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Line d is parallel to line c in the figure below.

Parallel lines d and c are intersected by lines q and p to form 2 triangles. At lines d and p, the angle is 2, at d and q, the angle is 1, and at q and p the angle is 3. At lines c and q, the angle is 4, at p and c, the angle is 5, and the third angle is 6.

Which statements about the figure are true? Select three options.
Vertical angles prove that Angle 2 is congruent to angle 5.
In the two similar triangles, Angle 1 and Angle 4 are alternate interior angles.
Vertical angles prove that Angle 3 is congruent to angle 6.
The triangles are similar because alternate interior angles are congruent.
In the two similar triangles, Angle 2 and Angle 4 are corresponding angles.
The triangles are similar because corresponding sides are congruent.

1 answer

To analyze the given statements about the figure with parallel lines d and c intersected by lines p and q, we can use the properties of angles formed by parallel lines and transversals:

  1. Vertical angles are congruent.
  2. Alternate interior angles are congruent when a transversal intersects two parallel lines.
  3. Corresponding angles are congruent as well.

Now, let's evaluate each statement:

  1. Vertical angles prove that Angle 2 is congruent to angle 5.

    • False. Angle 2 and Angle 5 are not vertical angles.

  2. In the two similar triangles, Angle 1 and Angle 4 are alternate interior angles.

    • True. Angle 1 and Angle 4 are formed by line q (transversal) intersecting lines c and d (which are parallel), making them alternate interior angles that are congruent.

  3. Vertical angles prove that Angle 3 is congruent to angle 6.

    • True. Angle 3 and Angle 6 are vertical angles, thus they are congruent.

  4. The triangles are similar because alternate interior angles are congruent.

    • True. Since one pair of alternate interior angles is congruent (Angle 1 and Angle 4), and we also have other pairs of angles congruent, the triangles are similar by the Angle-Angle (AA) similarity criterion.

  5. In the two similar triangles, Angle 2 and Angle 4 are corresponding angles.

    • True, but it's worth noting that Angle 2 and Angle 4 are not directly corresponding angles since they do not lie on the same sides of the transversal. However, they are part of two triangles, where congruent angles suggest similarity of triangles.

  6. The triangles are similar because corresponding sides are congruent.

    • False. While the triangles may be similar, the statement cannot be deemed true without affirming that sides are given. The similarity is established through angles rather than sides in this context.

Based on this analysis, the three true statements you should select are:

  • In the two similar triangles, Angle 1 and Angle 4 are alternate interior angles.
  • Vertical angles prove that Angle 3 is congruent to angle 6.
  • The triangles are similar because alternate interior angles are congruent.

These statements reflect the properties of the various angles formed by the interaction of parallel and transversal lines in the figure.