Question

2-2 lesson quiz
proving lines parallel

1. select all the true statements.

□ a. p || q because ∠2 ≅ ∠3.
□ b. p || q because ∠5 ≅ ∠7.
□ c. r || s because ∠2 ≅ ∠4.
□ d. r || s because ∠5 ≅ ∠6.
□ e. r || s because ∠5 ≅ ∠7.

use the figure shown for items 2 and 3.

1. if m∠1 = m∠2, which of the following statements is true?

a. k || j c. ℓ || k
b. n || m d. ℓ || m

1. which statement must be true to prove j || k?

a. ∠2 ≅ ∠3 c. m∠2 + m∠5 = 180
b. ∠1 ≅ ∠4 d. ∠6 ≅ ∠4

use the figure shown for items 4 and 5.

1. if ∠1 ≅ ∠2, can you conclude that any of the lines are parallel? explain.

a. yes; lines n and p are parallel because corresponding angles are congruent.
b. no; ∠1 and ∠2 show no relationship.
c. yes; lines ℓ and m are parallel because corresponding angles are congruent.
d. no; neither angle is formed by the transversal, line q.

1. if m∠3 + m∠4 = 180, which lines can you conclude are parallel? explain.

a. lines n and p are parallel because alternate interior angles are congruent.
b. lines n and p are parallel because same-side interior angles are supplementary.
c. lines ℓ and m are parallel because same-side interior angles are supplementary.
d. lines ℓ and m are parallel because alternate interior angles are congruent.

Explanation:

Item 1

To determine the true statements, we analyze each option using angle - parallel line theorems (corresponding angles, alternate interior angles, same - side interior angles):

• Option A: $\angle2$ and $\angle3$ are alternate interior angles. If alternate interior angles are congruent, then the lines cut by the transversal are parallel. So, if $\angle2\cong\angle3$, then $p\parallel q$. This statement is true.
• Option B: $\angle5$ and $\angle7$ are vertical angles (not related to proving $p\parallel q$). Vertical angles are always congruent, but this does not imply $p\parallel q$. This statement is false.
• Option C: $\angle2$ and $\angle4$ are corresponding angles. If corresponding angles are congruent, then the lines cut by the transversal are parallel. So, if $\angle2\cong\angle4$, then $r\parallel s$. This statement is true.
• Option D: $\angle5$ and $\angle6$ are same - side interior angles. For lines to be parallel, same - side interior angles should be supplementary, not congruent. So this statement is false.
• Option E: $\angle5$ and $\angle7$ are not related to proving $r\parallel s$ in a way that would imply parallelism. This statement is false.

So the true statements are A and C.

Item 2

If $m\angle1 = m\angle2$, $\angle1$ and $\angle2$ are corresponding angles. The lines cut by the transversal that form these corresponding angles are $n$ and $m$. By the corresponding angles postulate, if corresponding angles are congruent, then the lines are parallel. So $n\parallel m$. The answer is B.

Item 3

To prove $j\parallel k$, we analyze each option:

• Option A: $\angle2$ and $\angle3$ are adjacent angles, not related to proving $j\parallel k$.
• Option B: $\angle1$ and $\angle4$ are not related to the parallelism of $j$ and $k$.
• Option C: $\angle2$ and $\angle5$ are same - side interior angles. If $m\angle2 + m\angle5=180^{\circ}$ (same - side interior angles are supplementary), then $j\parallel k$. This statement is true.
• Option D: $\angle6$ and $\angle4$ are not related to proving $j\parallel k$.

The answer is C.

Item 4

$\angle1$ and $\angle2$ are corresponding angles formed by the transversal that cuts lines $\ell$ and $m$. By the corresponding angles postulate, if corresponding angles are congruent, then the lines are parallel. So lines $\ell$ and $m$ are parallel because corresponding angles are congruent. The answer is C.

Item 5

$\angle3$ and $\angle4$ are same - side interior angles formed by the transversal that cuts lines $n$ and $p$. If $m\angle3 + m\angle4 = 180^{\circ}$ (same - side interior angles are supplementary), then lines $n$ and $p$ are parallel. The answer is B.

Final Answers

1. A, C
2. B
3. C
4. C
5. B

Answer:

Item 1

To determine the true statements, we analyze each option using angle - parallel line theorems (corresponding angles, alternate interior angles, same - side interior angles):

• Option A: $\angle2$ and $\angle3$ are alternate interior angles. If alternate interior angles are congruent, then the lines cut by the transversal are parallel. So, if $\angle2\cong\angle3$, then $p\parallel q$. This statement is true.
• Option B: $\angle5$ and $\angle7$ are vertical angles (not related to proving $p\parallel q$). Vertical angles are always congruent, but this does not imply $p\parallel q$. This statement is false.
• Option C: $\angle2$ and $\angle4$ are corresponding angles. If corresponding angles are congruent, then the lines cut by the transversal are parallel. So, if $\angle2\cong\angle4$, then $r\parallel s$. This statement is true.
• Option D: $\angle5$ and $\angle6$ are same - side interior angles. For lines to be parallel, same - side interior angles should be supplementary, not congruent. So this statement is false.
• Option E: $\angle5$ and $\angle7$ are not related to proving $r\parallel s$ in a way that would imply parallelism. This statement is false.

So the true statements are A and C.

Item 2

If $m\angle1 = m\angle2$, $\angle1$ and $\angle2$ are corresponding angles. The lines cut by the transversal that form these corresponding angles are $n$ and $m$. By the corresponding angles postulate, if corresponding angles are congruent, then the lines are parallel. So $n\parallel m$. The answer is B.

Item 3

To prove $j\parallel k$, we analyze each option:

• Option A: $\angle2$ and $\angle3$ are adjacent angles, not related to proving $j\parallel k$.
• Option B: $\angle1$ and $\angle4$ are not related to the parallelism of $j$ and $k$.
• Option C: $\angle2$ and $\angle5$ are same - side interior angles. If $m\angle2 + m\angle5=180^{\circ}$ (same - side interior angles are supplementary), then $j\parallel k$. This statement is true.
• Option D: $\angle6$ and $\angle4$ are not related to proving $j\parallel k$.

The answer is C.

Item 4

$\angle1$ and $\angle2$ are corresponding angles formed by the transversal that cuts lines $\ell$ and $m$. By the corresponding angles postulate, if corresponding angles are congruent, then the lines are parallel. So lines $\ell$ and $m$ are parallel because corresponding angles are congruent. The answer is C.

Item 5

$\angle3$ and $\angle4$ are same - side interior angles formed by the transversal that cuts lines $n$ and $p$. If $m\angle3 + m\angle4 = 180^{\circ}$ (same - side interior angles are supplementary), then lines $n$ and $p$ are parallel. The answer is B.

Final Answers

1. A, C
2. B
3. C
4. C
5. B