Math Problem Statement

The figure to the right shows two parallel lines intersected by a transversal. Let x = 109°. Find the measure of each of the other seven angles.

Solution

The problem in the image involves two parallel lines intersected by a transversal, and you're given one angle x=109x = 109^\circ. The goal is to find the measures of the other seven angles.

Key concepts for solving:

  • Corresponding angles: Equal when two parallel lines are cut by a transversal.
  • Alternate interior angles: Equal on opposite sides of the transversal.
  • Supplementary angles: Two angles that add up to 180180^\circ (angles on a straight line).

Let's label the angles according to the diagram:

  • Angle x=109\text{Angle } x = 109^\circ (already given)
  • Angle 11, which is adjacent to xx, is a supplementary angle: Angle 1=180109=71\text{Angle } 1 = 180^\circ - 109^\circ = 71^\circ
  • Angle 33 is vertically opposite to xx, so: Angle 3=109\text{Angle } 3 = 109^\circ
  • Angle 22 is vertically opposite to angle 11, so: Angle 2=71\text{Angle } 2 = 71^\circ
  • Since angles on the parallel lines correspond, the lower set of angles (4, 5, 6, 7) are the same as the upper set:
    • Angle 4=109\text{Angle } 4 = 109^\circ
    • Angle 5=71\text{Angle } 5 = 71^\circ
    • Angle 6=109\text{Angle } 6 = 109^\circ
    • Angle 7=71\text{Angle } 7 = 71^\circ

Summary of all angles:

  • x=109x = 109^\circ
  • Angle 1=71\text{Angle } 1 = 71^\circ
  • Angle 2=71\text{Angle } 2 = 71^\circ
  • Angle 3=109\text{Angle } 3 = 109^\circ
  • Angle 4=109\text{Angle } 4 = 109^\circ
  • Angle 5=71\text{Angle } 5 = 71^\circ
  • Angle 6=109\text{Angle } 6 = 109^\circ
  • Angle 7=71\text{Angle } 7 = 71^\circ

Let me know if you'd like more details or have any questions.


Here are 5 related questions to consider:

  1. What is the relationship between alternate interior angles when a transversal cuts parallel lines?
  2. How do corresponding angles behave in a transversal setup?
  3. What would change if the lines were not parallel?
  4. How can you identify supplementary angles in different geometric setups?
  5. Can you use this method for more complex figures involving multiple transversals?

Tip: In problems with transversals and parallel lines, look for angle pairs such as corresponding, alternate interior, and supplementary angles to simplify the process of finding unknown angles.

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Math Problem Analysis

Mathematical Concepts

Parallel Lines
Transversals
Angles

Formulas

Supplementary Angles: two angles on a straight line sum to 180°
Corresponding Angles: equal when two parallel lines are intersected by a transversal
Alternate Interior Angles: equal when on opposite sides of a transversal

Theorems

Corresponding Angles Postulate
Alternate Interior Angles Theorem
Linear Pair Postulate

Suitable Grade Level

Grades 8-10