Question

in the diagram below, lines m and n are crossed by line t. which set of angles below creates an alternate interior - angle pair? (1) ∠1 and ∠2 (2) ∠2 and ∠4 (3) ∠3 and ∠4 (4) ∠1 and ∠3 12. in a flat plane, given line m and point p not on line m, which of the following best describes the number of lines that can be drawn through point p that are parallel to line m? (1) an infinite number (2) exactly two (3) exactly one (4) at least one 13. given that line m is parallel to line n, which of the following is the correct value of x? (1) 15 (2) 18 (3) 26 (4) 31 14. the lines (overline{ab}) and (overline{cd}) pass through the points (a(-3,7)), (b(9,3)), (c(3, - 2)), and (d(7,6)). at which of the following points do the two lines intersect? (1) ((3,7)) (2) ((-1,5)) (3) ((8,-2)) (4) ((6,4)) 15. which of the following statements is false? (1) every angle has a unique line that bisects it. (2) every segment has a unique line that bisects it. (3) every linear - angle pair is also a supplementary - angle pair. (4) through any point not on a line, there is only one line that can be drawn through the point parallel to the line.

Explanation:

Step1: Recall alternate - interior angles definition

Alternate - interior angles are between the two lines and on opposite sides of the transversal. In the given diagram, $\angle3$ and $\angle4$ are alternate - interior angles.

Step2: Recall the parallel - line postulate

In a plane, given a line $m$ and a point $P$ not on line $m$, exactly one line can be drawn through point $P$ that is parallel to line $m$.

Step3: Use the property of corresponding angles

Since line $m$ is parallel to line $n$, the corresponding angles are equal. The angle $4x - 2$ and the $58^{\circ}$ angle are corresponding angles. So, $4x-2 = 58$. Solving for $x$:
$4x=58 + 2=60$, then $x = 15$.

Step4: Find the equations of the lines

The slope of line $\overline{AB}$ with $A(-3,7)$ and $B(9,3)$ is $m_{AB}=\frac{3 - 7}{9+3}=\frac{-4}{12}=-\frac{1}{3}$. Using the point - slope form $y - y_1=m(x - x_1)$ with point $A(-3,7)$, the equation of line $\overline{AB}$ is $y - 7=-\frac{1}{3}(x + 3)$, which simplifies to $y=-\frac{1}{3}x+6$.
The slope of line $\overline{CD}$ with $C(3,-2)$ and $D(7,6)$ is $m_{CD}=\frac{6 + 2}{7 - 3}=\frac{8}{4}=2$. Using the point - slope form with point $C(3,-2)$, the equation of line $\overline{CD}$ is $y+2 = 2(x - 3)$, which simplifies to $y=2x-8$.
To find the intersection point, set $-\frac{1}{3}x+6=2x - 8$.
Multiply through by 3 to get $-x + 18=6x-24$.
Combining like terms gives $7x=42$, so $x = 6$. Substitute $x = 6$ into $y=2x-8$ gives $y=2\times6-8 = 4$. The intersection point is $(6,4)$.

Step5: Analyze each statement

(1) Every angle has a unique line that bisects it. This is true.
(2) Every segment has infinitely many lines that bisect it (any line passing through the mid - point of the segment), not a unique line. This statement is false.
(3) Every linear angle pair is supplementary. This is true.
(4) Through any point not on a line, there is only one line that can be drawn through the point parallel to the line. This is true.

Answer:

1. (3)
2. (3)
3. (1)
4. (4)
5. (2)