Lesson objective: Make a connection to a relationship between the size of an angle and the size of the side opposite that angle.

This lesson provides an opportunity for students to apply their knowledge and understanding of the sum of the angles of a triangle equaling 180\(^\circ\) and the Triangle Inequality Theorem to a real-life situation. Students are asked to consider two bicyclists who have traveled the same number of miles from a starting point and end up at a different distance from the starting point.

Key Concept students will use:

- The sum of the lengths of two sides of a triangle must be greater than the length of the third side, and this can be informally proven.
- The sum of the angles of a triangle is 180 degrees, and this can be informally proven using the idea of parallel lines.

Skills students will use:

- their knowledge of determining the length of the missing side of a triangle.
- their knowledge of determining the missing angle in a triangle.

Students engage in Mathematical Practice #3 (construct viable arguments and critique the reasoning of others) as they make conjectures and build a logical progression of statements to explore the truth of their conjectures by analyzing a situation involving the route traveled by two bicyclists.

**Key vocabulary: **

- acute triangle
- angle
- obtuse angle
- obtuse triangle
- right angle
- right triangle
- side
- triangle angle sum
- Triangle Inequality Theorem
- vertex

**Special materials needed:**