Lesson 11-2: Solve Problems Involving Angle Relationships

Property

Adjacent Angles are two angles that are side-by-side. They must meet two rules:

1. They share a common vertex (corner).
2. They share a common side (ray) between them, with no overlapping insides.
3. Key Rule: The measures of two adjacent angles add up to the measure of the larger total angle they create.

Examples

• Basic: $\angle AOB$ and $\angle BOC$ share vertex O and side OB. If $\angle AOB$ is 30° and $\angle BOC$ is 45°, the total large angle $\angle AOC$ is 30 + 45 = 75°.
• Forming a Right Angle: Two adjacent angles form a 90° corner. One is $(3x + 10)$° and the other is $(2x - 5)$°.

Equation:

• Forming a Straight Angle: Two adjacent angles make a flat 180° line. If one is 65°, the other must be 180 - 65 = 115°.

Explanation

Think of adjacent angles like two adjoining rooms in a house that share a single wall. They don't overlap, but together they make up the total space! When you know what kind of larger angle they form together (like a 90° corner or a 180° flat line), you can simply add their two expressions together and set them equal to that total number to solve the math puzzle.

Property

Vertical angles at the point of intersection of two lines have the same measure. A rotation with the intersection point as the vertex through a straight angle ($180^\circ$) takes one angle and its rays to the other. This rotation shows that the vertical angles are congruent, and thus have the same measure. Traditionally, vertical angles are shown to be equal because they are both supplementary to the same adjacent angle.

Examples

• Two lines intersect. One of the angles formed is $45^\circ$. The angle directly opposite to it is also $45^\circ$ because they are vertical angles.

• An angle measures $110^\circ$. The angle adjacent to it on a straight line is $180^\circ - 110^\circ = 70^\circ$. The angle vertical to this $70^\circ$ angle is also $70^\circ$.

Property

To find an unknown angle measure, set up an equation using the sum property:

• For complementary angles: x + known angle = 90°
• For supplementary angles: x + known angle = 180°

Examples

• If two angles are complementary and one measures 35°, find the other: x + 35 = 90, so x = 55°.
• Two supplementary angles where one is (2x + 10)° and the other is 70°. Find x: (2x + 10) + 70 = 180, so 2x + 80 = 180, therefore x = 50.
• Two complementary angles expressed as (3x - 5)° and (x + 15)°. Find both angles: (3x - 5) + (x + 15) = 90, so 4x + 10 = 90, therefore x = 20. The angles are 55° and 35°.

Explanation

When one angle measure is known, you can find the unknown angle by setting up a simple equation. Subtract the known angle from 90° for complementary angles or from 180° for supplementary angles. If the angles contain variables, set up the addition equation and solve it using basic algebra.