Solving Triangles: Which Law, When?

Lesson 2 of 2: Applying the Laws

In this lesson:

• Choose the right law for each triangle case
• Solve complete triangles step by step
• Handle the ambiguous SSA case

Learning Objectives

By the end of this lesson, you should be able to:

5. Use the Law of Sines to solve triangles given AAS, ASA, or SSA information
6. Use the Law of Cosines to solve triangles given SAS or SSS information
7. Distinguish between when to apply Law of Sines vs. Law of Cosines based on given information

From Proof to Practice

You can now prove both laws work.

The question is: which one do you reach for first?

The answer is determined entirely by what information you're given:

• Two angles + a side → one law
• Two sides + included angle → the other
• Three sides → the other

Not all triangle problems are the same — identify the case first.

Which Law Do You Use?

The case is determined by what information is given — not by preference.

The Key Discriminator

Is the known angle between the two known sides?

• Yes (SAS) → Law of Cosines — included angle connects directly to the formula
• No (AAS, ASA) → Law of Sines — you have a matched side-angle pair
• Two sides, non-included angle (SSA) → Law of Sines — but check for ambiguity

At the start of every problem: identify the case, then choose the law.

Example 1 — ASA: First Steps

Given: , , . Find all unknown parts.

Identify the case: Two angles + one side → ASA → Law of Sines

Step 1: Find the third angle using angle sum

Now we have all three angles and one side — ready for Law of Sines.

Example 1 — ASA: Find the Sides

Step 2: Apply Law of Sines for side

Step 3: Apply Law of Sines for side

Quick Check

Given: , , .

• What case is this?
• Which law do you use first?

Don't solve — just identify the case and first law.

Example 2 — SAS: Find Side

Given: , , . Find all unknown parts.

Identify the case: Two sides + included angle → SAS → Law of Cosines

Step 1: Find side

Example 2 — SAS: Find the Angles

Step 2: Apply Law of Sines for angle

Step 3: Angle sum for

Strategy: Law of Cosines for the unknown side → then Law of Sines for angles.

Quick Check

Given: , , .

• What case is this?
• Which law first?
• Which angle should you find first, and why?

Hint: the reason for finding the largest angle first matters.

Example 3 — SSS: Find Angle

Given: , , . Find all angles.

Identify the case: Three sides → SSS → Law of Cosines

Find angle (opposite shortest side):

Example 3 — SSS: Find Remaining Angles

Find angle :

Find angle by angle sum:

Check: is the longest side and is the largest angle. ✓

Your Turn — Identify the Case

For each, write the case and the first law to use. Don't solve.

1. , ,
2. , ,
3. , ,

Write your case and law for each before advancing.

Answers

1. , , → AAS → Law of Sines (find first)
2. , , → SAS → Law of Cosines (find side )
3. , , → SSS → Law of Cosines

Bonus for (3): — this is a Pythagorean triple!

The Law of Cosines would confirm . Recognizing triples saves time.

SSA: The Ambiguous Case

Given: two sides and a non-included angle (SSA).

This is the only case where a unique triangle is not guaranteed.

• Fix angle and side (adjacent to )
• Side "hangs" from the far end of and can swing
• Depending on the length of , it may reach the base ray at:
  • 0 points → no triangle
  • 1 point → one triangle
  • 2 points → two triangles

Three Possible Outcomes

The "swinging" side may intersect the angle's ray at zero, one, or two points.

When Does Each Outcome Occur?

After applying Law of Sines to find :

Checking Procedure for SSA

When , always check both candidates:

1. Compute — the value your calculator returns (0° to 90°)
2. Compute — the supplementary angle (same sine)
3. For each: check if
   • If yes → valid triangle (solve it fully)
   • If no → discard that candidate

Always check both. Never stop after finding .

Quick Check

Using Law of Sines on an SSA problem, you find .

• How many triangles exist?
• What does this mean geometrically?

Think about what means before advancing.

Example: Two Solutions

Given: , , . Find all triangles.

Apply Law of Sines:

Since , we must check both candidates. Don't assume one solution.

Example: Two Solutions (continued)

→ ✓

→ ✓

Triangle 1: ,

Triangle 2: ,

Both are valid. Report all solutions.

Example: No Solution

Given: , , .

Since : no triangle exists.

Side is too short to reach the base ray when and .

Check Your Understanding

Given: , , .

1. Find using the Law of Sines
2. How many triangles exist?
3. Find all valid triangles completely

Work through all three steps before advancing.

Key Takeaways

✓ AAS/ASA → Law of Sines → use angle sum for the third angle first
✓ SAS → Law of Cosines (third side) → then Law of Sines for angles
✓ SSS → Law of Cosines → find largest angle first, then proceed
✓ SSA → Law of Sines → always check before concluding

Wrong law = dead end. Identify the case before calculating.

SSA always requires checking both angles. Never stop at .

Side pairs with angle . Never mix up the side-angle pairing.

Coming Up: Real-World Applications

You can now solve any triangle given sufficient information.

Next lesson — HSG.SRT.D.11:

• Surveying: triangulation from two observation points
• Navigation: ship bearing and distance problems
• Engineering: structural force diagrams

The laws you proved and applied here are used every day in these fields.