• MAFS.912.G-SRT.2.4 : Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity.
• MAFS.912.G-SRT.2.5 : Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.
  
  

• 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.

• Proving Theorems About Triangles: 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.

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

• 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.

• 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.

• 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.

• 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.

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

• Prove Rhombus Diagonals Bisect Angles: Students are asked to prove a specific diagonal of a rhombus bisects a pair of angles.

• 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.

• 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.

• What's the Problem: Students solve problems using triangle congruence postulates and theorems.

• 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.

• 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.

• 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.

• Proofs of the Pythagorean Theorem: This lesson enriches the students' perspective of geometric proofs that are not in two-column form. Students will apply algebaic skills and geometric properties to prove the Pythagorean Theorem in a variety of ways. This unit is designed 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.
• 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.

• Solving Geometry Problems: Floodlights: This resource is designed to support how well students can identify and use geometrical knowledge to solve a problem. Specifically, identify similar triangles and use their properties to prove and solve problems. Students will also be provided with examples of other students’ work to critique. The lesson closes with a whole-group discussion where students explain and compare the alternative approaches they have seen and used.

• 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 can be used as a review or summative assessment after teaching similar triangles.

• Measuring Height with Triangles and Mirrors: <p>Reflect for a moment on how to measure tall objects with mirrors and mathematics.</p>

• 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.

• 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.

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

• Another Pythagorean Theorem Proof: This video visually proves the Pythagorean Theorem using triangles and parallelograms.

• Pythagorean Theorem Proof Using Similar Triangles: This video shows a proof of the Pythagorean Theorem using similar triangles.