Triangle Angle Calculation: Solving for Angle C Using 1:3 Ratio

Q: How do I set up variables when given an angle ratio?

When you see $ \frac{∢B}{∢C} = \frac{1}{3} $, this means ∢B is 1 part and ∢C is 3 parts. Set ∢B = x (the smaller part) and ∢C = 3x (the larger part).

Q: Why is angle C larger than angle B in this problem?

Because the ratio tells us that ∢C is 3 times larger than ∢B. When we have 1:3, the second number (3) corresponds to the larger angle, which is ∢C.

Q: What if I get a decimal answer for an angle?

Decimal angles like 82.5° are perfectly valid! Many geometry problems result in non-whole number angles. Just make sure your decimals add up correctly with the other angles.

Q: How do I check if my ratio is correct in the final answer?

Divide the angles: $ \frac{27.5°}{82.5°} = \frac{1}{3} $. If this matches your original ratio, you solved it correctly!

Triangle Angle Sum with Ratio Relationships

The triangle ABC is shown below.

angle $∢A=70°$ .

$\frac{∢B}{∢C}=\frac{1}{3}$

Calculate angle $∢C$ .

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Calculate angle C

00:03 Angle ratio according to the given data

00:05 Isolate angle C

00:21 Let's mark angle B's value as A

00:31 Substitute the angle expressions in the triangle

00:40 Sum of angles in a triangle equals 180

00:45 Group terms and isolate the value of angle A

01:11 This is the value of angle A, now let's substitute in the expression for angle C

01:20 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

The triangle ABC is shown below.

angle $∢A=70°$ .

$\frac{∢B}{∢C}=\frac{1}{3}$

Calculate angle $∢C$ .

Step-by-step solution

To solve this problem, we'll use the properties of a triangle and given ratio:

Step 1: Let $∢B = x$ and $∢C = 3x$ as per the given ratio $\frac{∢B}{∢C} = \frac{1}{3}$ .
Step 2: Use the triangle sum property: $∢A + ∢B + ∢C = 180^\circ$ .
Step 3: Substitute known values: $70^\circ + x + 3x = 180^\circ$ .
Step 4: Simplify: $4x + 70^\circ = 180^\circ$ .
Step 5: Solve for $x$ : $4x = 110^\circ$ .
Step 6: Determine $x$ : $x = 27.5^\circ$ .
Step 7: Calculate $∢C$ : $∢C = 3x = 3 \times 27.5^\circ = 82.5^\circ$ .

Therefore, the measure of angle $∢C$ is $82.5^\circ$ .

Final Answer

82.5°

Key Points to Remember

Essential concepts to master this topic

Triangle Sum Rule: All three interior angles always equal 180°
Ratio Technique: If ∢B:∢C = 1:3, then ∢B = x and ∢C = 3x
Verification: Check that 70° + 27.5° + 82.5° = 180° ✓

Common Mistakes

Avoid these frequent errors

Setting up the ratio incorrectly
Don't write ∢B = 3x and ∢C = x when the ratio is 1:3 = wrong angle assignments! This reverses which angle is larger. Always match the first number in the ratio to the first letter: ∢B/∢C = 1/3 means ∢B = x and ∢C = 3x.

Practice Quiz

Test your knowledge with interactive questions

Indicates which angle is greater

FAQ

Everything you need to know about this question

How do I set up variables when given an angle ratio?

When you see $\frac{∢B}{∢C} = \frac{1}{3}$ , this means ∢B is 1 part and ∢C is 3 parts. Set ∢B = x (the smaller part) and ∢C = 3x (the larger part).

Why is angle C larger than angle B in this problem?

Because the ratio tells us that ∢C is 3 times larger than ∢B. When we have 1:3, the second number (3) corresponds to the larger angle, which is ∢C.

What if I get a decimal answer for an angle?

Decimal angles like 82.5° are perfectly valid! Many geometry problems result in non-whole number angles. Just make sure your decimals add up correctly with the other angles.

Can I solve this without using variables?

Using variables like x is the most reliable method because it keeps track of the ratio relationship. Without variables, it's easy to lose track of which angle should be 3 times the other.

How do I check if my ratio is correct in the final answer?

Divide the angles: $\frac{27.5°}{82.5°} = \frac{1}{3}$ . If this matches your original ratio, you solved it correctly!

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Triangle Angle Calculation: Solving for Angle C Using 1:3 Ratio

Triangle Angle Sum with Ratio Relationships

❤️ Continue Your Math Journey!

Step-by-step video solution

Step-by-step written solution

Understand the problem

Step-by-step solution

Final Answer

Key Points to Remember

Common Mistakes

Practice Quiz

FAQ

How do I set up variables when given an angle ratio?

Why is angle C larger than angle B in this problem?

What if I get a decimal answer for an angle?

Can I solve this without using variables?

How do I check if my ratio is correct in the final answer?

🌟 Unlock Your Math Potential

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