Question

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.
(A) Vertical angles prove that Angle 2 is congruent to angle 5.
(B) In the two similar triangles, Angle 1 and Angle 4 are alternate interior angles.
(C) Vertical angles prove that Angle 3 is congruent to angle 6.
(D) The triangles are similar because alternate interior angles are congruent.
(E) In the two similar triangles, Angle 2 and Angle 4 are corresponding angles.
(F) The triangles are similar because corresponding sides are congruent.

Answer 1

(B) In the two similar triangles, Angle 1 and Angle 4 are alternate interior angles.

(C) Vertical angles prove that Angle 3 is congruent to Angle 6.

(D) The triangles are similar because alternate interior angles are congruent.

Explanation:

Alternate Interior Angles Theorem

If a line intersects a set of parallel lines in the same plane at two distinct points, the alternate interior angles that are formed are congruent.

As line q has intersected the set of parallel lines c and d, angles 1 and 4 are alternate interior angles, and are therefore congruent.

As line p has intersected the set of parallel lines c and d, angles 2 and 5 are alternate interior angles, and are therefore congruent.

Angle-Angle similarity states that if any two angles of a triangle are equal to any two angles of another triangle, then the two triangles are similar to each other. Therefore, the triangles are similar because alternate interior angles are congruent.

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Vertical Angles Theorem

When two straight lines intersect, the opposite vertical angles are congruent.

The intersection of lines p and q created two opposite vertical angles: angle 3 and angle 6 are opposite vertical angles. Therefore:

• m∠3 ≅ m∠6

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Triangle similarity

Two triangles are similar if their corresponding angles are the same size.

As angles 1 and 4 are alternate interior angles and are therefore congruent, they are corresponding angles. Similarly, as angles 2 and 5 are alternate interior angles and are therefore congruent, they are corresponding angles

Therefore this proves that the two triangles are similar and Angle 1 and Angle 4 are corresponding angles.