Angle Relationships: Step-by-Step Guide
When lines cross, or when a line crosses a pair of parallel lines, the angles it makes follow patterns. Complementary angles add up to 90 degrees. Supplementary angles add up to 180 degrees. Vertical angles (across from each other where two lines cross) are equal. When the lines are parallel, corresponding angles and alternate interior angles are equal too.
Math Racer skill guide · Geometry · Grade 10
How to solve it, step by step
- Figure out the angle pair. Look at the picture. Are the angles vertical (across an X from each other), a linear pair (side by side on a straight line), corresponding (in the same spot at each crossing), alternate interior (between the parallel lines on opposite sides), or co-interior (between the parallel lines on the same side)?\[\text{vertical} \;|\; \text{linear} \;|\; \text{corresponding} \;|\; \text{alternate} \;|\; \text{co-interior}\]
- Use the rule for that pair. Vertical angles are equal. A linear pair adds up to 180 degrees. Corresponding and alternate interior angles are equal (when the lines are parallel). Co-interior angles add up to 180 degrees.\[\text{vertical: } \angle 1 = \angle 2,\quad \text{linear: } \angle 1 + \angle 2 = 180^\circ\]
- Write an equation. Turn the rule into an equation using the angle measures or expressions you are given.\[(2x + 10) + (3x - 5) = 180\]
- Solve and plug back in. Solve for the variable, then put it back in to find each angle.\[x = \frac{180 - 5}{5} \Rightarrow \angle = (2x+10)^\circ\]
- Check. Make sure the angles do what the rule says. They should be equal, or add up to the right total.\[\angle 1 + \angle 2 \stackrel{?}{=} 180^\circ\]
Worked example
Two parallel lines are cut by a transversal. One angle measures (2x + 10) degrees and its co-interior angle measures (3x - 5) degrees. Find both angles.
Step 1: Co-interior angles between parallel lines add up to 180 degrees.
Step 2: Write the equation: (2x + 10) + (3x - 5) = 180.
Step 3: Combine like terms: 5x + 5 = 180.
Step 4: Solve: 5x = 175, so x = 35.
Step 5: Put x back in. First angle: 2(35) + 10 = 80 degrees. Second angle: 3(35) - 5 = 100 degrees.
Check: 80 + 100 = 180. Correct.
Tips and common mistakes
- Draw and label the picture if one is not given. It helps you see which angles go together.
- Vertical angles are always equal, and a linear pair always adds up to 180 degrees, even if there are no parallel lines.
- Corresponding, alternate interior, and alternate exterior angles are only equal if the lines are parallel. Look for arrows or the word 'parallel'.
- Co-interior angles (also called same-side interior angles) add up to 180 degrees. They are not equal.
- If the angles are given as expressions with x, write an equation first and then solve.
- Complementary angles add up to 90 degrees. Supplementary angles add up to 180 degrees. Do not mix them up.
- After you solve, put x back into both expressions. The angles should be positive and less than 180 degrees.
- Label every angle you can figure out. One angle often leads to the next.