# What You Will Learn In this lesson you will understand the definitions of **complementary** and **supplementary** angle pairs, learn how to find a missing angle when you know its complement or supplement, and explore related ideas including **vertical angles** and **linear pairs**. ## Theory ### Complementary Angles Two angles are **complementary** when their measures add up to exactly 90°. \[ α + β = 90° \] Each angle is called the **complement** of the other. For example, 35° and 55° are complementary because 35° + 55° = 90°. **Key fact:** Complementary angles do not need to be next to each other. Any two angles whose measures sum to 90° are complementary, whether they share a side or are in completely different diagrams. A real-world example: if a ramp rises at a 25° angle from the ground, the angle between the ramp and a vertical wall is 90° − 25° = 65°. These two angles are complementary. ### Supplementary Angles Two angles are **supplementary** when their measures add up to exactly 180°. \[ α + β = 180° \] Each angle is the **supplement** of the other. For example, 110° and 70° are supplementary because 110° + 70° = 180°. **Memory trick:** **C** omplementary = **C** orner (90°); **S** upplementary = **S** traight line (180°). ### Linear Pairs A **linear pair** is a specific case of supplementary angles: two adjacent angles that together form a straight line. Because a straight angle is 180°, the two angles in a linear pair are always supplementary. If one angle in a linear pair is 130°, its partner is: \[ 180° − 130° = 50° \] ### Vertical Angles When two straight lines cross, they create four angles. The angles that sit directly across from each other (sharing only the vertex, not a side) are called **vertical angles**. Vertical angles are always equal. If two lines intersect and one angle is 72°, then: * The vertical angle across from it is also 72°. * Each of the other two angles is \[ 180° − 72° = 108° \] (since they form linear pairs with the 72° angle). * Those two 108° angles are also vertical angles to each other. ## Worked Examples ### Example 1: Finding a complement Find the complement of 57°. \[ 90° − 57° = 33° \] The complement is 33°. Check: 57° + 33° = 90°. Correct. ### Example 2: Finding a supplement Find the supplement of 143°. \[ 180° − 143° = 37° \] The supplement is 37°. Check: 143° + 37° = 180°. Correct. ### Example 3: Algebraic — complementary angles Two complementary angles are in the ratio 2:3. Find both angles. Let the angles be 2x and 3x. \[ 2x + 3x = 90° \] \[ 5x = 90° \] \[ x = 18° \] The angles are 2(18°) = 36° and 3(18°) = 54°. Check: 36° + 54° = 90°. Correct. ### Example 4: Vertical angles and linear pairs combined Two lines intersect. One of the four angles measures (3x+10)° and the angle adjacent to it measures (2x+20)°. Find x and all four angle measures. Adjacent angles at an intersection form a linear pair, so they are supplementary: \[ (3x+10) + (2x+20) = 180 \] \[ 5x + 30 = 180 \] \[ 5x = 150 \] \[ x = 30 \] The first angle is \[ 3(30) + 10 = 100° \]. The adjacent angle is \[ 2(30) + 20 = 80° \]. By vertical angles: the four angles are 100°, 80°, 100°, 80°. Check: 100° + 80° = 180°. Correct. ## Common Mistakes ### Mistake 1: Mixing up 90 and 180 ❌ "The supplement of 40° is 90° − 40° = 50°." ✅ **Supplement** means the angles sum to 180°, not 90°. The supplement of 40° is \[ 180° − 40° = 140° \]. The number 50° would be the **complement**. ### Mistake 2: Assuming complementary angles must be adjacent ❌ "These two angles are not next to each other, so they cannot be complementary." ✅ Complementary (and supplementary) angles only need their measures to add up to the target sum. They do **not** need to share a side or even be in the same figure. ### Mistake 3: Confusing vertical angles with adjacent angles ❌ "Vertical angles are the ones right next to each other." ✅ **Vertical** angles are across from each other at an intersection point (they share only the vertex). The angles **next to** each other are **adjacent** angles and form linear pairs. ## Practice Problems **1.** Find the complement of 76°. Show Answer \[ 90°−76°=14° \]. The complement is 14°. **2.** Find the supplement of 25°. Show Answer \[ 180°−25°=155° \]. The supplement is 155°. **3.** Angle A and angle B are supplementary. If A=(4x−10)° and B=(2x+40)°, find both angles. Show Answer \[ (4x−10)+(2x+40)=180 \] \[ 6x+30=180 \] \[ 6x=150 \] \[ x=25 \] Angle A=4(25)−10=90° and Angle B=2(25)+40=90°. Both angles are 90°. **4.** Two lines intersect. One angle is 65°. Find the other three angles. Show Answer The vertical angle is also 65°. Each of the two remaining angles is \[ 180°−65°=115° \]. **5.** The complement of an angle is three times the angle itself. Find the angle. Show Answer Let the angle be x. Its complement is 3x. \[ x + 3x = 90° \] \[ 4x = 90° \] \[ x = 22.5° \] The angle is 22.5° and its complement is 67.5°. Check: 22.5° + 67.5° = 90°. Correct. ## Summary - **Complementary** angles sum to 90°; **supplementary** angles sum to 180°. - A **linear pair** is two adjacent angles forming a straight line — they are always supplementary. - **Vertical angles** are the opposite angles formed when two lines cross — they are always equal. - To find a missing complement, subtract from 90°. To find a missing supplement, subtract from 180°. - These relationships are the foundation for solving more advanced geometry problems involving parallel lines, triangles, and polygons.