Math Problem Statement

If l || m, solve for x and y. The given angles are (8x - 14)°, (5x + 34)°, and (5y + 16)°.

Solution

In the given problem, lines ll and mm are parallel, and the angles provided correspond to those formed by a transversal crossing these parallel lines.

The key angles are:

  • (8x14)(8x - 14)^\circ
  • (5x+34)(5x + 34)^\circ
  • (5y+16)(5y + 16)^\circ

Step 1: Identify Angle Relationships

Since lml \parallel m, we can use the following angle properties:

  1. Alternate interior angles: These angles are equal.

    • (8x14)(8x - 14)^\circ and (5x+34)(5x + 34)^\circ are alternate interior angles. Therefore, we set them equal: 8x14=5x+348x - 14 = 5x + 34
  2. Same-side interior angles: These angles are supplementary (sum to 180180^\circ).

    • (5y+16)(5y + 16)^\circ and (5x+34)(5x + 34)^\circ are on the same side of the transversal, so: (5y+16)+(5x+34)=180(5y + 16) + (5x + 34) = 180

Step 2: Solve for xx

From the first equation 8x14=5x+348x - 14 = 5x + 34, solve for xx: 8x5x=34+148x - 5x = 34 + 14 3x=483x = 48 x=16x = 16

Step 3: Solve for yy

Substitute x=16x = 16 into the second equation: (5y+16)+(5(16)+34)=180(5y + 16) + (5(16) + 34) = 180 (5y+16)+(80+34)=180(5y + 16) + (80 + 34) = 180 5y+16+114=1805y + 16 + 114 = 180 5y+130=1805y + 130 = 180 5y=505y = 50 y=10y = 10

Final Answer:

  • x=16x = 16
  • y=10y = 10

Would you like further clarification on any part of the process? Here are five related questions to explore this topic further:

  1. How can you use corresponding angles to solve similar problems with parallel lines and a transversal?
  2. What other properties of angles are useful when working with parallel lines?
  3. How would the solution change if the lines were not parallel?
  4. Can you apply these angle properties to polygons, and how?
  5. What are exterior angles, and how do they relate to the sum of interior angles in polygons?

Tip: Always remember that alternate interior angles are equal when two parallel lines are intersected by a transversal.

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Math Problem Analysis

Mathematical Concepts

Geometry
Algebra
Parallel Lines and Angles

Formulas

Alternate Interior Angles: equal angles
Same-Side Interior Angles: supplementary angles

Theorems

Alternate Interior Angle Theorem
Same-Side Interior Angle Theorem

Suitable Grade Level

Grades 8-10