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Congruence and Transformations

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  1. Congruence and Transformations Chapter 7, Lesson 1

  2. Congruent = Same Size & Shape Determine if the two figures are congruent by using transformations. Explain you reasoning. STEP 2: Translate ΔA’B’C’ until all sides and angles match ΔXYZ. STEP 1: Reflect ΔABC over a vertical line. So, the two triangles are congruent because a reflection followed by a translation will map ΔABC to ΔXYZ.

  3. Example 1 Determine if these two figures are congruent by using transformations. Explain your reasoning. STEP 2: Translate the red figure up to the green shape. STEP 1: Reflect the red figure over a vertical line. Even if the reflected figure is translated up and over, it will not match the green figure exactly.

  4. Got it? 1 Congruent; A reflection followed by a translation maps figure A onto Figure B. a. b. Not congruent; No transformations will match the two figures exactly.

  5. Determine the Transformations If you have two congruent figures, analyze the orientation of the figures.

  6. Example 2 Ms. Martinez created the logo shown. What transformations did she use if the letter “d” is the preimage and the letter “p” is the image? Are the two figures congruent? STEP 2: Translate the new image up. STEP 1: Start with the pre-image. Rotate the letter “d” 180 about point A. Ms. Martinez used a rotation and translation to create the logo. The letters are congruent because rotation and translation do not change the shape or size.

  7. Got it? 2 What transformations could be used if the letter “W” is the preimage and the letter “M” is the image in the logo shown? Are the two images congruent? Explain. A vertical rotation followed by a translation. Yes, images produce by a reflection and translation are congruent.

  8. Congruence Chapter 7, Lesson 2

  9. Real-World Link Lauren is creating a quilt using the geometric pattern shown. She wants to make sure all of the triangles in the pattern are the same shape and size. • What would Lauren need to do to show the two triangles are congruent? Measure the sides and angles of each triangle and compare them. • Suppose you cut out the two triangles and laid one on top of the other so the parts of the same measure were matched up. What is true about the triangles? They are congruent.

  10. Corresponding Parts In the figure below, we can proof why the two figures are congruent. The parts of each triangle that match, or correspond, are called corresponding parts.

  11. Example 1 Write congruence statements comparing the corresponding parts in the congruent triangles. Corresponding Angles: Corresponding Sides: JKGH **Make sure to add the line for the corresponding sides.**

  12. Got it? 1 Write congruence statements comparing the corresponding parts in the congruent quadrilaterals.

  13. Example 2 Triangle ABC is congruent to XYZ. Write congruence statements comparing the corresponding parts. Then determine which transformations map ΔABC toΔXYZ. The transformations from ΔABC to ΔXYZ consist of a reflection over the y-axis followed by a translation of 2 units down.

  14. Stop and Reflect… You can determine which points correspond by using the congruence statements. If AB MN, then point A corresponds with point M.

  15. Got it? 2 Parallelogram WXYZ is congruent to parallelogram KLMN. Write congruence statements. Determine which transformation(s) map parallelogram WXYZ to parallelogram KLMN. If you reflect KLMN over the x-axis and then translate it to the right 5 units, it coincides with WXYZ.

  16. Finding Missing Measures Miley is using a brace to support a tabletop. In the figure, ΔBCE ΔDFG if the length of CE is 2 feet, what is the length of FG? 2 feet

  17. Got it? 3 Miley is using a brace to support a tabletop. In the figure, ΔBCE ΔDFG. If mCEB = 50, what is the measure of FGD? Since CEB and FGD are corresponding parts in congruent figures, they are congruent. So, FGD measures 50.

  18. Similar Triangles Inquiry Lab

  19. Measure and record the lengths and angles. 78 78  18 cm 9 cm 21 cm 58  58 10.5 cm 44 44  25 cm 12.5 cm What do you notice about the measure of the corresponding angles of the triangles? They are equal. What do you notice about the ratios of the corresponding sides of the triangles? They are equal.

  20. Similarity and Transformations Chapter 7, Lesson 3

  21. Vocabulary Rating Scale Copy and complete the table. Place a check mark in the appropriate box next to the word. If you do not know the meaning, use your iPad to look up the word. (Use http://www.mathematicsdictionary.com) an enlargement or reduction of a figure the ratio of two similar figures two figures that have the same shapes, but different size

  22. Two figures are a similar if the second can be obtained from the first by a transformation and dilation. Determine if the two triangles are similar by using transformations. STEP 1: Translate ΔDEF down 2 units and 5 units to the right so D maps onto G. STEP 2: Write ratios comparing the sides of each side. or 2 or 2 or 2 Since the ratios are equal, ΔHGI is a dilated image of ΔEDF. So, the two triangles are similar because of a translation and dilation.

  23. Example 1- Determine if the two rectangles are similar using transformations. STEP 1: Rotate rectangle VWTU 90 clockwise about W so that it is orientated the same way as rectangle WXYZ. STEP 2: Write ratios comparing the lengths of each side. The ratios are not equal, so the two rectangles are not similar.

  24. Got it? 1 a. b.

  25. Scale Factor Similar figures have the same shape, but have different sizes. They sizes of the two figures are related to the scale factor of the dilation.

  26. Example 2 Ken enlarges the photo shown by a scale factor of 2 for his webpage. He then enlarges the webpage photo by a scale factor of 1.5 to print. If the original photo is 2 inches by 3 inches, what are the dimensions of the print? Are the enlarged photos similar to the original? Size of webpage photo: 2 in x 2 = 4 3 in x 2 = 6 Size of print: 4 in x 1.5 = 6 6 in x 1.5 = 9 The printed photo is a 6 x 9. All three photos are similar.

  27. Got it? 2 An art show offers different size prints of the same painting. The original print measures 24 centimeters by 30 centimeters. A printer enlarges the original by a scale factor of 1.5, and then enlarges the second image by a scale factor of 3. What are the dimensions of the largest print? Are both prints similar to the original? Size of second photo: 36 cm x 3 = 108 cm 45 cm x 3 = 135 cm. Size of printed photo: 24 cm x 1.5 = 36 cm 30 cm x 1.5 = 45 cm The largest size will be 108 x 135 cm. Yes, all three sizes are similar.

  28. Properties of Similar Polygons Chapter 7, Lesson 4

  29. ΔABC ΔXYZ is read as “triangle ABC is similar to triangle XYZ.”

  30. Example 1 – Determine if the figures are similar. Ask: Are the angles congruent? YES Then ask: Are the sides proportional? Since and are not equal, the rectangles are not similar.

  31. Got it? 1 Determine if the figures are similar. No; the corresponding angles are not congruent, and

  32. Scale Factor A ratio of the lengths of two corresponding sides of two similar polygons. Example: The two squares have a scale factor of 1.5 or .

  33. Example 2 Quadrilateral WXYZ is similar to quadrilateral ABCD. Describe the transformation that map WXYZ onto ABCD. Since the figures are similar and not congruent, a translation followed by a dilation would map WXYZ onto ABCD.

  34. Example 2 Quadrilateral WXYZ is similar to quadrilateral ABCD. Find the missing measure. METHOD 1: Find the scale factor between the two figures. scale factor = YZ is 1.5 times larger than CD. So, m would be 1.5 times larger than 12. m = 12(1.5) m = 18

  35. Example 2 Quadrilateral WXYZ is similar to quadrilateral ABCD. Find the missing measure. METHOD 2: Setup a proportion to find the missing measure. m ∙ 10 = 12 ∙ 15 10m = 180 m = 18

  36. Got it? 2 Find each missing measure. • WZ 19.5 • AB 16

  37. Similar Triangles and Indirect Measure Chapter 7, Lesson 5

  38. Angle-Angle (AA) Similarity

  39. Proof of the AA Similiary Rule In the figure below, and If you extend the lines, you can see the two triangles are similar.

  40. Example 1 Determine whether the triangles are similar. If so, write a similarity statement. Angle A and E have the same measure. Since 180 – 62 – 48 = 70, G measures 70. Angle G and C have the same measure. Two angles in ΔABC and ΔEFG are congruent. ΔABC ΔEFG

  41. Got it? 1 Determine whether the triangles are similar. If so, write a similarity statement. Angle H and L have the same measure. Angle JKH and MKL are the same. Two angles in ΔHJK and ΔLKM are congruent. ΔHJK ΔLMK

  42. Indirect Measure Used to measure very large or very small items. Since the two shapes are similar, then the angles are congruent. We can use proportions to find the missing length.

  43. Example 2 A fire hydrant 2.5 feet high casts a 5 foot shadow. How tall is a street light that casts a 26-foot shadow at the same time? Let h represent the height of the street light. 5h = 26(2.5) 5h = 65 h = 13 The street light is 13 feet tall.

  44. Example 3 In the figure at the right, triangle DBA is similar to triangle ECA. Ramon wants to know the distance across the lake. 320d = 482(40) 320d = 19,280 d = 60.25 The distance across the lake is 60.25 meters.

  45. Got it? 2 & 3 At the same time a 2-meter street sign casts a 3-meter shadow, a nearby telephone pole casts a 12.3 meter shadow. How tall is the telephone pole? The telephone pole is 12 meters.

  46. Slope and Similar Triangles Chapter 7, Lesson 6

  47. Similar Triangles and the Coordinate Plane

  48. Example 1 Write a proportion comparing the rise to the run for each of the similar slope triangles shown above. Then find the numeric value. Slope = 2

  49. Got it? 1 Write a proportion comparing the rise to the run for each of the similar slope triangles shown above. Then find the numeric value.