**Subject Area:**Mathematics (B.E.S.T.)

**Grade:**912

**Strand:**Geometric Reasoning

**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.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 $k$ = $\frac{\text{PQ}}{\text{AB}}$. 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 Δ*PQR*≅ Δ*A*′*B*′*C*′ by Side-Angle-Side Congruence Criterion. Additionally, since Δ*ABC*~Δ*A*′*B*′*C*′ and Δ*PQR*≅ Δ*A*′*B*′*C*′, it can be concluded that Δ*ABC*~Δ*PQR*. Justification of the other criteria can be done in a similar manner.

- For example, if one wants to justify the Angle-Angle criterion, one method is as follows: Start with triangles

### 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 Δ
*ACB*~Δ*LMN*be proved using one of the similarity criteria? - Part B. How can Δ
*ACB*~Δ*LMN*be proved using rigid and non-rigid transformations?

- Part A. How can Δ

### 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 Δ*XYZ*onto Δ*RST*.

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

## Related Courses

## Related Access Points

## Related Resources

## Formative Assessments

## Lesson Plans

## Problem-Solving Task

## MFAS Formative Assessments

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

## Problem-Solving Task

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

## Problem-Solving Task

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