Question
Carmen is planning rail lines for a new train station. Help her find mes002-1.jpg1. Explain how you found that solution.
Parallel lines a and b are cut by transversal t to form 8 angles. Clockwise from top left, the angles formed with line a are blank, 17 degrees, 2, blank; with line b are 1, blank, blank, blank.
Answers
GPT-4o mini
To solve the problem of finding the missing angles when two parallel lines (lines a and b) are cut by a transversal (line t), we can use the properties of angles formed by transversals with parallel lines.
Let's label the angles as follows based on your description:
- At line a (from top left, clockwise):
- Angle 1: blank
- Angle 2: 17 degrees
- Angle 3: blank
- Angle 4: blank
- At line b (from top left, clockwise):
- Angle 5: 1 degree
- Angle 6: blank
- Angle 7: blank
- Angle 8: blank
### Step-by-step explanation:
1. **Identify relationships between angles**:
- The angles formed by a transversal cutting through parallel lines exhibit several relationships including corresponding angles, alternate interior angles, and consecutive interior angles.
2. **Corresponding Angles Theorem**:
- This theorem states that corresponding angles are equal when lines are parallel. In this arrangement, if Angle 2 (17 degrees) at line a has a corresponding Angle 6 at line b, then Angle 6 = 17 degrees.
3. **Vertical Angles**:
- Angles that are opposite each other when two lines intersect are also equal. Angle 2 (17 degrees) has a vertical angle with Angle 4. Thus, Angle 4 = 17 degrees.
4. **Sum of Angles in a Line**:
- Since angles on a straight line (like Angle 2 and Angle 3 at line a) add up to 180 degrees, we apply this property:
- Angle 3 (blank) = 180 degrees - Angle 2 (17 degrees)
- Therefore, Angle 3 = 180 - 17 = 163 degrees.
5. **Finding Missing Angles Related to Angle 5 (1 degree)**:
- Since Angle 5 (1 degree) at line b is an alternate interior angle to Angle 3 (163 degrees), we understand that:
- Angle 1 (blank) = 180 degrees - Angle 5 (1 degree)
- Hence, Angle 1 = 180 - 1 = 179 degrees.
6. **Finding Other Angles**:
- Now knowing Angle 1 (179 degrees) and Angle 6 (17 degrees), we can use the same principles to find the last angles.
- For Angle 7, which is alternate to Angle 2 (17 degrees), therefore Angle 7 = 17 degrees.
- For Angle 8, which is a vertical angle to Angle 5 (1 degree), hence Angle 8 = 1 degree.
### Final Angles:
- Angle 1: 179 degrees
- Angle 2: 17 degrees
- Angle 3: 163 degrees
- Angle 4: 17 degrees
- Angle 5: 1 degree
- Angle 6: 17 degrees
- Angle 7: 1 degree
- Angle 8: 1 degree
Thus, by applying the properties of angles created by a transversal cutting through parallel lines, we found all the missing angles.
Let's label the angles as follows based on your description:
- At line a (from top left, clockwise):
- Angle 1: blank
- Angle 2: 17 degrees
- Angle 3: blank
- Angle 4: blank
- At line b (from top left, clockwise):
- Angle 5: 1 degree
- Angle 6: blank
- Angle 7: blank
- Angle 8: blank
### Step-by-step explanation:
1. **Identify relationships between angles**:
- The angles formed by a transversal cutting through parallel lines exhibit several relationships including corresponding angles, alternate interior angles, and consecutive interior angles.
2. **Corresponding Angles Theorem**:
- This theorem states that corresponding angles are equal when lines are parallel. In this arrangement, if Angle 2 (17 degrees) at line a has a corresponding Angle 6 at line b, then Angle 6 = 17 degrees.
3. **Vertical Angles**:
- Angles that are opposite each other when two lines intersect are also equal. Angle 2 (17 degrees) has a vertical angle with Angle 4. Thus, Angle 4 = 17 degrees.
4. **Sum of Angles in a Line**:
- Since angles on a straight line (like Angle 2 and Angle 3 at line a) add up to 180 degrees, we apply this property:
- Angle 3 (blank) = 180 degrees - Angle 2 (17 degrees)
- Therefore, Angle 3 = 180 - 17 = 163 degrees.
5. **Finding Missing Angles Related to Angle 5 (1 degree)**:
- Since Angle 5 (1 degree) at line b is an alternate interior angle to Angle 3 (163 degrees), we understand that:
- Angle 1 (blank) = 180 degrees - Angle 5 (1 degree)
- Hence, Angle 1 = 180 - 1 = 179 degrees.
6. **Finding Other Angles**:
- Now knowing Angle 1 (179 degrees) and Angle 6 (17 degrees), we can use the same principles to find the last angles.
- For Angle 7, which is alternate to Angle 2 (17 degrees), therefore Angle 7 = 17 degrees.
- For Angle 8, which is a vertical angle to Angle 5 (1 degree), hence Angle 8 = 1 degree.
### Final Angles:
- Angle 1: 179 degrees
- Angle 2: 17 degrees
- Angle 3: 163 degrees
- Angle 4: 17 degrees
- Angle 5: 1 degree
- Angle 6: 17 degrees
- Angle 7: 1 degree
- Angle 8: 1 degree
Thus, by applying the properties of angles created by a transversal cutting through parallel lines, we found all the missing angles.