Welcome to Math 6 Today’s subject is: Proportions and Similar Figures

Objectives for this lesson: Each student will: 1- Use proportions to determine if two figures are similar. 2-Use proportions to make similar figures. 3-Use proportions to find the length of corresponding sides of similar figures.

Key Vocabulary Matching sides of two or more polygons are called corresponding sides, and matching angles are called corresponding angles. 𝑸 𝑨 𝑹 𝑺 𝑪 𝑩

Use ratios to determine if two figures are similar. 𝐒 𝐄 𝟕 𝐢𝐧 10 𝐢𝐧 𝟑𝟎 𝐢𝐧 21 𝐢𝐧 𝐃 𝐅 𝟖 𝐢𝐧 𝐐 𝐑 24 𝐢𝐧

DE corresponds to RS. EF corresponds to SQ QR corresponds to FD 24 𝐢𝐧 21 𝐢𝐧 𝟑𝟎 𝐢𝐧 𝐐 𝐒 𝐑 𝟖 𝐢𝐧 𝟕 𝐢𝐧 10 𝐢𝐧 𝐅 𝐃 𝐄 DE corresponds to RS. EF corresponds to SQ QR corresponds to FD

24 𝐢𝐧 21 𝐢𝐧 𝟑𝟎 𝐢𝐧 𝐐 𝐒 𝐑 𝟖 𝐢𝐧 𝟕 𝐢𝐧 10 𝐢𝐧 𝐅 𝐃 𝐄 𝐷𝐸 𝑅𝑆 = 𝐸𝐹 𝑆𝑄 = 𝐹𝐷 𝑄𝑅

24 𝐢𝐧 21 𝐢𝐧 𝟑𝟎 𝐢𝐧 𝐐 𝐒 𝐑 𝟖 𝐢𝐧 𝟕 𝐢𝐧 10 𝐢𝐧 𝐅 𝐃 𝐄 7 21 = 10 30 = 8 24

Similar figures have corresponding sides and corresponding angles. Similar figures have the same shape, but not necessarily the same size. Similar figures have corresponding sides and corresponding angles. The symbol ~ means “is similar to.” ABC ~ QRS. 𝑺 𝑹 𝑸 𝑨 𝑪 𝑩

If two figures are similar, then their corresponding angles are congruent, and the lengths of the corresponding sides are proportional. 𝒊 𝒋 𝒘 𝒛 𝒌 𝒉 𝒚 𝒙

To find out if triangles are similar, determine whether the ratios of the lengths of their corresponding sides are proportional. If the ratios are proportional, then the corresponding angles must have equal measures.

Similar figures have two properties: The corresponding angles have equal measures. The lengths of the corresponding sides are in proportion.

In figures with four or more sides, it is possible for the corresponding side lengths to be proportional and the figures to have different shapes. To find out if these figures are similar, first check that their corresponding angles have equal measures. 10 10 7 7 7 7 10 10

When you know two figures are similar, you can find the measure of a missing side by setting up a proportion. Parallelogram ABCD ~ Parallelogram EFGH. Find the value of x. 24 24 18 𝑥

Create a proportion:

Guided Practice

Guided Practice #1 Identify the corresponding sides in each pair of triangles. Then use ratios to find whether the triangles are similar. 3𝟎 𝐢𝐧 20 𝐢𝐧 4𝟎 𝐢𝐧 𝐐 𝐒 𝐑 6 𝐢𝐧 9 𝐢𝐧 12 𝐢𝐧 𝐅 𝐃 𝐄

Guided Practice #1 Notes to instructor Using doc, the instructor will present the previous slide and make the following points: In order to identify which sides are corresponding, compare the relative sizes of the sides. Ie. The two shortest sides must be corresponding, the two longest sides must be corresponding, etc. Then model how to set up a pair of ratios and tell if they are equal by using cross-multiplying. If the cross-products are equal, then the sides are in proportion. YES! Guided practice # 1 shows SIMILAR FIGURES. 3𝟎 𝐢𝐧 20 𝐢𝐧 4𝟎 𝐢𝐧 𝐐 𝐒 𝐑 6 𝐢𝐧 9 𝐢𝐧 12 𝐢𝐧 𝐅 𝐃 𝐄

Guided Practice #2 Identify the corresponding sides in each pair of triangles. Then use ratios to find whether the triangles are similar. 𝑩 28 𝑺 𝑹 𝑸 15 7 3 𝑨 5 𝑪 20

𝑩 𝑺 𝑹 𝑸 𝑨 28 15 7 3 5 𝑪 20 Guided Practice #2 Notes to instructor Again, demonstrate how you can use the relative lengths of the sides to find corresponding sides. Order the lengths of sides: 15, 20, 28 and 3, 5, 7. Cross multiply. Since the cross products are not equal, these figures are not similar. 𝑩 𝑺 𝑹 𝑸 𝑨 28 15 7 3 5 𝑪 20

Guided Practice #3 Use the properties of similarity to determine whether the figures are similar. 4 𝑩 𝑪 𝑨 𝑫 𝑭 𝑮 𝑬 𝑯 6 2 2 3 3 4 6

4 𝑩 𝑪 𝑨 𝑫 𝑭 𝑮 𝑬 𝑯 6 2 2 3 3 4 6 Guided Practice #3 Notes to instructor Same procedure, these figures ARE similar. 4 𝑩 𝑪 𝑨 𝑫 𝑭 𝑮 𝑬 𝑯 6 2 2 3 3 4 6

Guided Practice #4 Use the properties of similarity to determine whether the figures are similar. 9 10 𝒉 𝒊 𝒋 𝒘 𝒛 𝒙 10 10 14 14 𝒌 18 y 24

Guided Practice #4 Notes to instructor Same procedure, these figures ARE NOT similar. 9 10 𝒉 𝒊 𝒋 𝒘 𝒛 𝒙 10 10 14 14 𝒌 18 y 24

Guided Practice #5 These two triangles are similar Guided Practice #5 These two triangles are similar. Use a proportion to find the length of the missing side. 𝟓 𝟐𝟎 𝒚 𝟏𝟔 𝟐 𝟖

Guided Practice #5 Notes to instructor Show how the measure of y = 4.

Independent Practice

Independent Practice #1 Use the properties of similarity to determine whether the figures are similar. 6 10 𝒉 𝒘 𝒙 𝒊 18 25 25 18 𝒌 y 15 𝒋 𝒛 24

Independent Practice #2 Identify the corresponding sides in each pair of triangles. Then use ratios to find whether the triangles are similar. 𝒃 𝑺 𝑹 𝑸 𝒂 𝟏𝟎 𝟖 𝟏𝟒 𝟔 𝟒 𝒄 𝟕

Independent Practice #3 Identify the corresponding sides in each pair of triangles. Then use ratios to find whether the triangles are similar. 𝟏𝟓 𝟏𝟖 𝟏𝟎 𝟏𝟐 𝟖 𝟏𝟐

Independent Practice #4 Identify the corresponding sides in each pair of triangles. Then use ratios to find whether the triangles are similar. 𝟏𝟖 𝟏𝟎 𝟏𝟎 𝟏𝟓 𝟏𝟓 𝟏𝟐

Independent Practice #5 These two triangles are similar Independent Practice #5 These two triangles are similar. Use a proportion to find the length of the missing side. 𝟏𝟓 𝟐𝟎 𝟏𝟐 𝟏𝟔 𝒏 𝟖

Independent Practice #6 These two triangles are similar Independent Practice #6 These two triangles are similar. Use a proportion to find the length of the missing side. 𝒔 𝟏𝟐 𝟔 𝟒

Independent Practice #7 Use the properties of similarity to determine whether the figures are similar. 𝒊 24 12 𝒉 6 𝒘 𝒙 𝒋 18 18 24 24 𝒛 y 24 𝒌

Independent Practice #8 Write a proportion for corresponding sides which would make these similar figures. 𝑭 𝑬 G 𝑯 𝑪 𝑩 𝑨 𝑫

Independent Practice #9 Write a proportion for corresponding sides which would make these similar figures. Trapezoid EFGH ~ trapezoid MNOP

Independent Practice #10 Write a proportion for corresponding sides which would make these similar figures.

Conclusion In this lesson we: 1- Used proportions to determine if two figures are similar. 2-Used proportions to make similar figures. 3-Used proportions to find the length of corresponding sides of similar figures.

Assignments- Complete the attached exercise entitled, “Worksheet 4.2: Proportions and Similar Figures.” Look at Math website for more information about Proportions and similar figures. http://www.math.com/school/subject3/lessons/S3U3L2GL.html

Assignments- 3. Here is one more website to check out for more information. This one actually has a quiz at the end. You can test yourself to find out how much of this you have retained. The website is http://studyjams.scholastic.com/studyjams/jams/math/geometry/similar-figures.htm

See you next time! See you next time! See you next time! See you next time!