent triangles, write a congruence statement (such as ΔPQR ΔXYZ). If the triangles are not congruent or if there is not enough information to determine congruence, write "cannot be determined." a. b.
9 2-21.A team is working together to try to prove SAS. Given the triangles shown below, they want to prove that ΔABC ΔDEF. Congruent means that two triangles have the same size and shape, so we have to be able to move ΔABC right on top of ΔDEF using transformations, since they preserve lengths and angles, to prove they are congruent. a. Describe the transformations used to move ΔABC right on top of ΔDEF. b. Copy ΔABC and ΔDEF on your paper and draw the transformations you used to move ΔABC right on top of ΔDEF In problem 2-21 you proved SAS by finding a sequence of transformations that would move one triangle onto another. A similar strategy can be used to prove the ASA. a. Describe the transformations used to move ΔABC right on top of ΔDEF. b. Copy ΔABC and ΔDEF on your paper and draw the transformations you used to move ΔABC right on top of ΔDEF.
10 2-23. Examine the two triangles below.. Are the triangles congruent? Justify your conclusion. If they are congruent, complete the congruence statement ΔDEF. What series of transformation(s) are needed to transform ΔDEF to ΔLJK? Use your triangle congruence theorems to determine if the following pairs of triangles must be congruent. Write the triangle congruence statement and the reason they are congruent. If the triangles are not congruent write "cannot be determined
11 2-25. Consider the triangles below. a. Which triangle congruence theorem shows that these triangles are congruent? b. Copy and complete the flowchart showing that these triangles are congruent. AB DF given given given ABC theorem Make a flowchart showing that the triangles below are congruent.
12 2-27. Determine whether or not the two triangles in each part below are congruent. If they are congruent, show your reasoning in a flowchart. If the triangles are not congruent or you cannot determine that they are, justify your conclusion Use a flowchart to show the triangles are congruent Copy each pair of triangles and name its triangle congruence theorem.
13 2-30. Raj is solving a problem about three triangles. He is trying to find the measure of H and the length of HI. Raj summarizes the relationships he has found so far in the diagrams below: Assuming everything marked in the diagram is true, find m H and the length of HI. Use flowcharts to prove ΔABC ΔDEF ΔGHI then use congruent parts of congruent triangles are congruent (CPCTC) to find m H and the length of HI Decide if each triangle below is congruent to ΔABC. Justify each answer. If you decide that they are congruent, organize your reasoning into a flowchart Examine ΔABC and ΔDEF below. Assume the triangles above are not drawn to scale. Complete a flowchart to justify the relationship between the two triangles. Find AC and DF.
14 2-33. Determine if the pair of triangles below are congruent. If they are congruent, organize your reasoning into a flowchart to prove PR MN Determine if the pair of triangles below are congruent. If they are congruent, organize your reasoning into a flowchart to prove A D The following pairs of triangles are not necessarily congruent even though they appear to be. Use the information provided in the diagram to show why. Justify your statements. a. b.
15 2-36. Jose started to prove that the triangles below are congruent. He was only told that point E is the midpoint of segments and. Copy and complete his flowchart below. Be sure that a reason is provided for every statement How can you tell which angles have equal measure? For example, in the diagram below, which angles must have equal measure? Name the angles and explain how you know. a. If you know that m B = 62, then what is m C, m A? Explain how you know. b. If you know that m B = x, then what is m C, m A? Explain how you know.
16 2-38. Consider the isosceles right triangle below. Find the measures of all its angles. What if you only know one angle of an isosceles triangle? For example, if m A = 34, what are the measures of the other two angles? Explain how you got your answers For each diagram below, set up an equation and solve for x.
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