Math Problem Statement

solve for all the variables

solve for all the variables in the geometric diagram

Solution

To solve the variables in this diagram, we will examine the angles provided, as well as use properties of parallel lines and the geometry of intersecting lines (e.g., vertically opposite angles, corresponding angles, supplementary angles).

Here’s how to approach solving the problem:

  1. Angle Properties of Parallel Lines:
    • When two parallel lines are cut by a transversal, the corresponding angles are equal, and the alternate interior angles are also equal.
    • Vertically opposite angles (formed by the intersection of two lines) are equal.
    • Adjacent angles that form a straight line (linear pair) sum to 180180^\circ.

Analyzing the Diagram:

  • Angle 1 (given as 42°)
  • Angle 2 (given as 110°)
  • Angle 3 (given as 124°)

Key Observations:

  1. At the intersection at point X:

    • 1\angle 1 is 4242^\circ.
    • Vertically opposite angles are equal, so the angle opposite 1\angle 1 is also 4242^\circ.
  2. At the intersection of lines u and w with line a:

    • One of the angles is 115115^\circ.
    • Since adjacent angles along a straight line sum to 180180^\circ, the adjacent angle to 115115^\circ will be: 180115=65.180^\circ - 115^\circ = 65^\circ. So, the angle opposite 115\angle 115^\circ is 6565^\circ.
  3. At the intersection where angle 124124^\circ is given:

    • The adjacent angle is: 180124=56.180^\circ - 124^\circ = 56^\circ.
  4. At the intersection where angle 135135^\circ is given:

    • The adjacent angle is: 180135=45.180^\circ - 135^\circ = 45^\circ.

With this approach, you can use the same principles for every other set of angles and intersections.

Would you like more detailed steps, or should I solve specific variables from the diagram?

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Math Problem Analysis

Mathematical Concepts

Geometry
Angles
Parallel Lines
Transversals
Vertically Opposite Angles
Supplementary Angles

Formulas

Vertically opposite angles are equal
Sum of angles on a straight line (linear pair) is 180°
Corresponding angles between parallel lines are equal
Alternate interior angles between parallel lines are equal

Theorems

Angle Sum Property of a Triangle
Linear Pair Theorem
Parallel Lines and Transversal Theorem

Suitable Grade Level

Grades 8-10