Question

In \(\triangle X Y Z, \overline{Z W}\) bisects \(\angle Y Z X\). If \(m \angle 1=5 x+9\) and \(m \angle 2=39\), find \(x\).

Short answer

The value of \( x \) is 6.

Step-by-step solution

Understand the Angle Bisector Property

In any triangle, an angle bisector divides the angle into two equal parts. Here, since \( \overline{Z W} \) is the angle bisector of \( \angle YZX \), it divides it into two equal angles \( \angle 1 \) and \( \angle 2 \). Therefore, \( m \angle 1 = m \angle 2 \).

Set Up the Equation

Since \( m \angle 1 = m \angle 2 \), we can write the equation: \( 5x + 9 = 39 \). This equation equates the expression for \( m \angle 1 \) with the given measurement of \( m \angle 2 \).

Solve for \( x \)

To solve for \( x \), begin by subtracting 9 from both sides of the equation: \( 5x + 9 - 9 = 39 - 9 \). This simplifies to \( 5x = 30 \). Next, divide both sides by 5: \( x = \frac{30}{5} \), resulting in \( x = 6 \).

Key concepts

Understanding Triangle Properties

Triangles are a basic shape in geometry formed by three sides and three angles. Each triangle has several properties that are crucial when solving related problems.

• **Interior Angles**: The sum of all interior angles in a triangle is always 180 degrees. This is a fundamental property used to find unknown angles when the measures of other angles are given.
• **Types of Triangles**: Triangles are classified based on their sides and angles. For instance, equilateral triangles have all sides and angles equal, while isosceles triangles have two equal sides and angles.
• **Angle Bisectors**: An angle bisector in a triangle divides an angle into two equal smaller angles. This property is essential in many geometry problems, like the one in the exercise, revealing equal angles that help set up equations for solutions.

Understanding these properties is helpful in identifying relationships within triangles and solving geometric problems.

Equation Solving Techniques

Equation solving is a critical skill in mathematics, helping us find the value of unknown variables. Here, the problem involves setting up and solving a simple linear equation.

• **Setting Up the Equation**: Start by translating the problem's narrative into a mathematical equation, like equating two expressions known to be equal based on geometric properties.
• **Solving Linear Equations**: To solve for a variable, the goal is to isolate it on one side of the equation. Techniques include:
• Adding or subtracting the same value from both sides to maintain equality.
• Multiplying or dividing both sides of the equation to simplify and solve.
• **Checking Solutions**: After finding the value of the variable, substitute it back into the original equation to verify correctness.
  

This systematic approach leads to accurate solutions and helps students build confidence in solving equations efficiently.

The Concept of Angle Measurement

Angle measurement in geometry is the degree of rotation from one arm of the angle to another, always measured in degrees.

• **Units of Measure**: Angles are typically measured in degrees (\( ^\circ \)), with a full circle being 360 degrees. Other units like radians may be used in advanced topics.
• **Protractors**: A common tool used in schools to measure angles, protractors have scales for directly reading the degree measure of an angle.
• **Applications**: Accurate angle measurement is crucial in various real-life applications, including construction, navigation, and design, where precise angles are necessary for structural integrity and alignment.

By understanding how angles are measured and utilized, students can apply geometric principles effectively to solve practical and theoretical problems.