Find X in Circle Sectors: Solving 3x+4 and 5x+16 Angle Relationship

Q: How do I know these angles are supplementary?

Look at the diagram! The angles are positioned on a straight line, which means they form a linear pair. Linear pairs are always supplementary, so they must add up to $ 180° $.

Q: What if I set the angles equal to each other instead?

That would only work if the angles were vertical angles or marked as equal. Since these angles are on a straight line, they're supplementary, not equal. Always identify the angle relationship first!

Q: Why do we get 8x + 20 = 180?

When you expand $ (3x+4) + (5x+16) = 180 $, you combine like terms: Combine x terms: $ 3x + 5x = 8x $Combine constants: $ 4 + 16 = 20 $

Q: How can I check if x = 20 is correct?

Substitute back into both original expressions: $ 3(20) + 4 = 60 + 4 = 64° $$ 5(20) + 16 = 100 + 16 = 116° $Check: $ 64° + 116° = 180° $ ✓

Linear Equations with Supplementary Angles

Find the value of the parameter x.

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Determine X

00:03 Adjacent angles add up to 180 degrees

00:09 Collect like terms

00:16 Isolate X

00:34 This is the solution

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Find the value of the parameter x.

Step-by-step solution

To solve this problem, we'll follow these steps:

Step 1: Apply the triangle angle sum property.
Step 2: Combine like terms in the equation.
Step 3: Solve the resulting linear equation for $x$ .

Now, let's work through each step:
Step 1: Based on our assumption, the angles $3x + 4$ and $5x + 16$ should sum to 180 degrees. We write the equation:
$(3x + 4) + (5x + 16) = 180$ .

Step 2: Combine like terms:
$3x + 5x + 4 + 16 = 180$ .

Step 3: Simplify the equation:
$8x + 20 = 180$ .

Subtract 20 from both sides:
$8x = 160$ .

Finally, divide by 8 to solve for $x$ :
$x = \frac{160}{8} = 20$ .

Therefore, the solution to the problem is $x = 20$ .

Final Answer

Key Points to Remember

Essential concepts to master this topic

Supplementary Angles: Two angles that sum to exactly 180 degrees
Technique: Set up equation (3x+4) + (5x+16) = 180
Check: Substitute x=20: (60+4) + (100+16) = 64+116 = 180 ✓

Common Mistakes

Avoid these frequent errors

Assuming angles are vertical or equal
Don't set 3x+4 = 5x+16 just because angles look similar = wrong relationship! This ignores that these are supplementary angles forming a straight line. Always check if angles add to 180° when they form a linear pair.

Practice Quiz

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Is DE side in one of the triangles?

FAQ

Everything you need to know about this question

How do I know these angles are supplementary?

Look at the diagram! The angles are positioned on a straight line, which means they form a linear pair. Linear pairs are always supplementary, so they must add up to $180°$ .

What if I set the angles equal to each other instead?

That would only work if the angles were vertical angles or marked as equal. Since these angles are on a straight line, they're supplementary, not equal. Always identify the angle relationship first!

Why do we get 8x + 20 = 180?

When you expand $(3x+4) + (5x+16) = 180$ , you combine like terms:

Combine x terms: $3x + 5x = 8x$
Combine constants: $4 + 16 = 20$

How can I check if x = 20 is correct?

Substitute back into both original expressions:

$3(20) + 4 = 60 + 4 = 64°$
$5(20) + 16 = 100 + 16 = 116°$
Check: $64° + 116° = 180°$ ✓

What if my answer doesn't make the angles add to 180°?

Go back and check your algebra! Common errors include:

Wrong signs when combining terms
Calculation mistakes when solving
Setting up the wrong equation initially

The final check should always give you 180°.

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