Question

1. which statement is true about parallel lines?

a they are coplanar and intersect.
b they are not coplanar and intersect.
c they are coplanar and do not intersect.
d they are not coplanar and do not intersect.
items 2 - 5. lines $ell$ and $m$ are intersected by transversal $t$. $ellparallel m$

1. which of the following angles are supplementary to $angle2$? select all that apply.

a $angle5$ b $angle6$ c $angle7$ d $angle8$

1. which of the following angles are congruent to $angle6$? select all that apply.

a $angle1$ b $angle2$ c $angle3$ d $angle4$

1. by which postulate or theorem is $angle1congangle8$?

a alternate exterior angles theorem
b alternate interior angles theorem
c corresponding angles theorem
d same - side interior angles postulate

1. if $mangle4 = 105$, what is $mangle5$?
2. write two equations relating the measure of $angle4$ to the measures of $angle1$, $angle2$, and $angle3$.

items 7 - 9. a triangle is shown.

1. what is $x$?

a 32 b 62 c 118 d 242

1. what is $y$?
2. which of the following statements are true? select all that apply.

a $x = y$ b $x + y=180$ c $y = z$ d $x + z = 180$

1. in $\triangle def$, $mangle d = 53$ and $mangle f = 68$. what is $mangle e$?

a 15 b 59 c 121 d 239

Explanation:

Step1: Recall definition of parallel lines

Parallel lines are coplanar and do not intersect.

Step2: Analyze supplementary angles

For two parallel lines cut by a transversal, same - side interior angles are supplementary. $\angle2$ and $\angle5$ are same - side interior angles, $\angle2$ and $\angle6$ are also same - side interior angles.

Step3: Analyze congruent angles

Corresponding angles are congruent for parallel lines cut by a transversal. $\angle6$ and $\angle2$ are corresponding angles.

Step4: Identify angle - congruence theorem

$\angle1$ and $\angle8$ are alternate exterior angles. By the Alternate Exterior Angles Theorem, they are congruent.

Step5: Use angle - sum properties

For a triangle, the sum of interior angles is 180 degrees. For the triangle with angles 43 and 75 degrees, $x=180-(43 + 75)=62$.

Step6: Use linear - pair property

$y$ and $x$ form a linear pair, so $y = 180 - x$. Since $x = 62$, $y=118$. Also, $y$ and $z$ are vertical angles, so $y = z$.

Step7: Use triangle angle - sum in $\triangle DEF$

In $\triangle DEF$, $m\angle D+m\angle E+m\angle F=180$. Given $m\angle D = 53$ and $m\angle F=68$, then $m\angle E=180-(53 + 68)=59$.

Answer:

1. C. They are coplanar and do not intersect.
2. A. $\angle5$, B. $\angle6$
3. B. $\angle2$
4. A. Alternate Exterior Angles Theorem
5. No information about $\angle5$'s relation to other angles given in the problem - setup for 5 is incomplete.
6. No diagram information about $\angle1,\angle2,\angle3,\angle4$'s relation is sufficient to write equations.
7. B. 62
8. C. 118
9. C. $y = z$
10. B. 59