Missing Angle Measures with a Right Angle and Straight Line

💡 Main Idea

The two outside angles and the center right angle together form a straight angle measuring \(180^\circ\). Since the center angle already measures \(90^\circ\), the two outside angles must combine to make the remaining \(90^\circ\). This gives the equation \((3c-8)+(2c+8)=90\). Combining like terms produces \(5c=90\), so \(c=18\).

✏️ Worked Example

Find the Value of \(c\)

Use the straight angle and the marked right angle to write and solve an equation.

Step 1: Determine the missing total.

The entire straight angle measures \(180^\circ\), and the center angle measures \(90^\circ\).

\(180^\circ-90^\circ=90^\circ\)

Therefore, the two outside angles must add to \(90^\circ\).

Step 2: Write the equation.

Add the expressions representing the two outside angles.

\((3c-8)+(2c+8)=90\)

Step 3: Combine like terms.

\(3c+2c-8+8=90\)

\(5c=90\)

Step 4: Solve for \(c\).

\(\displaystyle \frac{5c}{\color{red}{5}}=\frac{90}{\color{red}{5}}\)

\(c=18\)

Step 5: Check the angle measures.

\(3(18)-8=46^\circ\)

\(2(18)+8=44^\circ\)

\(46^\circ+90^\circ+44^\circ=180^\circ\)

Final Answer

\(\boxed{c=18}\)

🧠 Math Vocabulary

• Right angle: An angle that measures exactly \(90^\circ\).
• Straight angle: An angle that measures exactly \(180^\circ\).
• Adjacent angles: Angles that share a common vertex and a common side without overlapping.
• Complementary angles: Two angles whose measures add to \(90^\circ\).
• Angle expression: An algebraic expression used to represent an unknown angle measure.
• Like terms: Terms containing the same variable raised to the same power.

📚 What You Should Already Know

You should know that a right angle measures \(90^\circ\), a straight angle measures \(180^\circ\), and adjacent angles can be added to determine a larger angle measure. You should also be able to combine like terms and solve a one-step equation by dividing both sides by the coefficient of the variable.

🚀 What Comes Next

Next, you can apply the same reasoning to more complex diagrams involving complementary angles, supplementary angles, vertical angles, and parallel lines cut by a transversal. Some problems will require solving multi-step equations before determining the individual angle measures.

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