Math Problem Statement

Identify each pair of angles as corresponding, alternate interior, alternate exterior, consecutive interior, vertical, or adjacent. Also, find the measure of the indicated angles in various figures.

Solution

Let's break down the problems one by one based on the image.

9-12) Identifying angle relationships:

You are asked to identify the relationship between pairs of angles as:

  • Corresponding
  • Alternate interior
  • Alternate exterior
  • Consecutive interior
  • Vertical
  • Adjacent

Here’s the breakdown for each:

9) The angles xx and yy are vertical angles because they are across from each other when two lines intersect.

10) xx and yy are alternate exterior angles because they lie on opposite sides of the transversal and are outside the two parallel lines.

11) xx and yy are alternate interior angles because they lie on opposite sides of the transversal and between the two parallel lines.

12) xx and yy are corresponding angles because they are on the same side of the transversal and occupy corresponding positions (one on the top left, one on the top right).


13-16) Finding angle measures:

13)

  • Given that one angle is 48°, the unknown angle is also 48°.
  • This is because these are corresponding angles, and corresponding angles are congruent when the lines are parallel.

14)

  • The given angle is 120°.
  • The unknown angle is supplementary to 120° (because they are on a straight line), meaning the two angles add up to 180°.
  • So, 180°120°=60°180° - 120° = 60°.

15)

  • The given angle is 85°.
  • The unknown angle is a vertical angle to the given angle.
  • Vertical angles are congruent, so the unknown angle is also 85°.

16)

  • The given angle is 62°.
  • The unknown angle is corresponding to the 62° angle.
  • Corresponding angles are congruent, so the unknown angle is also 62°.

Would you like further clarification or any more details on these concepts?


Here are some related questions to extend your understanding:

  1. What is the difference between alternate interior and alternate exterior angles?
  2. How can you identify adjacent angles in any given figure?
  3. Why are corresponding angles congruent when two lines are parallel?
  4. What is the mathematical relationship between supplementary angles?
  5. Can two vertical angles ever be supplementary?

Tip: Always check whether lines are parallel, as angle relationships like corresponding, alternate interior, and alternate exterior depend on this condition.

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Math Problem Analysis

Mathematical Concepts

Angle Relationships
Corresponding Angles
Vertical Angles
Alternate Interior Angles
Alternate Exterior Angles
Supplementary Angles

Formulas

Vertical Angles are congruent: ∠x = ∠y
Corresponding Angles are congruent when lines are parallel: ∠x = ∠y
Supplementary Angles add to 180°: ∠x + ∠y = 180°

Theorems

Vertical Angles Theorem
Alternate Interior Angles Theorem
Corresponding Angles Postulate

Suitable Grade Level

Grades 7-9