Lesson 14 | Dilations and Similarity | 10th Grade Mathematics

Objective

Develop the side splitter theorem and side-angle-side similarity criteria, and use these in the solution of problems.

Common Core Standards

Core Standards

The core standards covered in this lesson

G.SRT.B.5 — Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.

Similarity, Right Triangles, and Trigonometry

G.SRT.B.5 — Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.

Foundational Standards

The foundational standards covered in this lesson

8.G.A.4

Geometry

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.

Criteria for Success

The essential concepts students need to demonstrate or understand to achieve the lesson objective

Use the angle-angle criteria for similarity to develop the side splitter theorem.
Explain the side splitter theorem as “If a line is parallel to a side of a triangle and intersects the other two sides, then this line divides those two sides proportionally” and the converse: “If a line segment divides two sides of a triangle proportionally, it is parallel to the third side.”
Extend the side splitter theorem and the angle-angle criteria for similarity to establish the theorem that if a segment joins two midpoints of a triangle, it is parallel to the third side and half the length.
Identify the side-angle-side criteria for similarity through the side splitter theorem and the angle-angle criteria for similarity.

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Anchor Problems

Problems designed to teach key points of the lesson and guiding questions to help draw out student understanding

Problem 1

In the diagram below, $${\overline{DE} \parallel \overline{BC}}$$.

What is the relationship between $${\overline{AD}}$$ and $${\overline{DB}}$$?
What is the relationship between $${\overline{AE}}$$ and $${\overline{EC}}$$?

Explain how you know.

Guiding Questions

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Problem 2

Below are two triangles that have two proportional sides with the corresponding included angle congruent.

Use what you know from Anchor Problem #1 to explain and write a side-angle-side criterion for similarity.

Guiding Questions

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References

GeoGebra CCSS High School: Geometry (Similarity, Right T, Trig.) — 4.1 SAS ~ Theorem

CCSS High School: Geometry (Similarity, Right T, Trig.) by Tim Brzezinski is made available by GeoGebra under the CC BY-NC-SA 3.0 license. Copyright © International GeoGebra Institute, 2013. Accessed Oct. 2, 2017, 10:33 a.m..

Target Task

A task that represents the peak thinking of the lesson - mastery will indicate whether or not objective was achieved

In the triangle shown below:

$${AD=2.5}$$

$${BC=9}$$

What do the lengths of $${\overline{DB}}$$ and $${\overline{BE}}$$ need to be so that triangles $${DBE}$$ and $${ABC}$$ are similar? Explain your reasoning.
How can you use SAS to prove the triangles similar? How can you use AA to prove the triangles similar?

Additional Practice

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.

EngageNY Mathematics Geometry > Module 2 > Topic C > Lesson 17 — Only use problems that have to do with SAS similarity
CK-12 SAS Similarity
Mathematics Vision Project: Secondary Mathematics Two Module 6: Similarity and Right Triangle Trigonometry — Lesson 6.3 (Focus on SAS)
MARS Formative Assessment Lesson for Grade 8 Identifying Similar Triangles — Focus on SAS

Lesson 13

Lesson 15

Lesson Map

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.

G.CO.A.2 G.SRT.A.2 G.SRT.A.3

Verify that dilations result in congruent angles and proportional line segments.

G.SRT.A.2

Divide a line segment into equal sections using dilation.

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.

G.SRT.A.2

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.

G.CO.A.2 G.SRT.A.2

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

Prove angle-angle criterion for two triangles to be similar.

G.SRT.A.3

Use angle-angle criterion to prove two triangles to be similar.

G.SRT.A.3

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.

G.SRT.B.5

Use the side-side-side criteria for similarity and other similarity and congruence theorems in the solution of problems.

G.SRT.B.5

Solve for measurements involving right triangles using scale factors and ratios.

G.SRT.B.5

Solve real-life problems with two different centers of dilation.

G.SRT.B.4 G.SRT.B.5

Dilations and Similarity

Objective

Common Core Standards

Core Standards

Similarity, Right Triangles, and Trigonometry

Foundational Standards

Geometry

Criteria for Success

Anchor Problems

Problem 1

Guiding Questions

Problem 2

Guiding Questions

References

Target Task

Additional Practice

Lesson Map

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