The missing reason in Step 3 is that alternate interior angles are congruent. This property is crucial in proving that the sum of angles ∠5, ∠2, and ∠6 in triangle ABC is 180°, given the parallel lines y and z.
In the given geometric proof, we aim to show that the sum of angles ∠5, ∠2, and ∠6 in triangle ABC is equal to 180°. The proof involves the fact that lines y and z are parallel. The missing reason in Step 3 is that alternate interior angles are congruent.
When two parallel lines are intersected by a transversal, alternate interior angles are formed on opposite sides of the transversal and are congruent. In this case, angles ∠1 and ∠5, as well as angles ∠3 and ∠6, are alternate interior angles. Therefore, ∠1 is congruent to ∠5 and ∠3 is congruent to ∠6.
The subsequent steps use the congruence of these angles to establish that the sum of angles ∠5, ∠2, and ∠6 is equal to 180°. This is achieved through the angle addition postulate and the definition of a straight angle.
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