Vertical Angles - Setting Equations Equal To Each Other

🔑 Key Concepts

• When two lines intersect, opposite angles are called vertical angles.
• Vertical angles are congruent, so their measures are equal.
• Expressions representing vertical angles can be set equal to form an equation.
• When the variable appears on both sides, use inverse operations to collect the variable terms on one side.
• Substitute the solution into an original expression to determine the requested angle measure.

✏️ Worked Example

Find the Measure of \(\angle ebc\)

The angles labeled \(x+25\) and \(2x+5\) are opposite angles formed by two intersecting lines.

Step 1: Identify the relationship.

The angles labeled \(x+25\) and \(2x+5\) are vertical angles. Vertical angles have equal measures.

Step 2: Write the equation.

\(x+25=2x+5\)

Step 3: Subtract \(x\) from both sides.

\(x+25\ {\color{red}{-x}}=2x+5\ {\color{red}{-x}}\)

\(25=x+5\)

Step 4: Subtract \(5\) from both sides.

\(25\ {\color{red}{-5}}=x+5\ {\color{red}{-5}}\)

\(20=x\)

Step 5: Find \(m\angle ebc\).

Substitute \(x=20\) into the expression \(2x+5\).

\(2(20)+5\)

\(40+5=45^\circ\)

Therefore, the angle formed by rays \(BE\) and \(BC\) measures \(45^\circ\).

Check the vertical angles.

\(20+25=45^\circ\)

\(2(20)+5=45^\circ\)

Both expressions produce the same angle measure.

Final Answer

\(x=20\)

\(\boxed{m\angle ebc=45^\circ}\)

🧠 Math Vocabulary

• Intersecting lines: Lines that cross at a common point.
• Vertical angles: Opposite angles formed when two lines intersect.
• Congruent angles: Angles that have equal measures.
• Angle expression: An algebraic expression representing an unknown angle measure.
• Inverse operations: Operations that undo one another and help isolate a variable.
• Substitution: Replacing a variable with a known value to evaluate an expression.
• Angle notation: A method of naming an angle with three points, placing the vertex in the middle.

💡 Main Idea

When two lines intersect, opposite angles are vertical angles and have equal measures. This allows the expressions \(x+25\) and \(2x+5\) to be set equal. Solving the equation gives \(x=20\). Substituting this value into either expression shows that both vertical angles measure \(45^\circ\), so \(m\angle ebc=45^\circ\).

📚 What You Should Already Know

You should know that vertical angles are opposite angles formed by two intersecting lines and that they have equal measures. You should also be able to solve equations with variables on both sides, use inverse operations, substitute a value into an expression, and name an angle by placing its vertex as the middle letter.

🚀 What Comes Next

These same strategies can be applied to complementary angles, supplementary angles, linear pairs, and angles formed when parallel lines are cut by a transversal. More advanced problems may require solving multi-step equations before determining the requested angle measure.

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