Question
Which diagram represents the hypothesis of the converse of corresponding angles theorem?
Parallel lines m and n are cut by a transversal. The angles at the top right of both intersections are congruent.
Parallel lines p and q are cut by a transversal. The angles at the top left of the intersection with the first line and the bottom right of the intersection with the second line are congruent.
Parallel lines x and y are cut by a transversal. The angles at the bottom left of the intersection with the first line and the top right of the intersection with the second line are congruent.
Parallel lines m and n are cut by a transversal. The angles at the top right of both intersections are congruent.
Parallel lines p and q are cut by a transversal. The angles at the top left of the intersection with the first line and the bottom right of the intersection with the second line are congruent.
Parallel lines x and y are cut by a transversal. The angles at the bottom left of the intersection with the first line and the top right of the intersection with the second line are congruent.
Answers
Answered by
GPT-4o mini
The converse of the Corresponding Angles Theorem states that if two lines (in this case, parallel lines) are cut by a transversal and the corresponding angles are equal, then the lines are parallel.
To find the diagram that represents the hypothesis of this theorem:
1. The first diagram describes parallel lines m and n with congruent angles at the top right of both intersections. This directly implies that the angles are corresponding angles.
2. The second diagram describes parallel lines p and q, but mentions angles that are not at corresponding positions (top left and bottom right). This does not satisfy the condition of the converse of the Corresponding Angles Theorem.
3. The third diagram describes parallel lines x and y with congruent angles at the bottom left of one intersection and the top right of the other. These angles also represent corresponding angles at their respective intersections, suggesting that the lines are parallel.
Given that we want the hypothesis where corresponding angles are congruent, the first and third diagrams fit the hypothesis, since they involve angles at corresponding positions. However, only the first scenario describes the overall setup properly for the hypothesis to lead to the conclusion about the lines being parallel directly based on angles in corresponding positions.
Therefore, the first diagram represents the hypothesis of the converse of the corresponding angles theorem.
To find the diagram that represents the hypothesis of this theorem:
1. The first diagram describes parallel lines m and n with congruent angles at the top right of both intersections. This directly implies that the angles are corresponding angles.
2. The second diagram describes parallel lines p and q, but mentions angles that are not at corresponding positions (top left and bottom right). This does not satisfy the condition of the converse of the Corresponding Angles Theorem.
3. The third diagram describes parallel lines x and y with congruent angles at the bottom left of one intersection and the top right of the other. These angles also represent corresponding angles at their respective intersections, suggesting that the lines are parallel.
Given that we want the hypothesis where corresponding angles are congruent, the first and third diagrams fit the hypothesis, since they involve angles at corresponding positions. However, only the first scenario describes the overall setup properly for the hypothesis to lead to the conclusion about the lines being parallel directly based on angles in corresponding positions.
Therefore, the first diagram represents the hypothesis of the converse of the corresponding angles theorem.
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