Question

4 5 6 7 c b z statements reasons 1. abc is a triangle 1. given 2. ( y parallel z ) 2. given 3. ( angle 1 cong angle 5 ); ( angle 3 cong angle 6 ) 3. ? 4. ( mangle 1 = mangle 5 ); ( mangle 3 = mangle 6 ) 4. def. ( cong ) 5. ( mangle 1 + mangle 2 + mangle 3 = mangle lam ) 5. ( angle ) addition postulate 6. ( mangle 1 + mangle 2 + mangle 3 = 180^circ ) 6. def. straight angle 7. ( mangle 5 + mangle 2 + mangle 6 = 180^circ ) 7. substitution which could be the missing reason in step 3? (\bigcirc) alternate interior angles are congruent (\bigcirc) alternate exterior angles are congruent (\bigcirc) vertical angles are congruent (\bigcirc) corresponding angles are congruent

Explanation:

To determine the missing reason in Step 3, we analyze the given information and the relationships between the angles. We know that \( y \parallel z \) (given in Step 2).

• Option 1: Alternate interior angles are congruent – Alternate interior angles are formed when a transversal crosses two parallel lines, and they lie between the two lines. However, \( \angle 1 \) and \( \angle 5 \), \( \angle 3 \) and \( \angle 6 \) are not between the parallel lines \( y \) and \( z \), so this is incorrect.
• Option 2: Alternate exterior angles are congruent – Alternate exterior angles lie outside the two parallel lines and on opposite sides of the transversal. \( \angle 1 \) and \( \angle 5 \), \( \angle 3 \) and \( \angle 6 \) do not fit the description of alternate exterior angles, so this is incorrect.
• Option 3: Vertical angles are congruent – Vertical angles are opposite angles formed by intersecting lines. \( \angle 1 \) and \( \angle 5 \), \( \angle 3 \) and \( \angle 6 \) are not vertical angles (vertical angles would be, e.g., \( \angle 4 \) and \( \angle 5 \) if they were formed by intersecting lines, but here \( y \parallel z \) is the key), so this is incorrect.
• Option 4: Corresponding angles are congruent – Corresponding angles are in the same position relative to the parallel lines and the transversal. Since \( y \parallel z \), when a transversal (e.g., the sides of the triangle) intersects them, corresponding angles are congruent. \( \angle 1 \) and \( \angle 5 \), \( \angle 3 \) and \( \angle 6 \) are corresponding angles, so this reason applies.

Answer:

corresponding angles are congruent