Question
Horizontal and parallel lines e and f are cut by transversals c and d. All angles are described clockwise, from uppercase left. The intersection of lines d and c form angles: 1, 2, 4, 3. The intersection of lines f and c form angles 5, 6, 8, 7. The intersection of lines e and d form angles 9, 10, 12, 11. The intersection of lines f and d form angles 13, 14, 16, 15.
In the diagram, m∠3 = 120° and m∠12 = 80°. Which angle measures are correct? Check all that apply.
m∠1 = 60°
m∠13 = 80°
m∠6 = 80°
m∠5 = 60°
m∠10 = 120°
m∠ 14 = 100°
In the diagram, m∠3 = 120° and m∠12 = 80°. Which angle measures are correct? Check all that apply.
m∠1 = 60°
m∠13 = 80°
m∠6 = 80°
m∠5 = 60°
m∠10 = 120°
m∠ 14 = 100°
Answers
GPT-4o mini
Given the relationships and properties of angles formed by parallel lines and transversals, we can use the information provided, including \( m\angle 3 = 120^\circ \) and \( m\angle 12 = 80^\circ \), to determine the measures of the other angles.
1. **At the intersection of lines \( d \) and \( c \)**, we have angles \( 1, 2, 4, 3 \). Since \( m\angle 3 = 120^\circ \):
- Angles \( 1 \) and \( 3 \) are supplementary (they form a linear pair), so:
\[
m\angle 1 + m\angle 3 = 180^\circ \implies m\angle 1 + 120^\circ = 180^\circ \implies m\angle 1 = 60^\circ
\]
- Therefore, **\( m\angle 1 = 60^\circ \)** is correct.
2. **At the intersection of lines \( e \) and \( d \)**, we have angles \( 9, 10, 12, 11 \). Since \( m\angle 12 = 80^\circ \):
- Angles \( 12 \) and \( 10 \) are alternate interior angles (parallel lines), so:
\[
m\angle 10 = m\angle 12 = 80^\circ
\]
- Therefore, **\( m\angle 10 = 80^\circ \)** is correct.
3. **At the intersection of lines \( f \) and \( c \)**, we have angles \( 5, 6, 8, 7 \). Angle \( 6 \) is corresponding to angle \( 10 \) (both are in the same position with respect to the transversal), thus:
- \[
m\angle 6 = m\angle 10 = 80^\circ
\]
- Therefore, **\( m\angle 6 = 80^\circ \)** is correct.
4. At the intersection of lines \( f \) and \( d \) with angles \( 13, 14, 16, 15 \):
- Angle \( 14 \) is alternate interior to angle \( 10 \), thus:
\[
m\angle 14 = m\angle 10 = 80^\circ
\]
- Angle \( 16 \) is corresponding to angle \( 3 \):
\[
m\angle 16 = m\angle 3 = 120^\circ
\]
5. Finally, from angle \( 5 \) (referring to \( 6 \)), since \( 5 + 6 = 180^\circ \):
\[
m\angle 5 + m\angle 6 = 180^\circ \Rightarrow m\angle 5 + 80^\circ = 180^\circ \implies m\angle 5 = 100^\circ
\]
Summarizing the findings:
- \( m\angle 1 = 60^\circ \) (correct)
- \( m\angle 13 = 80^\circ \) (uncertain without further information)
- \( m\angle 6 = 80^\circ \) (correct)
- \( m\angle 5 = 100^\circ \) (correct - if we interpret your message it seems this is incorrect; it contradicts the m∠5 measure)
- \( m\angle 10 = 80^\circ \) (incorrect, expectation of \( m(10) = 80^\circ \); hence \( m(12) = 80^\circ \))
- \( m\angle 14 = 80^\circ \) (correct)
So the final correct measurements based on the information given are:
- \( m\angle 1 = 60^\circ \)
- \( m\angle 6 = 80^\circ \)
- \( m\angle 10 = 80 \)
- \( m\angle 14 \)
Note: If we consider 15 and 16 from the statements in your message as confirmations, those results may lead to further deductions.
1. **At the intersection of lines \( d \) and \( c \)**, we have angles \( 1, 2, 4, 3 \). Since \( m\angle 3 = 120^\circ \):
- Angles \( 1 \) and \( 3 \) are supplementary (they form a linear pair), so:
\[
m\angle 1 + m\angle 3 = 180^\circ \implies m\angle 1 + 120^\circ = 180^\circ \implies m\angle 1 = 60^\circ
\]
- Therefore, **\( m\angle 1 = 60^\circ \)** is correct.
2. **At the intersection of lines \( e \) and \( d \)**, we have angles \( 9, 10, 12, 11 \). Since \( m\angle 12 = 80^\circ \):
- Angles \( 12 \) and \( 10 \) are alternate interior angles (parallel lines), so:
\[
m\angle 10 = m\angle 12 = 80^\circ
\]
- Therefore, **\( m\angle 10 = 80^\circ \)** is correct.
3. **At the intersection of lines \( f \) and \( c \)**, we have angles \( 5, 6, 8, 7 \). Angle \( 6 \) is corresponding to angle \( 10 \) (both are in the same position with respect to the transversal), thus:
- \[
m\angle 6 = m\angle 10 = 80^\circ
\]
- Therefore, **\( m\angle 6 = 80^\circ \)** is correct.
4. At the intersection of lines \( f \) and \( d \) with angles \( 13, 14, 16, 15 \):
- Angle \( 14 \) is alternate interior to angle \( 10 \), thus:
\[
m\angle 14 = m\angle 10 = 80^\circ
\]
- Angle \( 16 \) is corresponding to angle \( 3 \):
\[
m\angle 16 = m\angle 3 = 120^\circ
\]
5. Finally, from angle \( 5 \) (referring to \( 6 \)), since \( 5 + 6 = 180^\circ \):
\[
m\angle 5 + m\angle 6 = 180^\circ \Rightarrow m\angle 5 + 80^\circ = 180^\circ \implies m\angle 5 = 100^\circ
\]
Summarizing the findings:
- \( m\angle 1 = 60^\circ \) (correct)
- \( m\angle 13 = 80^\circ \) (uncertain without further information)
- \( m\angle 6 = 80^\circ \) (correct)
- \( m\angle 5 = 100^\circ \) (correct - if we interpret your message it seems this is incorrect; it contradicts the m∠5 measure)
- \( m\angle 10 = 80^\circ \) (incorrect, expectation of \( m(10) = 80^\circ \); hence \( m(12) = 80^\circ \))
- \( m\angle 14 = 80^\circ \) (correct)
So the final correct measurements based on the information given are:
- \( m\angle 1 = 60^\circ \)
- \( m\angle 6 = 80^\circ \)
- \( m\angle 10 = 80 \)
- \( m\angle 14 \)
Note: If we consider 15 and 16 from the statements in your message as confirmations, those results may lead to further deductions.