Geometry Unit 6: Right Triangle Trigonometry

Unit Driving Question 

What relationships exist between the sides of right triangles and their angles?

Essential Questions 

1. What relationships exist between the sides of similar right triangles? 
2. What is the relationship between angles and siade of right triangles? 

Launch Task 

Big Ideas for Development Lessons 

Closure & Assessment 

1 Lesson 

4 Weeks (approximately 1 week per big idea) 

1 Week 

Click on the links below to see each Big Idea's lesson overview (includes links to teacher notes and student activities)

1. Corresponding sides of similar triangles prove the Pythagorean Theorem is true for all right triangles

2. Corresponding Sides of Special right triangles are proportional
3. Sine, Cosine, and Tangent are constant ratios that relate the angles and sides of a right triangle 

1. Formative Assessment 1 (after Big Idea 3) 
2. Re-engagement Activity (Not Provided) 
3. Unit Assessment 

Big Idea 1: Corresponding sides of similar triangles prove the Pythagorean Theorem is true for all right triangles.

OAS-M: G.RT.1.1

Lessons and Additional Activities

Big Idea 1 Lessons 1-3 Overview (includes teacher notes and student activities)

Additional Collaborative Activities:

Stations Activity: Properties of Right Triangles- Students will work in collaborative groups to complete station activities providing opportunities to develop concepts and skills related to applying properties of right triangles, specifically the Pythagorean theorem (Students may have completed stations 1-3 in Pre-Algebra.)

Stations Activity: Pythagorean Theorem- Students will work in collaborative groups to complete station activities providing opportunities to develop concepts and skills related to understanding that the Pythagorean theorem is a statement about areas of squares on the sides of a right triangle.

Evidence of Understanding

Analyze characteristics of triangles using the Pythagorean Theorem

• Classify a triangle as acute, right, or obtuse using the Pythagorean theorem

• Use an area model to explain that when a2+b2>c2, the triangle is acute; that when a2+b2=c2, the triangle is right; and that when a2+b2<c2, the triangle is obtuse

Analyze and describe special Pythagorean triples using similarity and scale factor

• Explain that special triples occur when a2 + b2 is a perfect square and use this to strategically name Pythagorean triples

• Use similarity to explain and name other triangles that are similar to the 3-4-5, 5-12-13, 8-15-17, or other Pythagorean triples

Find the missing side length of a right triangle using Pythagorean Theorem

• Create diagrams to model situations and use Pythagorean Theorem to solve them
• Use right triangles and their properties to describe objects and/or situations
• Use properties of similar triangles and scale factor with the Pythagorean Theorem
• Use a Pythagorean triple, when applicable, to state missing side lengths
• Example: length measure vs.color of the segment, thickness, etc.

Big Idea 2: Corresponding sides of Special Right triangles are proportional.

Lessons and Additional Activities

Big Idea 2 Lessons 1-3 Overview (includes teacher notes and student activities)

Evidence of Understanding 

Identify 30-60-90 and 45-45-90 as special right triangles

• Use the patterns found in special right triangles to solve for missing sides and/or angles.

• Apply the properties of special right triangles to apply them to various diagrams and verbal descriptions.

Use examples and nonexamples to make conjectures about special right triangles

• The short side of a 30-60-90 triangle is always half the length of the hypotenuse

• The long side of a 30-60-90 triangle is 3times the length of the short side

• The sine of 30° is always 1/2; and, conversely,  if the sine is 1/2; then the angle is 30°

• The cosine of 60° is always 1/2; and, conversely,  if the cosine is 1/2; then the angle is 60°

• Both legs of a 45-45-90 triangle are congruent (the triangle is isosceles)

• The hypotenuse is 2 times the length of a side of the triangle

• The tangent of 45-45-90 is always 1 and, conversely, if the tangent is 1 then the angle is 45°

Big Idea 3: Sine, Cosine, and Tangent are constant ratios that relate the angles and sides of a right triangle. 

Lessons and Additional Activities

Big Idea 3 Lessons 1-4 Overview (includes teacher notes and student activities)

Evidence of Understanding 

Analyze characteristics of trigonometric ratios using right triangles

• Use similarity, dilations, and scale factor to justify why a trigonometric ratio is constant

Analyze the legs of a right triangle to describe sine and cosine

• Use the definition of the sine or cosine ratio to explain why neither value can exceed 1

• Describe how increasing or decreasing one leg impacts the sine or cosine value

• Explain how the sine and cosine of complementary angles are related and create an algebraic expression that generalizes the relationship between sine and cosine

Analyze the legs of a right triangle to describe the tangent

• Use the definition of the tangent ratio to explain why the tangent values increases without limitations as the angle increases