Writing Equations To Find Unknown Supplementary Angles

🔑 Key Concepts

• Supplementary angles have measures that add to \(180^\circ\).
• A straight angle measures \(180^\circ\).
• Adjacent angles whose nonshared sides form a straight line create a linear pair.
• Algebraic expressions representing supplementary angles can be added to form an equation.
• After solving for the variable, substitute its value into each expression to determine the angle measures.

✏️ Worked Example

Find \(x\) and Determine Both Angle Measures

The angles labeled \(2x+30\) and \(2x-10\) are adjacent angles that form a straight line.

Step 1: Identify the angle relationship.

The angles form a linear pair. Linear pairs are supplementary, so their measures add to \(180^\circ\).

Step 2: Write the equation.

\((2x+30)+(2x-10)=180\)

Step 3: Combine like terms.

\(2x+30+2x-10=180\)

\(4x+20=180\)

Step 4: Subtract \(20\) from both sides.

\(4x+20\ {\color{red}{-20}}=180\ {\color{red}{-20}}\)

\(4x=160\)

Step 5: Divide both sides by \(4\).

\(\displaystyle \frac{4x}{\color{red}{4}} = \frac{160}{\color{red}{4}}\)

\(x=40\)

Step 6: Find both angle measures.

\(2(40)+30=110^\circ\)

\(2(40)-10=70^\circ\)

Check the angle sum.

\(110^\circ+70^\circ=180^\circ\)

The two angle measures correctly form a straight angle.

Final Answers

\(x=40\)

\(110^\circ\) and \(70^\circ\)

🧠 Math Vocabulary

• Supplementary angles: Two angles whose measures add to \(180^\circ\).
• Straight angle: An angle that measures exactly \(180^\circ\).
• Adjacent angles: Angles that share a common vertex and a common side.
• Linear pair: Two adjacent angles whose nonshared sides form a straight line.
• Variable: A symbol used to represent an unknown number.
• Equation: A mathematical statement showing that two expressions have equal values.
• Substitution: Replacing a variable with a known value to evaluate an expression.

💡 Main Idea

Adjacent angles that form a straight line are supplementary, so their measures add to \(180^\circ\). The expressions \(2x+30\) and \(2x-10\) can therefore be added to form an equation. Solving gives \(x=40\). Substituting this value into the expressions shows that the angles measure \(110^\circ\) and \(70^\circ\).

📚 What You Should Already Know

You should know that a straight angle measures \(180^\circ\), recognize adjacent angles that form a linear pair, and understand that supplementary angles have a sum of \(180^\circ\). You should also be able to combine like terms, solve a two-step equation, and substitute a value into an expression.

🚀 What Comes Next

These same algebra and angle-reasoning strategies can be applied to complementary angles, vertical angles, and angles formed when parallel lines are crossed by a transversal. More advanced problems may require using several angle relationships within the same diagram.

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