Find Missing Angles Using Vertical, Complementary, and Supplementary Angles

🔑 Key Concepts

• Complementary angles have measures that add to \(90^\circ\).
• Supplementary angles have measures that add to \(180^\circ\).
• A right-angle symbol shows that an angle measures \(90^\circ\).
• Vertical angles are opposite angles formed by intersecting lines, and they have equal measures.
• A linear pair consists of two adjacent angles whose measures add to \(180^\circ\).
• Several angle relationships may be used together to determine all of the unknown measures in a diagram.

✏️ Worked Examples

Find the Measures of \(a\), \(b\), \(c\), \(d\), and \(e\)

Use the relationships shown in the diagram to determine each labeled angle measure.

Step 1: Find \(a\).

The \(41^\circ\) angle and angle \(a\) divide a right angle into two complementary angles.

\(41^\circ+a=90^\circ\)

\(a=90^\circ-41^\circ=49^\circ\)

Step 2: Find \(b\).

The right-angle symbol shows that the vertical and horizontal lines are perpendicular.

\(b=90^\circ\)

Step 3: Find \(c\).

Angle \(c\) and the \(139^\circ\) angle are vertical angles, so they have equal measures.

\(c=139^\circ\)

Step 4: Find \(d\).

Angle \(d\) and the \(139^\circ\) angle form a linear pair, so their measures add to \(180^\circ\).

\(d+139^\circ=180^\circ\)

\(d=180^\circ-139^\circ=41^\circ\)

Step 5: Find \(e\).

Angle \(e\) and angle \(d\) are vertical angles, so they have equal measures.

\(e=d=41^\circ\)

Final Answers

\(a=49^\circ,\quad b=90^\circ,\quad c=139^\circ\)
\(d=41^\circ,\quad e=41^\circ\)

🧠 Math Vocabulary

• Complementary angles: Two angles whose measures add to \(90^\circ\).
• Supplementary angles: Two angles whose measures add to \(180^\circ\).
• Right angle: An angle that measures exactly \(90^\circ\).
• Linear pair: Two adjacent angles whose nonshared sides form a straight line.
• Vertical angles: Opposite angles formed by two intersecting lines. Vertical angles have equal measures.
• Parallel lines: Lines in the same plane that never intersect.
• Transversal: A line that intersects two or more other lines at different points.

💡 Main Idea

Unknown angle measures can be found by identifying the relationships shown in a diagram. Right angles, complementary angles, supplementary angles, linear pairs, and vertical angles provide information that can be used to calculate missing measures. When parallel lines and a transversal are included, one known angle can often be used to determine several additional angles.

📚 What You Should Already Know

You should know that a right angle measures \(90^\circ\) and that a straight angle measures \(180^\circ\). It is also helpful to recognize adjacent angles, intersecting lines, perpendicular lines, and angles that share a common vertex. Basic subtraction with angle measures is used to find each missing value.

🚀 What Comes Next

These angle relationships can next be applied to diagrams in which the unknown measures are represented by algebraic expressions. You will also use corresponding angles, alternate interior angles, alternate exterior angles, and same-side interior angles when parallel lines are cut by a transversal.

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