Problem 35 - Intermediate Algebra

Question

Vertical Angles. When two lines intersect, four angles are formed. Angles that are side-by-side, such as \(\angle 1\) (angle 1 ) and \(\angle 2,\) are called adjacent angles. Angles that are nonadjacent, such as \(\angle 1\) and \(\angle 3\) or \(\angle 2\) and \(\angle 4,\) are called vertical angles. From geometry, we know that if two lines intersect, vertical angles have the same measure. If \(\angle 1=3 x+10^{\circ}\) and \(\angle 3=5 x-10^{\circ},\) find \(x\)

Step-by-Step Solution

Answer

The value of \(x\) is 10.

Step 1: Identify Vertical Angles

To solve the problem, first identify the vertical angles. According to the problem statement, \(\angle 1\) and \(\angle 3\) are vertical angles. Therefore, they must have the same measure.

Step 2: Set Angles Equal

Since vertical angles are equal, set \(\angle 1\) equal to \(\angle 3\): \[ 3x + 10 = 5x - 10 \]

Step 3: Solve for x

Rearrange the equation to solve for \(x\). Start by eliminating \(3x\) from the left side:\[ 10 = 5x - 3x - 10 \] which simplifies to:\[ 10 = 2x - 10 \]

Step 4: Isolate x

Add 10 to both sides to isolate terms with \(x\):\[ 20 = 2x \] Now divide by 2 to solve for \(x\):\[ x = 10 \]

Key Concepts

Adjacent Angles

When we talk about adjacent angles, we refer to angles that share a common arm and a common vertex. Imagine two hands of a clock placed closely together to form an angle; each hand represents an arm, and where they meet is the vertex. This is similar to what happens when two lines cross.
Adjacent angles are often easy to spot due to their side-by-side nature.
Here are some important points to remember:

• They are formed when two angles share a side.
• Adjacent angles add up to form a larger angle, such as supplementary angles summing to 180°.
• Knowing the measure of one adjacent angle can help deduce the measure of the other if the total angle sum is known.

In the context of vertical and adjacent angles, identifying which are adjacent helps us solve for unknowns based on complementary properties.

Geometry

The world of geometry is vast and exciting, encompassing everything from points to shapes. When lines intersect, geometry provides us with rules and properties that predict how angles relate to each other. This makes geometry a powerful tool for solving complex problems. Some core principles of geometry related to angles include:

• Complementary Angles: Two angles whose measures add up to 90°.
• Supplementary Angles: Two angles whose measures sum up to 180°.
• Vertical Angles: These are formed when two lines intersect and are always equal in measure.

Understanding these geometrical principles enables you to visualize and solve problems effectively and is especially useful when working with intersecting lines and angles.

Intersection of Lines

Lines intersecting creates a fascinating geometrical scenario where multiple angles form, bringing into play unique properties like vertical angles. When two lines cross each other, it results in:

• Four angles being formed at the intersection.
• Two pairs of vertical angles, each set having equal measures.
• Four pairs of adjacent angles, combining to be supplementary in each pairing.

Understanding intersections in geometry is crucial because they highlight symmetrical properties and relationships between angles. This concept is vital not only for solving problems in textbooks but also for practical applications in engineering and design. When lines meet, they provide a consistent pattern which can be readily interpreted, making calculations and predictions far easier.