# Cluster 2: Prove theorems involving similarity. (Geometry - Major Cluster)Archived

Clusters should not be sorted from Major to Supporting and then taught in that order. To do so would strip the coherence of the mathematical ideas and miss the opportunity to enhance the major work of the grade with the supporting clusters.

General Information
Number: MAFS.912.G-SRT.2
Title: Prove theorems involving similarity. (Geometry - Major Cluster)
Type: Cluster
Subject: Mathematics - Archived
Domain-Subdomain: Geometry: Similarity, Right Triangles, & Trigonometry

## Related Standards

This cluster includes the following benchmarks.

## Related Access Points

This cluster includes the following access points.

## Access Points

MAFS.912.G-SRT.2.AP.4a
Establish facts about the lengths of segments of sides of a triangle when a line parallel to one side of the triangles divides the other two sides proportionally.
MAFS.912.G-SRT.2.AP.5a
Apply the criteria for triangle congruence and/or similarity (angle-side-angle [ASA], side-angle-side [SAS], side-side-side [SSS], angle-angle [AA] to determine if geometric shapes that divide into triangles are or are not congruent and/or can be similar.

## Related Resources

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

## Formative Assessments

Pythagorean Theorem Proof:

Students are asked to prove the Pythagorean Theorem using similar triangles.

Type: Formative Assessment

Geometric Mean Proof:

Students are asked to prove that the length of the altitude to the hypotenuse of a right triangle is the geometric mean of the lengths of the two segments of the hypotenuse.

Type: Formative Assessment

Converse of the Triangle Proportionality Theorem:

Students are asked to prove that if a line intersecting two sides of a triangle divides those two sides proportionally, then that line is parallel to the third side.

Type: Formative Assessment

Triangle Proportionality Theorem:

Students are asked to prove that a line parallel to one side of a triangle divides the other two sides of the triangle proportionally.

Type: Formative Assessment

County Fair:

Students are given a diagram of a county fair and are asked to use similar triangles to determine distances from one location of the fair to another.

Type: Formative Assessment

Students are asked to decide if a basketball goal is regulation height and are given enough information to determine this using similar triangles.

Type: Formative Assessment

Prove Rhombus Diagonals Bisect Angles:

Students are asked to prove a specific diagonal of a rhombus bisects a pair of angles.

Type: Formative Assessment

Similar Triangles - 2:

Students are asked to locate a pair of similar triangles in a diagram, explain why they are similar, and use the similarity to find an unknown length in the diagram.

Type: Formative Assessment

Similar Triangles - 1:

Students are asked locate a pair of similar triangles in a diagram, explain why they are similar, and use the similarity to find two unknown lengths in the diagram.

Type: Formative Assessment

## Lesson Plans

What's the Problem:

Students solve problems using triangle congruence postulates and theorems.

Type: Lesson Plan

How Do You Measure the Immeasurable?:

Students will use similar triangles to determine inaccessible measurements. Examples include exploring dangerous caves and discovering craters on Mars.

Type: Lesson Plan

Let's Prove the Pythagorean Theorem:

Students will use Triangle Similarity to derive the proof of the Pythagorean Theorem and apply this method to develop the idea of the geometric mean with respect to the relationships in right triangles.

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

Proofs of the Pythagorean Theorem:

This lesson is intended to help you assess how well students are able to produce and evaluate geometrical proofs. In particular, this unit is intended to help you identify and assist students who have difficulties in:

• Interpreting diagrams.
• Identifying mathematical knowledge relevant to an argument.
• Linking visual and algebraic representations.
• Producing and evaluating mathematical arguments.

Type: Lesson Plan

Modeling: Rolling Cups:

This lesson unit is intended to help you assess how well students are able to choose appropriate mathematics to solve a non-routine problem, generate useful data by systematically controlling variables and develop experimental and analytical models of a physical situation.

Type: Lesson Plan

Solving Geometry Problems: Floodlights:

This lesson unit is intended to help you assess how well students are able to identify and use geometrical knowledge to solve a problem. In particular, this unit aims to identify and help students who have difficulty in making a mathematical model of a geometrical situation, drawing diagrams to help with solving a problem, identifying similar triangles and using their properties to solve problems and tracking and reviewing strategic decisions when problem-solving.

Type: Lesson Plan

Mirror, Mirror on the ... Ground?:

This activity allows students to go outdoors to measure the height of objects indirectly. Similar right triangles are formed when mirrors are placed on the ground between the object that needs to be measured and the student observing the object in the mirror. Students work in teams to measure distances and solve proportions.

This activity is meant to be a review activity after similar triangles have already been taught, and can be used as a summative assessment.

Type: Lesson Plan

Patterns in Fractals:

This lesson is designed to introduce students to the idea of finding patterns in the generation of several different types of fractals. This lesson provides links to discussions and activities related to patterns and fractals as well as suggested ways to work them into the lesson. Finally, the lesson provides links to follow-up lessons designed for use in succession with the current one.

Type: Lesson Plan

## Original Student Tutorial

Use properties, postulates, and theorems to prove a theorem about a triangle. In this interactive tutorial, you'll also learn how to prove that a line parallel to one side of a triangle divides the other two proportionally.

Type: Original Student Tutorial

## Perspectives Video: Teaching Idea

Measuring Height with Triangles and Mirrors:

Reflect for a moment on how to measure tall objects with mirrors and mathematics.

Type: Perspectives Video: Teaching Idea

## Presentation/Slideshow

The Pythagorean Theorem: Geometry’s Most Elegant Theorem:

This lesson teaches students about the history of the Pythagorean theorem, along with proofs and applications. It is geared toward high school Geometry students that have completed a year of Algebra and addresses the following national standards of the National Council of Teachers of Mathematics and the Mid-continent Research for Education and Learning: 1) Analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships; 2) Use visualization, spatial reasoning, and geometric modeling to solve problems; 3) Understand and apply basic and advanced properties of the concepts of geometry; and 4) Use the Pythagorean theorem and its converse and properties of special right triangles to solve mathematical and real-world problems. The video portion is about thirty minutes, and with breaks could be completed in 50 minutes. (You may consider completing over two classes, particularly if you want to allow more time for activities or do some of the enrichment material). These activities could be done individually, in pairs, or groups. I think 2 or 3 students is optimal. The materials required for the activities include scissors, tape, string and markers.

Type: Presentation/Slideshow

Bank Shot:

This task asks students to use similarity to solve a problem in a context that will be familiar to many, though most students are accustomed to using intuition rather than geometric reasoning to set up the shot.

Extensions, Bisections and Dissections in a Rectangle:

This task involves a reasonably direct application of similar triangles, coupled with a moderately challenging procedure of constructing a diagram from a verbal description.

Joining two midpoints of sides of a triangle:

Using a triangle with line through it, students are tasked to show the congruent angles, and conclude if one triangle is similar to the other.

Unit Squares and Triangles:

This problem solving task asks students to find the area of a triangle by using unit squares and line segments.

## Professional Development

Using Similarity to Prove the Pythagorean Theorem:

In the eighth grade, your students were introduced to the Pythagorean Theorem and its converse. They applied this theorem to a variety of real-world and mathematical problems to find the lengths of sides of right triangles. Now that your students understand concepts related to similarity, they are able to appreciate a simple proof of the Pythagorean Theorem. The focus of this tutorial is on describing and explaining a proof of the Pythagorean Theorem using similar triangles.

Type: Professional Development

## Tutorials

Bhaskara's Proof of the Pythagorean Theorem:

This video demonstrates Bhaskara's proof of the Pythagorean Theorem.

Type: Tutorial

Another Pythagorean Theorem Proof:

This video visually proves the Pythagorean Theorem using triangles and parallelograms.

Type: Tutorial

Pythagorean Theorem Proof Using Similar Triangles:

This video shows a proof of the Pythagorean Theorem using similar triangles.

Type: Tutorial

## Video/Audio/Animation

Annotated Proof of the Pythagorean Theorem :

This resource gives an animated and then annotated proof of the Pythagorean Theorem.

Type: Video/Audio/Animation

## Virtual Manipulative

Demonstrate the Pythagorean Theorem:

Representation to illustrate the Pythagorean Theorem.

Type: Virtual Manipulative

## Student Resources

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

## Original Student Tutorial

Use properties, postulates, and theorems to prove a theorem about a triangle. In this interactive tutorial, you'll also learn how to prove that a line parallel to one side of a triangle divides the other two proportionally.

Type: Original Student Tutorial

## Presentation/Slideshow

The Pythagorean Theorem: Geometry’s Most Elegant Theorem:

This lesson teaches students about the history of the Pythagorean theorem, along with proofs and applications. It is geared toward high school Geometry students that have completed a year of Algebra and addresses the following national standards of the National Council of Teachers of Mathematics and the Mid-continent Research for Education and Learning: 1) Analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships; 2) Use visualization, spatial reasoning, and geometric modeling to solve problems; 3) Understand and apply basic and advanced properties of the concepts of geometry; and 4) Use the Pythagorean theorem and its converse and properties of special right triangles to solve mathematical and real-world problems. The video portion is about thirty minutes, and with breaks could be completed in 50 minutes. (You may consider completing over two classes, particularly if you want to allow more time for activities or do some of the enrichment material). These activities could be done individually, in pairs, or groups. I think 2 or 3 students is optimal. The materials required for the activities include scissors, tape, string and markers.

Type: Presentation/Slideshow

Bank Shot:

This task asks students to use similarity to solve a problem in a context that will be familiar to many, though most students are accustomed to using intuition rather than geometric reasoning to set up the shot.

Extensions, Bisections and Dissections in a Rectangle:

This task involves a reasonably direct application of similar triangles, coupled with a moderately challenging procedure of constructing a diagram from a verbal description.

Joining two midpoints of sides of a triangle:

Using a triangle with line through it, students are tasked to show the congruent angles, and conclude if one triangle is similar to the other.

Unit Squares and Triangles:

This problem solving task asks students to find the area of a triangle by using unit squares and line segments.

## Tutorials

Bhaskara's Proof of the Pythagorean Theorem:

This video demonstrates Bhaskara's proof of the Pythagorean Theorem.

Type: Tutorial

Another Pythagorean Theorem Proof:

This video visually proves the Pythagorean Theorem using triangles and parallelograms.

Type: Tutorial

Pythagorean Theorem Proof Using Similar Triangles:

This video shows a proof of the Pythagorean Theorem using similar triangles.

Type: Tutorial

## Video/Audio/Animation

Annotated Proof of the Pythagorean Theorem :

This resource gives an animated and then annotated proof of the Pythagorean Theorem.

Type: Video/Audio/Animation

## Parent Resources

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

Bank Shot:

This task asks students to use similarity to solve a problem in a context that will be familiar to many, though most students are accustomed to using intuition rather than geometric reasoning to set up the shot.

Extensions, Bisections and Dissections in a Rectangle:

This task involves a reasonably direct application of similar triangles, coupled with a moderately challenging procedure of constructing a diagram from a verbal description.