Question
Consider Carmen’s new set of plans.
Parallel lines k and l are crossed by transversal t. Line m crosses lines t, k, and l. The angles formed by lines k, t, and m, clockwise from top left are 2, 1, blank, blank, blank (6 x + 5) degrees; formed by lines l and t are blank, (7x minus 4) degrees, blank, blank; formed by lines l and m are 48 degrees, blank, blank, blank.
Examine the lines that are cut by transversals to determine the measure of angle 1.
What is the measure of angle 1?
m <1=
°
Why?
Parallel lines k and l are crossed by transversal t. Line m crosses lines t, k, and l. The angles formed by lines k, t, and m, clockwise from top left are 2, 1, blank, blank, blank (6 x + 5) degrees; formed by lines l and t are blank, (7x minus 4) degrees, blank, blank; formed by lines l and m are 48 degrees, blank, blank, blank.
Examine the lines that are cut by transversals to determine the measure of angle 1.
What is the measure of angle 1?
m <1=
°
Why?
Answers
GPT-4o mini
To solve for the measure of angle 1, let's define the relationship of the angles formed by the intersections of the lines. Here's a breakdown of the information provided:
1. **Parallel Lines (k and l)**: Since lines k and l are parallel, angles formed by a transversal intersecting them will maintain certain relationships (alternate interior angles, corresponding angles, etc.).
2. **Angle Relationships**: Based on the way angles are numbered and the placement of the lines, we can deduce the relationships.
- From the description:
- Angle 2 is at the intersection of line k and transversal t.
- Angle 1 is adjacent to angle 2 at line k.
- The angles formed by lines l and t and the angles formed by lines l and m seem to complete the configurations at both sets of angles intersecting.
3. **Given Angles**:
- We know:
- Angle 2 = 1 (at t and k)
- The angle opposite to angle 2 taken by transversal t and line m corresponds to angle 1.
- Angle related to line l = 7x - 4 (basically opposite to angle 2 when considering alternate interior angles on parallel lines).
4. **Finding Measure of Angles**:
Since angles 2 and the angle opposing angle 1 are alternate interior angles, they are equal. Therefore:
- \( m \angle 2 = m \angle (7x - 4) \)
5. **From the Information**:
We know:
- Angle 2 = (6x + 5)
- Therefore, \( 6x + 5 = 7x - 4 \)
6. **Solving the Equation**:
Rearranging gives us:
\[
6x + 5 = 7x - 4 \\
5 + 4 = 7x - 6x \\
9 = x
\]
7. **Finding Measure of Angle 1**:
Now that we have the value of x, we can substitute x back into either angle 2 or angle 1:
\[
m \angle 2 = 6(9) + 5 = 54 + 5 = 59 \text{ degrees}
\]
Thus,
\[
m \angle 1 = m \angle 2 = 59 \text{ degrees}
\]
So, the measure of angle 1 is:
- **m <1 = 59°**
### Why?
This is because angle 1 and angle 2 are alternate interior angles formed by the transversal t intersecting the parallel lines k and l, hence they are equal. Therefore, by determining angle 2, we effectively found the measure of angle 1.
1. **Parallel Lines (k and l)**: Since lines k and l are parallel, angles formed by a transversal intersecting them will maintain certain relationships (alternate interior angles, corresponding angles, etc.).
2. **Angle Relationships**: Based on the way angles are numbered and the placement of the lines, we can deduce the relationships.
- From the description:
- Angle 2 is at the intersection of line k and transversal t.
- Angle 1 is adjacent to angle 2 at line k.
- The angles formed by lines l and t and the angles formed by lines l and m seem to complete the configurations at both sets of angles intersecting.
3. **Given Angles**:
- We know:
- Angle 2 = 1 (at t and k)
- The angle opposite to angle 2 taken by transversal t and line m corresponds to angle 1.
- Angle related to line l = 7x - 4 (basically opposite to angle 2 when considering alternate interior angles on parallel lines).
4. **Finding Measure of Angles**:
Since angles 2 and the angle opposing angle 1 are alternate interior angles, they are equal. Therefore:
- \( m \angle 2 = m \angle (7x - 4) \)
5. **From the Information**:
We know:
- Angle 2 = (6x + 5)
- Therefore, \( 6x + 5 = 7x - 4 \)
6. **Solving the Equation**:
Rearranging gives us:
\[
6x + 5 = 7x - 4 \\
5 + 4 = 7x - 6x \\
9 = x
\]
7. **Finding Measure of Angle 1**:
Now that we have the value of x, we can substitute x back into either angle 2 or angle 1:
\[
m \angle 2 = 6(9) + 5 = 54 + 5 = 59 \text{ degrees}
\]
Thus,
\[
m \angle 1 = m \angle 2 = 59 \text{ degrees}
\]
So, the measure of angle 1 is:
- **m <1 = 59°**
### Why?
This is because angle 1 and angle 2 are alternate interior angles formed by the transversal t intersecting the parallel lines k and l, hence they are equal. Therefore, by determining angle 2, we effectively found the measure of angle 1.