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Identify the Relationship Between ∠2 and ∠8 in Parallel Lines Cut by a Transversal
Mathematics (Geometry)
Grade 9 (Junior High School)
Question Content
∠2 and ∠8 are ________ because they are ________. (Given a diagram with two parallel lines j and k cut by a transversal l, with angles numbered 1-8 as shown.) Choose from the options: supplementary/alternate interior angles; supplementary/consecutive angles; congruent/consecutive angles; congruent/alternate interior angles
Correct Answer
congruent/alternate interior angles
Detailed Solution Steps
1
Step 1: Identify the type of angles ∠2 and ∠8 are. ∠2 lies between parallel lines j and k, on the right side of transversal l; ∠8 also lies between the two parallel lines, on the left side of the transversal. This matches the definition of alternate interior angles: angles that lie between two parallel lines, on opposite sides of the transversal.
2
Step 2: Apply the property of alternate interior angles. A key theorem states that when two parallel lines are cut by a transversal, alternate interior angles are congruent (have equal measure).
3
Step 3: Match with the given options. Only the option "congruent/alternate interior angles" aligns with the angle type and their relationship.
Knowledge Points Involved
1
Alternate Interior Angles
Alternate interior angles are a pair of angles formed when a transversal intersects two parallel (or non-parallel) lines. They lie between the two lines, on opposite sides of the transversal, and are not adjacent. When the two lines are parallel, these angles are congruent.
2
Congruent Angles
Congruent angles are angles that have exactly the same measure (in degrees or radians). In the context of parallel lines cut by a transversal, specific angle pairs (like alternate interior, alternate exterior, corresponding angles) are congruent.
3
Parallel Lines and Transversals Theorems
When a transversal intersects two parallel lines, several angle relationships hold: alternate interior angles are congruent, alternate exterior angles are congruent, corresponding angles are congruent, and consecutive interior angles are supplementary. These theorems are used to prove angle measures and line parallelism.
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