Question

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
b n || m
c ℓ || k
d ℓ || m

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

a ∠2 ≅ ∠3
b ∠1 ≅ ∠4
c m∠2 + m∠5 = 180
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:

Step1: Recall parallel - line theorems

If corresponding angles are congruent, alternate interior angles are congruent, or same - side interior angles are supplementary, then the lines are parallel.

Step2: Analyze question 1

• A. $\angle2$ and $\angle3$ are vertical angles, not related to parallel - line criteria for $p$ and $q$.
• B. $\angle5$ and $\angle7$ are vertical angles, not related to parallel - line criteria for $p$ and $q$.
• C. $\angle2$ and $\angle4$ are corresponding angles for lines $r$ and $s$. If $\angle2\cong\angle4$, then $r\parallel s$.
• D. $\angle5$ and $\angle6$ are same - side interior angles for lines $r$ and $s$. If $\angle5\cong\angle6$, it does not prove $r\parallel s$. For $r\parallel s$, $\angle5+\angle6 = 180^{\circ}$ would be the correct condition.
• E. $\angle5$ and $\angle7$ are vertical angles, not related to parallel - line criteria for $r$ and $s$.

Step3: Analyze question 2

If $m\angle1=m\angle2$, then $n\parallel m$ because $\angle1$ and $\angle2$ are corresponding angles.

Step4: Analyze question 3

To prove $j\parallel k$, we need same - side interior angles to be supplementary. $\angle2$ and $\angle5$ are same - side interior angles. If $m\angle2 + m\angle5=180$, then $j\parallel k$.

Step5: Analyze question 4

If $\angle1\cong\angle2$, then lines $\ell$ and $m$ are parallel because $\angle1$ and $\angle2$ are corresponding angles.

Step6: Analyze question 5

If $m\angle3 + m\angle4 = 180$, then lines $\ell$ and $m$ are parallel because $\angle3$ and $\angle4$ are same - side interior angles.

Answer:

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