MA.912.GR.2.9

loading spinner
Export Print
Justify the criteria for triangle similarity using the definition of similarity in terms of non-rigid transformations.
General Information
Subject Area: Mathematics (B.E.S.T.)
Grade: 912
Strand: Geometric Reasoning
Standard: Apply properties of transformations to describe congruence or similarity.
Date Adopted or Revised: 08/20
Status: State Board Approved

Benchmark Instructional Guide

Connecting Benchmarks/Horizontal Alignment

 

Terms from the K-12 Glossary

  • Coordinate Plane
  • Dilation
  • Origin
  • Reflection
  • Rigid Transformation
  • Rotation
  • Scale Factor
  • Translation

 

Vertical Alignment

Previous Benchmarks

Next Benchmarks

 

Purpose and Instructional Strategies

In grade 8, students first learned to associate congruence and similarity with reflections, rotations, translations and dilations. In Geometry, students learn that these transformations provide an equivalent alternative to using congruence and similarity criteria for triangles.
  • Instruction includes the connection to the Angle-Angle, Side-Angle-Side, Hypotenuse-Leg and Side-Side-Side similarity criteria and to justifying congruence criteria.
    • For example, if one wants to justify the Angle-Angle criterion, one method is as follows: Start with triangles ABC and PQR, with ∠A ≅ ∠P and ∠B ≅ ∠Q, as shown below.
      Triangle
      Students should be able to realize the need of a dilation, in this case with a scale factor k such that 0 < k < 1 and k = PQAB. After this dilation, triangle A’B’C’ is obtained, such that the length of A’B’ equals the length of PQ. Since dilations preserve angle measures, ∠A ≅ ∠A′ and ∠B ≅ ∠B′. Using the transitive property of congruence, if ∠A ≅ ∠P and ∠A ≅ ∠A′, then ∠P ≅ ∠A′, and if ∠B ≅ ∠Q and ∠B ≅ ∠B′, then ∠Q ≅ ∠B′. With PQ ≅ A′B′, ∠P ≅ ∠A′ and ∠Q ≅ ∠B′, we can prove ΔPQR ≅ ΔA′B′C′ by Side-Angle-Side Congruence Criterion. Additionally, since ΔABC~ΔA′B′C′ and ΔPQR ≅ ΔA′B′C′, it can be concluded that ΔABC~ΔPQR. Justification of the other criteria can be done in a similar manner.

 

Common Misconceptions or Errors

  • When determining the scale factor of a dilation, students may misidentify the preimage and image, leading to an incorrect scale factor.

 

Instructional Tasks

Instructional Task 1 (MTR.3.1, MTR.4.1)
  • Triangle XYZ has the coordinates (0,2), (2,4) and (6,0) and triangle DEF has the coordinates (4, −4), (8,0) and (16, −8).
    • Part A. How can ΔACB~ΔLMN be proved using one of the similarity criteria?
    • Part B. How can ΔACB~ΔLMN be proved using rigid and non-rigid transformations?

 

Instructional Items

Instructional Item 1
  • Shown below are two triangles where m∠X = m∠R, m∠Y = m∠S, and m∠Z = m∠T. Determine a dilation that maps ΔXYZ onto ΔRST.
    Triangle

 

*The strategies, tasks and items included in the B1G-M are examples and should not be considered comprehensive.

Related Courses

This benchmark is part of these courses.
1206320: Geometry Honors (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))

Related Access Points

Alternate version of this benchmark for students with significant cognitive disabilities.

Related Resources

Vetted resources educators can use to teach the concepts and skills in this benchmark.

Formative Assessments

Proving Similarity:

Students are asked to explain similarity in terms of transformations.

Type: Formative Assessment

The Consequences of Similarity:

Students are given the definition of similarity in terms of similarity transformations and are asked to explain how this definition ensures the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides.

Type: Formative Assessment

Lesson Plans

Transformation and Similarity:

Using non-rigid motions (dilations), students learn how to show that two polygons are similar. Students will write coordinate proofs confirming that two figures are similar.

Type: Lesson Plan

How Much Proof Do We Need?:

Students determine the minimum amount of information needed to prove that two triangles are similar.

Type: Lesson Plan

Congruence vs. Similarity:

Students will learn the difference between congruence and similarity of classes of figures (such as circles, parallelograms) in terms of the number of variable lengths in the class. A third category will allow not only rigid motions and dilations, but also a single one-dimensional stretch, allowing more classes of figures to share sufficient common features to belong.

Type: Lesson Plan

Dilation Transformation:

Students identify dilations, verify that polygons are similar, and use the dilation rule to map dilations. Task cards are provided for independent practice. The PowerPoint also includes detailed illustrations for constructing a dilation using a compass and a straight edge.

Type: Lesson Plan

Altitude to the Hypotenuse:

Students will discover what happens when the altitude to the hypotenuse of a right triangle is drawn. They learn that the two triangles created are similar to each other and to the original triangle. They will learn the definition of geometric mean and write, as well as solve, proportions that contain geometric means. All discovery, guided practice, and independent practice problems are based on the powerful altitude to the hypotenuse of a right triangle.

Type: Lesson Plan

Problem-Solving Task

Are They Similar?:

In this problem, students are given a picture of two triangles that appear to be similar, but whose similarity cannot be proven without further information. Asking students to provide a sequence of similarity transformations that maps one triangle to the other, using the definition of similarity in terms of similarity transformations.

Type: Problem-Solving Task

MFAS Formative Assessments

Proving Similarity:

Students are asked to explain similarity in terms of transformations.

The Consequences of Similarity:

Students are given the definition of similarity in terms of similarity transformations and are asked to explain how this definition ensures the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides.

Student Resources

Vetted resources students can use to learn the concepts and skills in this benchmark.

Problem-Solving Task

Are They Similar?:

In this problem, students are given a picture of two triangles that appear to be similar, but whose similarity cannot be proven without further information. Asking students to provide a sequence of similarity transformations that maps one triangle to the other, using the definition of similarity in terms of similarity transformations.

Type: Problem-Solving Task

Parent Resources

Vetted resources caregivers can use to help students learn the concepts and skills in this benchmark.

Problem-Solving Task

Are They Similar?:

In this problem, students are given a picture of two triangles that appear to be similar, but whose similarity cannot be proven without further information. Asking students to provide a sequence of similarity transformations that maps one triangle to the other, using the definition of similarity in terms of similarity transformations.

Type: Problem-Solving Task