Question

properties of parallel lines

1. select all the true statements.

□ a. ∠3≅∠2 because they are alternate interior angles.
□ b. m∠1 + m∠3 = 180 because they form a straight angle.
□ c. ∠3≅∠6 because they are alternate interior angles.
□ d. ∠1 and ∠6 are supplementary because ∠3≅∠6 and m∠1 + m∠3 = 180.
□ e. ∠1≅∠3 because they are vertical angles.
use the figure shown for items 2 - 4.

1. what is m∠2 + m∠3? explain.

a 180; m∠1 + 90 = 180, so m∠2 + m∠3 = 180.
b 180; m∠1 + 90 = m∠2 + m∠3, so m∠2 + m∠3 = 180.
c 90; m∠1 + m∠2 + m∠3 = 180. m∠1 = 90, so m∠2 + m∠3 = 90.
d 90; ∠2 and ∠3 are corresponding angles, so m∠2 + m∠3 = 90.

1. if m∠4 = 35, find m∠2. explain.

a 55; ∠2 and ∠4 are complementary angles, so m∠2 = 90 - m∠4.
b 35; ∠2 and ∠4 are corresponding angles, so m∠2 = m∠4.
c 145; ∠2 and ∠4 are supplementary angles, so m∠2 = 180 - m∠4.
d 35; ∠2 and ∠4 are alternate interior angles, so m∠2 = m∠4.

1. if m∠4 = 35, find m∠3. explain.

a 55; ∠3 and ∠4 are complementary angles, so m∠3 = 90 - m∠4.
b 35; ∠3 and ∠4 are corresponding angles, so m∠3 = m∠4.
c 145; ∠3 and ∠4 are supplementary angles, so m∠3 = 180 - m∠4.
d 35; ∠3 and ∠4 are alternate interior angles, so m∠3 = m∠4.

1. in the figure shown, t || x and k || w.

if m∠3 = 20, list all the angles that are 20° and all the angles that are 160°.

Explanation:

Step1: Analyze statement A

$\angle3$ and $\angle2$ are not alternate - interior angles. Alternate - interior angles are non - adjacent angles between two parallel lines and on opposite sides of a transversal. So, statement A is false.

Step2: Analyze statement B

$\angle1$ and $\angle3$ form a straight angle. The sum of angles on a straight line is $180^{\circ}$, so $m\angle1 + m\angle3=180$. Statement B is true.

Step3: Analyze statement C

$\angle3$ and $\angle6$ are alternate - interior angles when two parallel lines are cut by a transversal. So, $\angle3\cong\angle6$. Statement C is true.

Step4: Analyze statement D

Since $\angle3\cong\angle6$ and $m\angle1 + m\angle3 = 180$, by substitution, $\angle1$ and $\angle6$ are supplementary. Statement D is true.

Step5: Analyze statement E

$\angle1$ and $\angle3$ are vertical angles. Vertical angles are congruent. So, $\angle1\cong\angle3$. Statement E is true.

For question 2:
In the second figure, $\angle1 = 90^{\circ}$ and $\angle1+\angle2+\angle3 = 180^{\circ}$ (since they are angles on a straight - line). Given $\angle1 = 90^{\circ}$, then $m\angle2 + m\angle3=90^{\circ}$. So the answer is C.

For question 3:
$\angle2$ and $\angle4$ are corresponding angles when two parallel lines are cut by a transversal. Corresponding angles are congruent. If $m\angle4 = 35$, then $m\angle2 = 35$. So the answer is B.

For question 4:
$\angle3$ and $\angle4$ are complementary angles because $\angle1 = 90^{\circ}$ and $\angle1+\angle2+\angle3+\angle4 = 180^{\circ}$, and $\angle2=\angle4$ (corresponding angles). If $m\angle4 = 35$, then $m\angle3=90 - 35=55$. So the answer is A.

For question 5:
If $m\angle3 = 20^{\circ}$, the angles that are $20^{\circ}$ are $\angle1,\angle4,\angle5,\angle8,\angle9,\angle12,\angle13,\angle16$ (corresponding and vertical angles). The angles that are $160^{\circ}$ are $\angle2,\angle6,\angle7,\angle10,\angle11,\angle14,\angle15$ (supplementary to the $20^{\circ}$ angles).

Answer:

1. B, C, D, E
2. C
3. B
4. A
5. Angles of $20^{\circ}$: $\angle1,\angle4,\angle5,\angle8,\angle9,\angle12,\angle13,\angle16$; Angles of $160^{\circ}$: $\angle2,\angle6,\angle7,\angle10,\angle11,\angle14,\angle15$