Solving Equations With A Variable On Both Sides - Vertical Angles

🔑 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.
• Use inverse operations to isolate the variable.
• Substitute the variable’s value into each expression to determine the angle measures.
• Adjacent angles forming a linear pair have a sum of \(180^\circ\).

✏️ Worked Example

Find \(x\) and Determine All Four Angle Measures

The angles labeled \(9x-18\) and \(6x+3\) are opposite angles formed by two intersecting lines.

Step 1: Identify the relationship.

The two labeled angles are vertical angles. Vertical angles are congruent, so their expressions have equal values.

Step 2: Write the equation.

\(9x-18=6x+3\)

Step 3: Subtract \(6x\) from both sides.

\(9x-18\ {\color{red}{-6x}}=6x+3\ {\color{red}{-6x}}\)

\(3x-18=3\)

Step 4: Add \(18\) to both sides.

\(3x-18\ {\color{red}{+18}}=3\ {\color{red}{+18}}\)

\(3x=21\)

Step 5: Divide both sides by \(3\).

\(\displaystyle \frac{3x}{\color{red}{3}} = \frac{21}{\color{red}{3}}\)

\(x=7\)

Step 6: Find the vertical angle measures.

\(9(7)-18=63-18=45^\circ\)

\(6(7)+3=42+3=45^\circ\)

Both expressions produce \(45^\circ\), confirming that the vertical angles are congruent.

Find the adjacent angle measures.

Each adjacent angle forms a linear pair with a \(45^\circ\) angle.

\(180^\circ-45^\circ=135^\circ\)

The upper and lower vertical angles both measure \(135^\circ\).

Final Answers

\(\boxed{x=7}\)

Acute angles: \(45^\circ\)
Obtuse angles: \(135^\circ\)

🧠 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.
• Adjacent angles: Angles that share a common vertex and a common side.
• Linear pair: Two adjacent angles whose nonshared sides form a straight line.
• Supplementary angles: Two angles whose measures have a sum of \(180^\circ\).
• 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 expressions \(9x-18\) and \(6x+3\) can be set equal. Solving the equation gives \(x=7\), and substitution shows that both labeled vertical angles measure \(45^\circ\). The remaining angles form linear pairs with the \(45^\circ\) angles, so each one measures \(135^\circ\).

📚 What You Should Already Know

You should know that opposite angles formed by intersecting lines are vertical angles and that vertical angles have equal measures. You should also understand that adjacent angles forming a straight line are supplementary. Algebra skills needed for this lesson include combining like terms, using inverse operations, solving a two-step equation, and substituting a value into an expression.

🚀 What Comes Next

These same equation-solving strategies can be used when parallel lines are crossed by a transversal. Corresponding angles, alternate interior angles, alternate exterior angles, and same-side interior angles can all be represented by algebraic expressions and used to determine unknown values.

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