Triangle ABC: Solving for Angle C When A+B=2C and B=3A

Q: Why can't I just solve A + B = 2C by itself?

That equation alone has infinitely many solutions! You need the triangle angle sum (A + B + C = 180°) to find the specific values. Think of it as needing two equations to solve for three unknowns.

Q: How do I know which variable to use for substitution?

Since ∠B = 3∠A, it's easiest to let ∠A = α (alpha). This makes ∠B = 3α, and from the constraint A + B = 2C, we get ∠C = 2α. Now you have everything in terms of one variable!

Q: What if I get a negative angle or angle greater than 180°?

Check your algebra! In a triangle, all angles must be positive and less than 180°. If you get impossible values, you likely made an error in setting up or solving your equations.

Q: Can I use degrees or do I need radians?

For this problem, degrees are perfectly fine! The angle relationships work the same way. Just make sure to be consistent - don't mix degrees and radians in the same calculation.

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

Verify both conditions: Triangle sum: 30° + 90° + 60° = 180° ✓Given constraint: 30° + 90° = 2(60°) = 120° ✓

Q: What type of triangle is this?

With angles 30°, 60°, and 90°, this is a 30-60-90 right triangle - one of the most important special triangles in geometry!

Triangle Angle Relationships with Algebraic Constraints

Look at triangle ABC below.

$∢A+∢B=2∢C$

$∢B=3∢A$

Calculate the size of angle $\sphericalangle C\text{.}$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Calculate angle C

00:04 Let's substitute the angle ratio according to the given data

00:14 Substitute A for angle A value

00:26 Group terms

00:33 Isolate angle C

00:39 Substitute the expression for angle C in the triangle

00:45 Sum of angles in a triangle equals 180

00:49 Group terms and isolate the value of angle A

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

01:12 And this is the solution to the problem

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Look at triangle ABC below.

$∢A+∢B=2∢C$

$∢B=3∢A$

Calculate the size of angle $\sphericalangle C\text{.}$

Step-by-step solution

To find the value of $\angle C$ , follow these steps:

Step 1: Set up the equations.
We know:
- $\angle A = \alpha$
- $\angle B = 3\alpha$

Using the given condition $\angle A + \angle B = 2\angle C$ :
$\alpha + 3\alpha = 2\angle C \implies 4\alpha = 2\angle C \implies \angle C = 2\alpha$

Step 2: Use the triangle angle sum property.
From the triangle angle sum, we have:
$\angle A + \angle B + \angle C = 180^\circ$ Substituting the expressions for the angles:
$\alpha + 3\alpha + 2\alpha = 180^\circ$ $6\alpha = 180^\circ$ Solving for $\alpha$ :
$\alpha = \frac{180^\circ}{6} = 30^\circ$

Step 3: Calculate $\angle C$ .
Since $\angle C = 2\alpha$ :
$\angle C = 2 \times 30^\circ = 60^\circ$ Therefore, the size of angle $\angle C$ is $\boxed{60^\circ}$ .

Final Answer

60°

Key Points to Remember

Essential concepts to master this topic

Triangle Sum Rule: All three angles must sum to 180°
Substitution Method: Replace angles with α: α + 3α + 2α = 180°
Verification Check: Confirm 30° + 90° + 60° = 180° and A + B = 2C ✓

Common Mistakes

Avoid these frequent errors

Forgetting the triangle angle sum equals 180°
Don't solve A + B = 2C alone without using the triangle sum = wrong answer! This gives infinitely many solutions, not the specific angle values. Always combine the given constraint with the triangle angle sum property (A + B + C = 180°).

Practice Quiz

Test your knowledge with interactive questions

Look at the angles shown in the figure below.

What is their relationship?

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FAQ

Everything you need to know about this question

Why can't I just solve A + B = 2C by itself?

That equation alone has infinitely many solutions! You need the triangle angle sum (A + B + C = 180°) to find the specific values. Think of it as needing two equations to solve for three unknowns.

How do I know which variable to use for substitution?

Since ∠B = 3∠A, it's easiest to let ∠A = α (alpha). This makes ∠B = 3α, and from the constraint A + B = 2C, we get ∠C = 2α. Now you have everything in terms of one variable!

What if I get a negative angle or angle greater than 180°?

Check your algebra! In a triangle, all angles must be positive and less than 180°. If you get impossible values, you likely made an error in setting up or solving your equations.

Can I use degrees or do I need radians?

For this problem, degrees are perfectly fine! The angle relationships work the same way. Just make sure to be consistent - don't mix degrees and radians in the same calculation.

How do I check if my final answer is correct?

Verify both conditions:

Triangle sum: 30° + 90° + 60° = 180° ✓
Given constraint: 30° + 90° = 2(60°) = 120° ✓

What type of triangle is this?

With angles 30°, 60°, and 90°, this is a 30-60-90 right triangle - one of the most important special triangles in geometry!

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Triangle ABC: Solving for Angle C When A+B=2C and B=3A

Triangle Angle Relationships with Algebraic Constraints

❤️ 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

Why can't I just solve A + B = 2C by itself?

How do I know which variable to use for substitution?

What if I get a negative angle or angle greater than 180°?

Can I use degrees or do I need radians?

How do I check if my final answer is correct?

What type of triangle is this?

🌟 Unlock Your Math Potential

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