Using the alternate interior angles congruence property, the proof demonstrates that ∠5 + ∠2 + ∠6 in triangle ABC equals 180°, supporting the given parallel lines.
The proof involves establishing the relationship between the angles in triangle ABC and the parallel lines y and z. We are given that ABC is a triangle and lines y and z are parallel. The goal is to prove that the sum of angles ∠5, ∠2, and ∠6 is equal to 180°.
To justify the statement in Step 3, we can use the property that when two parallel lines are intersected by a transversal, alternate interior angles are congruent. This is the missing reason in Step 3. Therefore, ∠1 is congruent to ∠5, and ∠3 is congruent to ∠6.
Steps 4 and 5 use the definition of congruence and the angle addition postulate to establish the equality of angle measures. Finally, in Step 6, the definition of a straight angle is invoked, stating that the sum of angles in a straight line is 180°.
Step 7 then uses substitution to express the sum of angles ∠5, ∠2, and ∠6 as 180°. This completes the proof, demonstrating that the given angles in triangle ABC satisfy the condition.
In summary, by establishing that alternate interior angles are congruent, the proof shows that the sum of angles ∠5, ∠2, and ∠6 in triangle ABC is indeed 180°.
The question probable may be:
Given: Lines y and z are parallel, and ABC forms a triangle.
Prove: m∠5 + m∠2 + m∠6 = 180°
Statements
Reasons
1. ABC is a triangle 1. given
2. y ∥ z 2. given
3. ∠1 ≅ ∠5; ∠3 ≅ ∠6 3. ?
4. m∠1 = m∠5; m∠3 = m∠6 4. def. ≅
5. m∠1 + m∠2 + m∠3 = m∠LAM 5. ∠ addition postulate
6. m∠1 + m∠2 + m∠3 = 180° 6. def. straight angle
7. m∠5 + m∠2 + m∠6 = 180° 7. substitution
Which could be the missing reason in Step 3?
alternate interior angles are congruent
alternate exterior angles are congruent
vertical angles are congruent
corresponding angles are congruent