Mastering the Law of Sines: Unlocking AAS, ASA, and the Ambiguous SSA Case

Algebra 2 Grades High School 17:08 Video
Print Lesson Plan Watch Video

Lesson Description

Explore the Law of Sines to solve triangles, focusing on Angle-Angle-Side (AAS), Angle-Side-Angle (ASA), and the ambiguous Side-Side-Angle (SSA) cases. Determine the number of possible triangles and solve them completely.

Video Resource

Law of Sines AAS, ASA, SSA Ambiguous Case

Mario's Math Tutoring

Duration: 17:08
Watch on YouTube

Key Concepts

  • Law of Sines
  • Angle-Angle-Side (AAS)
  • Angle-Side-Angle (ASA)
  • Side-Side-Angle (SSA) - Ambiguous Case
  • Number of Possible Triangles (0, 1, or 2)
  • Altitude of a Triangle

Learning Objectives

  • Apply the Law of Sines to solve triangles given AAS, ASA, or SSA information.
  • Determine the number of possible triangles (0, 1, or 2) in the SSA ambiguous case.
  • Calculate missing angles and side lengths of triangles using the Law of Sines.
  • Use the altitude of a triangle to analyze the SSA ambiguous case.

Educator Instructions

  • Introduction (5 mins)
    Begin with a brief review of the Law of Sines formula. Explain the different cases: AAS, ASA, and SSA. Emphasize the 'ambiguous' nature of the SSA case and the potential for multiple solutions.
  • AAS and ASA Cases (10 mins)
    Work through examples of AAS and ASA cases, demonstrating how to find missing angles and sides using the Law of Sines. Emphasize that these cases result in a single, unique triangle.
  • SSA Ambiguous Case - Introduction (10 mins)
    Introduce the SSA ambiguous case. Explain why it's ambiguous (0, 1, or 2 possible triangles). Introduce the method of dropping an altitude to determine the number of triangles. Relate the side opposite the given angle to the altitude length.
  • SSA Ambiguous Case - Examples (20 mins)
    Work through several SSA examples, demonstrating how to calculate the altitude and determine the number of possible triangles. For cases with 1 or 2 triangles, solve for all missing angles and sides. Address obtuse triangle conditions.
  • Practice and Review (10 mins)
    Provide students with practice problems to solve independently or in small groups. Review the key concepts and strategies for each case.

Interactive Exercises

  • Triangle Solver
    Provide students with various triangle configurations (AAS, ASA, SSA) and have them determine the number of possible triangles and solve for all missing angles and sides. Use online tools or graphing calculators to verify solutions.
  • Ambiguous Case Challenge
    Present a series of SSA scenarios where students must first determine if there are 0, 1, or 2 possible triangles before attempting to solve. This reinforces the importance of analyzing the problem before applying the Law of Sines.

Discussion Questions

  • Why is the SSA case called the 'ambiguous case'?
  • How does the altitude of a triangle help determine the number of possible triangles in the SSA case?
  • What are some common mistakes to avoid when using the Law of Sines?
  • How does knowing if an angle is obtuse affect the possible number of triangles in SSA?

Skills Developed

  • Problem-solving
  • Critical Thinking
  • Application of Trigonometric Functions
  • Geometric Reasoning

Multiple Choice Questions

Question 1:

Which of the following cases might result in zero, one, or two possible triangles?

Correct Answer: SSA

Question 2:

In the ambiguous case (SSA), what is the first step to determine the number of possible triangles?

Correct Answer: Drop an altitude

Question 3:

If side 'a' is opposite angle A, and side 'b' is opposite angle B, the Law of Sines states:

Correct Answer: a/sin(A) = b/sin(B)

Question 4:

Given an angle and the side adjacent to it, as well as the side opposite, you are given the triangle case of:

Correct Answer: SSA

Question 5:

If the side opposite the given angle is shorter than the altitude, in the SSA case, how many triangles are possible?

Correct Answer: 0

Question 6:

Given AAS, which angle is used to find the third angle of the triangle?

Correct Answer: Add the angles and subtract from 180

Question 7:

What does the altitude of a triangle determine?

Correct Answer: The amount of solutions in SSA

Question 8:

For an obtuse triangle, how many solutions are possible for SSA?

Correct Answer: Zero or One

Question 9:

When doing Law of Sines, you must make sure your calculator is in:

Correct Answer: Degree Mode

Question 10:

When sides are equal in a triangle, what kind of triangle is it?

Correct Answer: Isosceles Triangle

Fill in the Blank Questions

Question 1:

The Law of Sines is used to solve _____ triangles.

Correct Answer: oblique

Question 2:

The ambiguous case arises when we are given the _____, _____, and _____ of a triangle.

Correct Answer: side, side, angle

Question 3:

If the side opposite the given angle in SSA is greater than the adjacent side, there is only _____ possible triangle.

Correct Answer: one

Question 4:

An altitude is drawn from the top _____ down to create a right angle.

Correct Answer: vertex

Question 5:

When the altitude is equal to the side opposite the angle, the triangle will form a _____ triangle.

Correct Answer: right

Question 6:

When an obtuse angle is given, and the side opposite the angle is not the longest side, there are _____ solutions.

Correct Answer: zero

Question 7:

When you find the angle measurement, it is important to take the _____ inverse of the number.

Correct Answer: sine

Question 8:

The angles in a triangle add up to _____ degrees.

Correct Answer: 180

Question 9:

In an isosceles triangle, the base angles are _____.

Correct Answer: congruent

Question 10:

The _____ side is across from the largest angle.

Correct Answer: longest

Educational Standards

A2.A.1:Represent and solve mathematical and real-world problems using nonlinear equations, systems of linear equations, and systems of linear inequalities; interpret the solutions in the original context. A2.F.1:Understand functions as descriptions of covariation (how related quantities vary together).

Teaching Materials

Download ready-to-use materials for this lesson:

Student Worksheet Classroom Slides Assessment Quiz Export to Google Docs
Add to Google Classroom

User Actions

Sign in to save this lesson plan to your favorites.

Sign In

Share This Lesson

Related Lesson Plans