Question
In each diagram, line p is parallel to line f, and line t intersects lines p and f.
Based on these diagrams, which statement is true?
Responses
A The value of x should be 45, because the angles shown in the diagram are complementary. The value of x should be 45, because the angles shown in the diagram are complementary.
B The value of x should be 45, because the angles shown in the diagram are congruent. The value of x should be 45, because the angles shown in the diagram are congruent.
C The value of x should be 135, because the angles shown in the diagram are congruent. The value of x should be 135, because the angles shown in the diagram are congruent.
D The value of x should be 135, because the angles shown in the diagram are supplementary.
Based on these diagrams, which statement is true?
Responses
A The value of x should be 45, because the angles shown in the diagram are complementary. The value of x should be 45, because the angles shown in the diagram are complementary.
B The value of x should be 45, because the angles shown in the diagram are congruent. The value of x should be 45, because the angles shown in the diagram are congruent.
C The value of x should be 135, because the angles shown in the diagram are congruent. The value of x should be 135, because the angles shown in the diagram are congruent.
D The value of x should be 135, because the angles shown in the diagram are supplementary.
Answers
GPT-4o mini
To determine which statement is true without the diagrams, we can analyze the relationships between angles formed when a transversal intersects two parallel lines.
1. **Complementary Angles**: Two angles are complementary if the sum of their measures is 90 degrees.
2. **Congruent Angles**: Two angles are congruent if they have the same measure.
3. **Supplementary Angles**: Two angles are supplementary if the sum of their measures is 180 degrees.
In the case of a transversal intersecting two parallel lines, several angle relationships can be employed:
- **Corresponding angles** are congruent.
- **Alternate interior angles** are congruent.
- **Same-side interior angles** are supplementary.
Let's evaluate the choices:
- **Option A** mentions complementary angles, suggesting that if \( x = 45 \), it implies that the angles sum up to 90 degrees. This situation might occur, but it’s not a standard case when dealing specifically with parallel lines crossed by a transversal.
- **Option B** states that \( x = 45 \) because the angles are congruent. This would only be correct if \( x \) represents a corresponding or alternate pair of angles, both measuring 45 degrees, which is plausible.
- **Option C** has \( x = 135 \) with a justification of being congruent. This would imply two angles formed by the transversal and parallel lines are equal to 135 degrees, which can happen if they are alternate interior angles.
- **Option D** states that \( x = 135 \) because the angles are supplementary. This would also be likely if representing same-side interior angles which sum up to 180 degrees, where one angle would be \( 45 \) degrees, and the supplementary angle would then be \( 135 \).
Given characteristics of transversal intersections:
- If the angles referenced are indeed congruent, then Option B or C could be true.
- If the angles are supplementary, then Option D could be true.
In most typical diagrams involving parallel lines and a transversal, corresponding angles or alternate interior angles tend to be congruent, leading us most likely to choose:
### **Answer: C** (if it is indeed showing congruent angles equal to 135), or D (if representing supplementary angles to sum to 180).
Without the specific diagrams in front of us, we would lean towards confirming the situation of \( x = 135 \) as congruent if it also accurately identifies supplementary cases which often arise in such scenarios.
1. **Complementary Angles**: Two angles are complementary if the sum of their measures is 90 degrees.
2. **Congruent Angles**: Two angles are congruent if they have the same measure.
3. **Supplementary Angles**: Two angles are supplementary if the sum of their measures is 180 degrees.
In the case of a transversal intersecting two parallel lines, several angle relationships can be employed:
- **Corresponding angles** are congruent.
- **Alternate interior angles** are congruent.
- **Same-side interior angles** are supplementary.
Let's evaluate the choices:
- **Option A** mentions complementary angles, suggesting that if \( x = 45 \), it implies that the angles sum up to 90 degrees. This situation might occur, but it’s not a standard case when dealing specifically with parallel lines crossed by a transversal.
- **Option B** states that \( x = 45 \) because the angles are congruent. This would only be correct if \( x \) represents a corresponding or alternate pair of angles, both measuring 45 degrees, which is plausible.
- **Option C** has \( x = 135 \) with a justification of being congruent. This would imply two angles formed by the transversal and parallel lines are equal to 135 degrees, which can happen if they are alternate interior angles.
- **Option D** states that \( x = 135 \) because the angles are supplementary. This would also be likely if representing same-side interior angles which sum up to 180 degrees, where one angle would be \( 45 \) degrees, and the supplementary angle would then be \( 135 \).
Given characteristics of transversal intersections:
- If the angles referenced are indeed congruent, then Option B or C could be true.
- If the angles are supplementary, then Option D could be true.
In most typical diagrams involving parallel lines and a transversal, corresponding angles or alternate interior angles tend to be congruent, leading us most likely to choose:
### **Answer: C** (if it is indeed showing congruent angles equal to 135), or D (if representing supplementary angles to sum to 180).
Without the specific diagrams in front of us, we would lean towards confirming the situation of \( x = 135 \) as congruent if it also accurately identifies supplementary cases which often arise in such scenarios.