Asked by alex
Line l is parallel to line e in the figure below.
Parallel lines e and l are crossed by lines m and n to form 2 triangles. At the intersection of parallel line e with line n is angle q, and with line m is angle 3. Angle 2 is the third angle. At the intersection of parallel line l and m is 6, at line n is 4. The third angle is 5.
Which statements about the figure are true? Check all that apply.
Vertical angles prove that Angle 1 is congruent to angle 4.
Vertical angles prove that Angle 2 is congruent to angle 5.
The triangles are similar because corresponding sides are congruent.
The triangles are similar because alternate interior angles are congruent.
In the similar triangles, Angle 3 and Angle 6 are alternate interior angles.
In the similar triangles, Angle 3 and Angle 4 are corresponding angles.
All Answers 1
Answered by
GPT-5 mini
AI
Answers (true/false) with brief reasons:
- Vertical angles prove that Angle 1 is congruent to angle 4. — False.
Angle 1 and Angle 4 are formed by the same transversal with two parallel lines (they are corresponding/alternate-pair type), not a vertical-angle pair.
- Vertical angles prove that Angle 2 is congruent to angle 5. — True.
Angles 2 and 5 are opposite angles where lines m and n cross, so they are vertical angles and therefore congruent.
- The triangles are similar because corresponding sides are congruent. — False.
Congruent corresponding sides would give congruence, not a justification for similarity here; no side lengths are given.
- The triangles are similar because alternate interior angles are congruent. — True.
The two parallel lines cut by transversals m and n produce pairs of equal angles (alternate interior/corresponding), yielding two equal angle pairs (AA) and thus similarity.
- In the similar triangles, Angle 3 and Angle 6 are alternate interior angles. — True.
Angle 3 and Angle 6 are the angles made by transversal m with the two parallel lines, so they form a congruent alternate-interior (or corresponding) pair used in the AA similarity.
- In the similar triangles, Angle 3 and Angle 4 are corresponding angles. — False.
Angle 3 corresponds to Angle 6 (the angles at the m–parallel intersections); Angle 4 corresponds to Angle 1 (the angles at the n–parallel intersections).
- Vertical angles prove that Angle 1 is congruent to angle 4. — False.
Angle 1 and Angle 4 are formed by the same transversal with two parallel lines (they are corresponding/alternate-pair type), not a vertical-angle pair.
- Vertical angles prove that Angle 2 is congruent to angle 5. — True.
Angles 2 and 5 are opposite angles where lines m and n cross, so they are vertical angles and therefore congruent.
- The triangles are similar because corresponding sides are congruent. — False.
Congruent corresponding sides would give congruence, not a justification for similarity here; no side lengths are given.
- The triangles are similar because alternate interior angles are congruent. — True.
The two parallel lines cut by transversals m and n produce pairs of equal angles (alternate interior/corresponding), yielding two equal angle pairs (AA) and thus similarity.
- In the similar triangles, Angle 3 and Angle 6 are alternate interior angles. — True.
Angle 3 and Angle 6 are the angles made by transversal m with the two parallel lines, so they form a congruent alternate-interior (or corresponding) pair used in the AA similarity.
- In the similar triangles, Angle 3 and Angle 4 are corresponding angles. — False.
Angle 3 corresponds to Angle 6 (the angles at the m–parallel intersections); Angle 4 corresponds to Angle 1 (the angles at the n–parallel intersections).
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