Math Problem Statement

In the figure below, h ∥ l. Find the values of y and z.

Solution

To solve for yy and zz in this diagram, we will use the properties of parallel lines and corresponding angles.

Step 1: Analyze the given angles and relationships

Given:

  • hlh \parallel l (the lines are parallel).
  • The angle marked as 8585^\circ is an interior angle with respect to the transversal crossing hh and ll.
  • There is an angle marked as yy^\circ and an angle (3z+22)(3z + 22)^\circ in the diagram.

Step 2: Set up equations using angle relationships

  1. Corresponding Angles (Parallel Lines): Since hlh \parallel l, the angle labeled 8585^\circ and the angle labeled (3z+22)(3z + 22)^\circ are corresponding angles. Thus, they must be equal: 3z+22=853z + 22 = 85

  2. Solve for zz: 3z=85223z = 85 - 22 3z=633z = 63 z=633=21z = \frac{63}{3} = 21

  3. Alternate Interior Angles: Since hlh \parallel l, the angle labeled yy^\circ is an alternate interior angle with the 8585^\circ angle. Thus: y=85y = 85

Final Answer:

y=85y = 85^\circ z=21z = 21

Would you like further details or have any questions about this solution?


Here are five related questions for practice:

  1. If hlh \parallel l and the angle between another transversal and hh is 7070^\circ, what would be the corresponding angle on ll?
  2. In a similar diagram, if y=90y = 90^\circ, what would be the value of an alternate interior angle?
  3. How would the solution change if 3z+22=903z + 22 = 90^\circ instead of 8585^\circ?
  4. What is the sum of angles formed at a point where a transversal intersects two parallel lines?
  5. Can we determine any angle if we only know zz but not yy in a similar setup?

Tip: For problems with parallel lines, always look for corresponding, alternate interior, or alternate exterior angles to simplify your calculations.

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Math Problem Analysis

Mathematical Concepts

Parallel Lines
Corresponding Angles
Alternate Interior Angles
Algebra

Formulas

Equation setup based on corresponding and alternate interior angles

Theorems

Corresponding Angles Postulate
Alternate Interior Angles Theorem

Suitable Grade Level

Grades 8-10