Question

Use the figure to answer the question that follows:

Segments UV and WZ are parallel with line ST intersecting both at points Q and R respectively

When written in the correct order, the two-column proof below describes the statements and reasons for proving that corresponding angles are congruent:


Statements Reasons
segment UV is parallel to segment WZ Given
Points S, Q, R, and T all lie on the same line. Given
m∠SQT = 180° Definition of a Straight Angle
m∠SQV + m∠VQT = m∠SQT Angle Addition Postulate
m∠SQV + m∠VQT = 180° Substitution Property of Equality
I m∠SQV + m∠VQT − m∠VQT = m∠VQT + m∠ZRS − m∠VQT
m∠SQV = m∠ZRS Subtraction Property of Equality
II m∠SQV + m∠VQT = m∠VQT + m∠ZRS Substitution Property of Equality
III m∠VQT + m∠ZRS = 180° Same-Side Interior Angles Theorem
∠SQV ≅ ∠ZRS Definition of Congruency

Answer 1

The order of the start of the proof seems fine; we're to choose the next steps I guess.

segment UV is parallel to segment WZ Given

Points S, Q, R, and T all lie on the same line. Given

m∠SQT = 180° Definition of a Straight Angle

m∠SQV + m∠VQT = m∠SQT Angle Addition Postulate

m∠SQV + m∠VQT = 180° Substitution Property of Equality

That's all valid up to here. It seems to me sort of the hard way to get to linear supplements but here we are.

ZRS is mentioned in the rest of the lines; let's find the one that comes first.

III m∠VQT + m∠ZRS = 180° Same-Side Interior Angles Theorem

Now we have two things equal to 180 degrees, so they're equal to each other.

II m∠SQV + m∠VQT = m∠VQT + m∠ZRS Substitution Property of Equality

Now comes

I m∠SQV + m∠VQT − m∠VQT = m∠VQT + m∠ZRS − m∠VQT

m∠SQV = m∠ZRS Subtraction Property of Equality

And we conclude,

∠SQV ≅ ∠ZRS Definition of Congruency

Answer 2

The corresponding angles formed by two parallel lines, having the same  transversal are congruent

The correct order of the statements are;

• III m∠VQT + m∠ZRS = 180°

• II m∠SQV + m∠VQT = m∠VQT + m∠ZRS

• I m∠SQV + m∠VQT - m∠VQT = m∠VQT  m∠ZRS - m∠VQT

The reason the above statements are in the correct order is as follows:

The two column proof is presented as follows;

Statements ${}$                                                          Reason

Segments UV is parallel to segment WZ ${}$           Given

Points S, Q, R, and T all lie on the same line ${}$      Given

m∠SQT = 180° ${}$                                                       Definition of a straight angle

m∠SQV + m∠VQT = m∠SQT  ${}$                               Angle Addition Postulate

m∠SQV + m∠VQT = 180° ${}$                            Substitution Property of Equality

III m∠VQT + m∠ZRS = 180°  ${}$                        Same-side interior angle theorem

II m∠SQV + m∠VQT = m∠VQT + m∠ZRS  ${}$  Substitution Property of Equality

I m∠SQV + m∠VQT - m∠VQT = m∠VQT

+ m∠ZRS - m∠VQT  ${}$                                     Subtraction Property of Equality

m∠SQV = m∠ZRS ${}$                                      

m∠SQV ≅ m∠ZRS   ${}$                                     Definition of Congruency

Therefore, the correct order is as follows;

First

• III m∠VQT + m∠ZRS = 180°; Given that m∠VQT and  m∠ZRS are same side interior angles

Second

• II m∠SQV + m∠VQT = m∠VQT + m∠ZRS;  ${}$  By Substitution Property of Equality given that both (m∠SQV + m∠VQT), (m∠VQT + m∠ZRS) are equal to 180°

Third

• I m∠SQV + m∠VQT - m∠VQT = m∠VQT  m∠ZRS - m∠VQT; The same value - m∠VQT is subtracted from both sides of the equation in Step II, therefore, both sides of the equation remains equal

Learn more about the angles formed by two parallel lines here:

https://brainly.com/question/13702600