Math Problem Statement

Find the values of x and y given the angles: \( \angle EBC = 3x + 6 \), \( \angle BDF = y \), \( \angle CAF = (x - 10) \), \( \angle DCF = (x - 11) \).

Solution

The problem presented in the image asks to find the values of xx and yy for the angles in the given diagram. Here is the breakdown:

The diagram consists of two intersecting lines forming vertical angles. We are given the following expressions for the angles:

  1. Angle EBC\angle EBC is labeled as 3x+63x + 6^\circ.
  2. Angle BDF\angle BDF is labeled as yy^\circ.
  3. Angle CAF\angle CAF is labeled as (x10)(x - 10)^\circ.
  4. Angle DCF\angle DCF is labeled as (x11)(x - 11)^\circ.

Step 1: Use Vertical Angles

We know from geometry that vertical angles are congruent, meaning their measures are equal. Using this:

  • EBC=DCF\angle EBC = \angle DCF, so: 3x+6=x113x + 6 = x - 11
  • BDF=CAF\angle BDF = \angle CAF, so: y=x10y = x - 10

Step 2: Solve for xx

First, solve the equation 3x+6=x113x + 6 = x - 11: 3xx=1163x - x = -11 - 6 2x=172x = -17 x=172=8.5x = -\frac{17}{2} = -8.5

Step 3: Solve for yy

Now, use the equation y=x10y = x - 10 to find yy: y=8.510=18.5y = -8.5 - 10 = -18.5

Final Answer:

  • x=8.5x = -8.5
  • y=18.5y = -18.5

Would you like more details or clarifications? Here are five questions to explore further:

  1. Why are vertical angles always congruent?
  2. What happens if the angles were supplementary instead of vertical?
  3. How can you verify if your solutions for xx and yy are correct?
  4. What is the importance of solving systems of equations in geometry?
  5. How would the problem change if the angle expressions were different?

Tip: Always check your solutions by substituting the values back into the original equations.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Algebra
Vertical Angles
Linear Equations

Formulas

Vertical Angles: Opposite angles are equal
Linear equation solution method

Theorems

Vertical Angles Theorem

Suitable Grade Level

Grade 8-10