About This Chapter
Standard: Verify experimentally the properties of dilations given by a center and a scale factor when: (CCSS.Math.Content.HSG-SRT.A.1) a dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged (CCSS.Math.Content.HSG-SRT.A.1a) or the dilation of a line segment is longer or shorter in the ratio given by the scale factor. (CCSS.Math.Content.HSG-SRT.A.1b)
Standard: Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides. (CCSS.Math.Content.HSG-SRT.A.2)
Standard: Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar. (CCSS.Math.Content.HSG-SRT.A.3)
Standard: Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. (CCSS.Math.Content.HSG-SRT.B.4)
Standard: Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. (CCSS.Math.Content.HSG-SRT.B.5)
About This Chapter
Students who understand the concept of similarity will be able to identify the properties of dilations and of congruent and similar shapes. They will have a grasp of similarity in terms of similarity transformations and be able to utilize this understanding to prove theorems that involve similarity. Students who comprehend these principles will be capable of using similarity and congruence to prove relationships in figures.
The lessons that meet this standard cover:
- Properties of dilations
- How to identify similar triangles
- Properties of congruent and similar shapes
- Similarity transformations
- Similar triangles and the AA criterion
- Applications of similar triangles
- Proving similarity theorems about triangles
- Proving relationships in figures using congruence and similarity
- Practice proving relationships
Students can demonstrate their mastery of these topics by calculating the properties of similar and congruent shapes. Armed with this knowledge, they can prove theorems about triangles, determine the similarity between two geometric figures and prove relationships between figures. This standard and these lessons can ready them for further studies and possible careers in fields such as mathematics, engineering and astronomy.
How to Use These Lessons in Your Classroom
Here are some tips on how you can incorporate our lessons on similarity into your curriculum to help meet the common core standards:
Congruent and Similar Shapes Lesson
Have the students watch the video lessons covering congruent and similar shapes and their properties. Give each student an assortment of geometric shapes cut from paper and have them determine, by flipping, rotating and measuring, whether shapes are congruent or similar.
AA Criterion Lesson Quizzes
Ask students to take the quiz on the AA Criterion, then watch the Similar Triangles and the AA Criterion lesson together in the classroom. Discuss the topic and let them test their comprehension by taking the quiz again. Review the answers together.
Properties of Dilations
Have students view the video lesson Properties of Dilations as homework or in class to introduce students to the concept of dilations. Give them coordinates and a scale factor and ask them to create a graph illustrating the dilation.
1. How to Identify Similar Triangles
Similar triangles have the same characteristics as similar figures but can be identified much more easily. Learn the shortcuts for identifying similar triangles here and test your ability with a quiz.
2. Properties of Congruent and Similar Shapes
In this lesson, we'll look at triangles, rectangles and other shapes that share properties. This includes both congruent and similar shapes. We'll also practice identifying the missing properties of these shapes.
3. Applications of Similar Triangles
Similar triangles are used to solve problems in everyday situations. Learn how to solve with similar triangles here, and then test your understanding with a quiz.
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