🔑 Key Concepts
- When two lines intersect, they form two pairs of vertical angles.
- Vertical angles are opposite one another and have equal measures.
- Expressions representing vertical angles can be set equal to form an equation.
- Inverse operations are used to isolate the variable.
- Substitution can be used to verify the solution and the angle measure.
✏️ Worked Example
🧠 Math Vocabulary
- Intersecting lines: Lines that cross at a common point.
- Vertical angles: Opposite angles formed by two intersecting lines. Vertical angles are congruent.
- Congruent angles: Angles that have equal measures.
- Supplementary angles: Two angles whose measures add to \(180^\circ\).
- Inverse operations: Operations that undo one another and are used to isolate a variable.
- Substitution: Replacing a variable with a known value to evaluate an expression.
💡 Main Idea
When two lines intersect, opposite angles are vertical angles and have equal measures. Therefore, the expression \(\displaystyle \frac{400}{x}+15\) can be set equal to \(95\). Solving the resulting equation determines the value of \(x\), and substitution verifies that both vertical angles measure \(95^\circ\).
📚 What You Should Already Know
You should know that vertical angles are opposite angles formed by intersecting lines and that they have equal measures. You should also be comfortable using inverse operations, simplifying fractions, multiplying both sides of an equation by the same quantity, and checking a solution through substitution.
🚀 What Comes Next
These strategies can be extended to equations involving linear pairs, complementary angles, supplementary angles, and parallel lines cut by a transversal. More advanced problems may contain variables in denominators, variables on both sides, or several related angles in the same diagram.
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