Math Problem Statement

Solve for x and y using the given angle expressions: (4y - 18)°, (y + 14)°, (6x - 11)°, 81°, and (x + 5)° in a geometric diagram involving linear pairs and vertical angles.

Solution

The image shows a geometric diagram with angles labeled as expressions involving variables xx and yy. This diagram likely involves solving for unknown variables using properties of angles, such as vertical angles or linear pairs.

To solve this, let's analyze the setup:

  1. Angle Relationships:

    • The pair of angles labeled (4y18)(4y - 18)^\circ and (y+14)(y + 14)^\circ form a straight line (linear pair), so their sum should be 180180^\circ.
    • Similarly, the angles labeled (6x11)(6x - 11)^\circ and (x+5)(x + 5)^\circ are also on a straight line and should sum to 180180^\circ.
    • The vertical angle property suggests that the angle (6x11)(6x - 11)^\circ is equal to the 8181^\circ angle directly opposite.
  2. Setting Up the Equations:

    • For the linear pair (4y18)+(y+14)=180(4y - 18)^\circ + (y + 14)^\circ = 180^\circ
    • For the linear pair (6x11)+(x+5)=180(6x - 11)^\circ + (x + 5)^\circ = 180^\circ
    • For the vertical angles (6x11)=81(6x - 11)^\circ = 81^\circ

Now let's solve these equations:

1. Solve for yy:

(4y18)+(y+14)=180(4y - 18) + (y + 14) = 180 Simplify: 4y+y18+14=1804y + y - 18 + 14 = 180 5y4=1805y - 4 = 180 5y=1845y = 184 y=36.8y = 36.8

2. Solve for xx using vertical angles:

6x11=816x - 11 = 81 6x=926x = 92 x=15.33x = 15.33

Now that xx and yy are known, these values can be used to find any angle expressions if needed.

Would you like further details or clarifications? Here are 5 related questions to consider:

  1. What are the properties of vertical angles, and how are they useful in solving geometric problems?
  2. How do linear pairs relate to supplementary angles?
  3. How would you verify that the solution for xx and yy satisfies all angle relationships in the diagram?
  4. Can the value of one angle change if one of the lines was not straight, and how would that affect the equation?
  5. What other geometric properties could be used to solve a similar problem with triangles or polygons?

Tip: Always verify if the angles are vertical or part of a linear pair before setting up equations; this helps ensure correct relationships.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Linear Pairs
Vertical Angles
Algebraic Equations

Formulas

Sum of linear pairs = 180°
Vertical angles are equal

Theorems

Linear Pair Theorem
Vertical Angle Theorem

Suitable Grade Level

Grades 8-10