Homework: Angles and Parallel Lines - Answer Key
This answer key provides solutions for common homework problems involving angles and parallel lines. Remember, always check your teacher's specific instructions and grading rubric as variations in problem presentation and required steps may exist. This key aims to illustrate the concepts and problem-solving techniques. Consult your textbook and class notes for further clarification and examples.
Understanding Key Concepts:
Before diving into specific problems, let's review some fundamental concepts:
- Parallel Lines: Two lines that never intersect, maintaining a constant distance apart.
- Transversal: A line that intersects two or more parallel lines.
- Corresponding Angles: Angles that are in the same relative position at an intersection when a line intersects two other lines. If the lines are parallel, corresponding angles are congruent (equal).
- Alternate Interior Angles: Angles on opposite sides of the transversal and inside the parallel lines. If the lines are parallel, alternate interior angles are congruent.
- Alternate Exterior Angles: Angles on opposite sides of the transversal and outside the parallel lines. If the lines are parallel, alternate exterior angles are congruent.
- Consecutive Interior Angles (Same-Side Interior Angles): Angles on the same side of the transversal and inside the parallel lines. If the lines are parallel, consecutive interior angles are supplementary (add up to 180°).
- Vertical Angles: Angles opposite each other when two lines intersect. Vertical angles are always congruent.
Sample Problems and Solutions:
Problem 1: Find the value of x if two parallel lines are intersected by a transversal, creating corresponding angles of 3x + 10 and 5x - 20 degrees.
Solution: Since corresponding angles are congruent when lines are parallel, we set the expressions equal to each other:
3x + 10 = 5x - 20
Solving for x:
2x = 30 x = 15
Therefore, x = 15 degrees.
Problem 2: Two parallel lines are cut by a transversal. One alternate interior angle measures 75 degrees. What is the measure of the other alternate interior angle?
Solution: Alternate interior angles are congruent when lines are parallel. Therefore, the other alternate interior angle also measures 75 degrees.
Problem 3: Find the value of x if two parallel lines are intersected by a transversal, creating consecutive interior angles of (2x + 30)° and (3x - 60)°.
Solution: Consecutive interior angles are supplementary. Therefore, their sum is 180°.
(2x + 30) + (3x - 60) = 180
5x - 30 = 180
5x = 210
x = 42
Therefore, x = 42 degrees.
Problem 4 (More Complex): In the diagram showing parallel lines intersected by a transversal, ∠A = 110°. Find the measures of ∠B, ∠C, ∠D, ∠E, ∠F, ∠G, and ∠H. (Assume standard angle labeling with angles A, B, C, D, E, F, G, and H appropriately situated).
Solution: This requires applying multiple angle relationships. For example:
- ∠B (vertical to ∠A) = 110°
- ∠C (corresponding to ∠A) = 110°
- ∠D (alternate interior to ∠A) = 70° (Supplementary to ∠C or ∠B)
- ∠E (vertical to ∠D) = 70°
- ∠F (corresponding to ∠D) = 70°
- ∠G (alternate interior to ∠D)=70°
- ∠H (vertical to ∠G) = 70°
Remember to clearly show your work and justify each step with the relevant geometric property. This approach ensures accuracy and full credit on your homework. If you encounter difficulties with specific problems, provide the diagram and problem statement, and I can offer more tailored assistance.
