Prove: ∠1 and ∠6 are supplementary.
Proof:
Since parallel lines make corresponding angles congruent, ∠1 ≅ ∠5.
By the definition of congruence, m∠1 = m∠5.
By the defintion of a linear pair, ∠5 and ∠6 are a linear pair and are supplementary.
By the definition of supplementary angles, m∠5 + m∠6 = 180°.
Applying the symmetric property, m∠1 + m∠6 = 180°.
So, by the definition of supplementary angles, ∠1 and ∠6 are supplementary.
In which step was the error made and how should the error be corrected?
A.
The error is in step 5, where the equation m∠1 + m∠6 = 180° should be obtained by the substitution property.
B.
The error is in step 4, where the equation m∠5 + m∠6 = 180° should be obtained by the angle addition postulate.
C.
The error is in step 1, where ∠1 ≅ ∠5 should be true because parallel lines make alternate interior angles congruent.
D.
The error is in step 3, where ∠5 and ∠6 are vertical angles and should be supplementary because of the definition of vertical angles.
B. The error is in step 4, where the equation m∠5 + m∠6 = 180° should be obtained by the angle addition postulate. The correction is to replace step 4 with the following:
By the angle addition postulate, m∠1 + m∠5 = m∠5 + m∠6.
By subtracting m∠5 from both sides, we have m∠1 = m∠6.
So, by the definition of supplementary angles, ∠1 and ∠6 are supplementary.
The error is in option C. The statement "∠1 ≅ ∠5 should be true because parallel lines make alternate interior angles congruent" is incorrect. Parallel lines make alternate interior angles congruent, not corresponding angles. To correct the error, we should say "Since parallel lines make alternate interior angles congruent, ∠1 ≅ ∠5" in step 3.
The error is in step 4, where the equation m∠5 + m∠6 = 180° should be obtained by the angle addition postulate.
To correct the error, we can use the definition of a linear pair to show that ∠5 and ∠6 are supplementary. The angle addition postulate states that if ∠A, ∠B, and ∠C are adjacent angles, then ∠A + ∠B = ∠C. In this case, we have ∠5 and ∠6 as adjacent angles, so we can use the angle addition postulate to rewrite the equation as m∠5 + m∠6 = m∠6.
By the definition of supplementary angles, we know that two angles are supplementary if their measures add up to 180°. Thus, we can rewrite the equation as m∠5 + m∠6 = 180°, which correctly shows that ∠5 and ∠6 are supplementary angles.
Therefore, the corrected statement would be:
By the angle addition postulate, m∠5 + m∠6 = 180°, and by the definition of supplementary angles, ∠5 and ∠6 are supplementary.