Math Problem Statement

In the diagram, solve for x given the angles: \( \angle ABE = (17x + 2)^\circ \), \( \angle ABC = (15x + 20)^\circ \), and \( \angle ECD = (9x + 10)^\circ \).

Solution

In this diagram, we observe three angles formed by intersecting lines at points BB, CC, and EE. These angles are labeled in terms of xx:

  1. ABE=(17x+2)\angle ABE = (17x + 2)^\circ,
  2. ABC=(15x+20)\angle ABC = (15x + 20)^\circ,
  3. ECD=(9x+10)\angle ECD = (9x + 10)^\circ.

To solve for xx, we need to apply the property that the sum of the angles along a straight line is 180180^\circ. Specifically, we are interested in:

  • ABE\angle ABE and ABC\angle ABC forming a straight line (linear pair),
  • ECD\angle ECD and its supplementary angle, which we might deduce based on other given information.

Let me solve this step by step.The value of xx is 7916\frac{79}{16} (approximately 4.94).

Now, using this value of xx, the angles are calculated as follows:

  1. ABE=13751685.94\angle ABE = \frac{1375}{16} \approx 85.94^\circ,
  2. ABC=15051694.06\angle ABC = \frac{1505}{16} \approx 94.06^\circ,
  3. ECD=8711654.44\angle ECD = \frac{871}{16} \approx 54.44^\circ.

Would you like further details or clarification on any of the steps?

Here are some related questions for further exploration:

  1. What other properties can we infer from the intersecting lines in the diagram?
  2. How do supplementary angles help in solving such problems?
  3. Could you verify the sum of angles in a triangle with these results?
  4. What role do linear pairs play in geometry?
  5. How does solving for variables in angles extend to more complex figures?

Tip: Always check for supplementary angles when dealing with intersecting lines to simplify calculations.

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Math Problem Analysis

Mathematical Concepts

Geometry
Angles
Linear Pairs
Supplementary Angles

Formulas

Sum of angles on a straight line: 180°
Angle-Sum Property of a Triangle: 180°

Theorems

Linear Pair Theorem
Supplementary Angles Theorem

Suitable Grade Level

Grades 9-11