Question

topic 2 proving lines are parallel using converse of theorems/postulates
name name redacted date 9/12/25 period
score:

1 prove the converse of same - side interior angles theorem

given: m∠1 + m∠2 = 180
prove: l || m
statements reasons

1. 1.
2. ∠1 & ∠3 are... 2.
3. ∠1 & ∠3 are... 3.
4. m∠1 + m∠3 = 180 4.
5. m∠1 + m∠2 = m∠ + m∠ 5.
6. m∠2 = m∠3 6.
7. ∠2 ≅ ∠3 7.
8. l || m 8.

2 complete the proof

given: m∠1 + m∠2 = 180 ; ab || dc
prove: bc || ef
statements reasons
1 ab || dc 1 given
2 ∠1 & ∠4 are... 2
3 3
4 m∠1 + m∠2 = 180 4 given
5 5
6 m∠2 = m∠4 6
7 ∠2 ≅ ∠4 7
8 8

Explanation:

Step1: Identify angle - pair relationship

$\angle1$ and $\angle3$ are linear - pair angles

Step2: State property of linear - pair angles

Linear - pair angles are supplementary, so $m\angle1 + m\angle3=180$ (Definition of supplementary angles)

Step3: Use given and substitution

Given $m\angle1 + m\angle2 = 180$, so $m\angle1 + m\angle2=m\angle1 + m\angle3$ (Substitution property of equality)

Step4: Solve for angle equality

Subtract $m\angle1$ from both sides, we get $m\angle2=m\angle3$ (Subtraction property of equality)

Step5: Convert to congruence

$\angle2\cong\angle3$ (Definition of congruent angles)

Step6: Apply parallel - line theorem

$l\parallel m$ (If alternate interior angles are congruent, then the lines are parallel)

For #2:

Step1: Identify angle - pair relationship

$\angle1$ and $\angle4$ are same - side interior angles (Since $\overline{AB}\parallel\overline{DC}$)

Step2: State property of same - side interior angles

Same - side interior angles are supplementary when lines are parallel, but we don't use this here directly. We know from the given $m\angle1 + m\angle2 = 180$ and $m\angle1 + m\angle4$ would also be related as $\overline{AB}\parallel\overline{DC}$

Step3: Use substitution

Since $m\angle1 + m\angle2 = 180$ and $m\angle1 + m\angle4$ (due to parallel lines $\overline{AB}\parallel\overline{DC}$), we can say $m\angle1 + m\angle2=m\angle1 + m\angle4$ (Substitution based on parallel - line angle relationships)

Step4: Solve for angle equality

Subtract $m\angle1$ from both sides, $m\angle2=m\angle4$ (Subtraction property of equality)

Step5: Convert to congruence

$\angle2\cong\angle4$ (Definition of congruent angles)

Step6: Apply parallel - line theorem

$\overline{BC}\parallel\overline{EF}$ (If corresponding angles are congruent, then the lines are parallel)

Answer:

For #1:

Statements	Reasons
2. $\angle1$ and $\angle3$ are linear - pair angles	2. Definition of linear - pair
3. $\angle1$ and $\angle3$ are supplementary	3. Linear - pair angles are supplementary
4. $m\angle1 + m\angle3 = 180$	4. Definition of supplementary angles
5. $m\angle1 + m\angle2=m\angle1 + m\angle3$	5. Substitution property of equality
6. $m\angle2=m\angle3$	6. Subtraction property of equality
7. $\angle2\cong\angle3$	7. Definition of congruent angles
8. $l\parallel m$	8. If alternate interior angles are congruent, then the lines are parallel

For #2:

Statements	Reasons
2. $\angle1$ and $\angle4$ are same - side interior angles	2. Definition of same - side interior angles for parallel lines $\overline{AB}\parallel\overline{DC}$
3. $m\angle1 + m\angle2 = 180$ and considering angle relationships with $\overline{AB}\parallel\overline{DC}$	3. Given and parallel - line angle relationships
4. $m\angle1 + m\angle2 = 180$	4. Given
5. $m\angle1 + m\angle2=m\angle1 + m\angle4$	5. Substitution based on parallel - line angle relationships
6. $m\angle2=m\angle4$	6. Subtraction property of equality
7. $\angle2\cong\angle4$	7. Definition of congruent angles
8. $\overline{BC}\parallel\overline{EF}$	8. If corresponding angles are congruent, then the lines are parallel