Curriculum / Math / 10th Grade / Unit 3: Dilations and Similarity / Lesson 12
Math
Unit 3
10th Grade
Lesson 12 of 18
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Prove angle-angle criterion for two triangles to be similar.
The core standards covered in this lesson
G.SRT.A.3 — Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar.
The foundational standards covered in this lesson
7.G.A.2 — Draw (freehand, with ruler and protractor, and with technology) geometric shapes with given conditions. Focus on constructing triangles from three measures of angles or sides, noticing when the conditions determine a unique triangle, more than one triangle, or no triangle.
8.G.A.4 — Understand that a two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations; given two similar two-dimensional figures, describe a sequence that exhibits the similarity between them.
The essential concepts students need to demonstrate or understand to achieve the lesson objective
Suggestions for teachers to help them teach this lesson
Criteria for Success #5 is continued from Lesson 12 through Lesson 13.Â
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Problems designed to teach key points of the lesson and guiding questions to help draw out student understanding
In the two triangles pictured below, $${m∠B=m∠E}$$ and $${m∠A=m∠D}$$.
Using a sequence of translations, rotations, reflections, and/or dilations, show that $${\triangle ABC \sim \triangle DEF}$$.
Similar Triangles, accessed on Oct. 19, 2017, 3:07 p.m., is licensed by Illustrative Mathematics under either the CC BY 4.0 or CC BY-NC-SA 4.0. For further information, contact Illustrative Mathematics.
Given: $${\overline{AB}\parallel\overline{CD}}$$
Prove: $${\triangle{ABE}\sim\triangle{DCE}}$$
A task that represents the peak thinking of the lesson - mastery will indicate whether or not objective was achieved
Triangles $${ABC}$$ and $${PQR}$$ below share two pairs of congruent angles as marked:
a.   Explain, using dilations, translations, reflections, and/or rotations why $$\triangle {PQR}$$ is similar to $$\triangle {ABC}$$.
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b.   Are angles $$C$$ and $$R$$ congruent? How do you know?
c.   Can you show the similarity in part a without using a reflection? What about without using a dilation? Explain.
d.   Suppose $${{DEF}}$$ and $${{KLM}}$$ are two triangles with $${m \angle D=m\angle K}$$ and $${m\angle E = m\angle L}$$. Are triangles $${{DEF}}$$ and $${{KLM}}$$ similar?
Similar Triangles II, accessed on Oct. 13, 2017, 4:25 p.m., is licensed by Illustrative Mathematics under either the CC BY 4.0 or CC BY-NC-SA 4.0. For further information, contact Illustrative Mathematics.
The following resources include problems and activities aligned to the objective of the lesson that can be used for additional practice or to create your own problem set.
Lesson 11
Lesson 13
Topic A: Dilations off the Coordinate Plane
Describe properties of scale drawings.
G.SRT.A.2
Define and describe the characteristics of dilations. Dilate figures using constructions when the center of dilation is not on the figure.Â
G.CO.A.2 G.SRT.A.2 G.SRT.A.3
Verify that dilations result in congruent angles and proportional line segments.
Divide a line segment into equal sections using dilation.Â
G.CO.D.12 G.SRT.A.1.A G.SRT.A.1.B
Dilate a figure from a point on the figure. Use the properties of dilations with respect to parallel lines to verify dilations and find the center of dilation.
G.CO.D.12 G.SRT.A.1.A G.SRT.A.2
Prove that a line parallel to one side of a triangle divides the other two sides proportionally.
G.CO.C.10 G.SRT.B.4 G.SRT.B.5
Identify measurements in dilated figures with the center of dilation on the figure directly and algebraically.
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Topic B: Dilations on the Coordinate Plane
Dilate a figure on the coordinate plane when the center of dilation is the origin.
G.CO.A.2 G.SRT.A.2
Dilate a figure when the center of dilation is not the origin. Determine center of dilation given the original and dilated figure.
Topic C: Defining Similarity
Define similarity transformation as the composition of basic rigid motions and dilations. Describe similarity transformation applied to an arbitrary figure.
G.SRT.A.2 G.SRT.B.5
Prove that all circles are similar.
G.C.A.1
G.SRT.A.3
Use angle-angle criterion to prove two triangles to be similar.
Develop the side splitter theorem and side-angle-side similarity criteria, and use these in the solution of problems.
G.SRT.B.5
Topic D: Similarity Applications
Develop the angle bisector theorem based on facts about similarity and congruence, and use this in the solution of problems.
Use the side-side-side criteria for similarity and other similarity and congruence theorems in the solution of problems.
Solve for measurements involving right triangles using scale factors and ratios.
Solve real-life problems with two different centers of dilation.
G.SRT.B.4 G.SRT.B.5
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