A chair rail is a type of wall trim that is placed about three feet above the floor to protect the walls. Part of the chair rail below has been replaced because it was damaged. The edges of the replacement piece were angled for a better fit. In the diagram, $\angle 1$ and $\angle 2$ are supplementary, $\angle 3$ and $\angle 4$ are supplementary, and $\angle 2$ and $\angle 3$ each have measures of $50^{\circ}$. Is $\angle 1 \cong \angle 4$ ? Explain.

Some facts about the maximum banking angles of Daytona International Speedway at corners 1, 2, 3, and 4 are at the right. Find $m \angle 3$. Explain your steps.

$$\begin{aligned} & m \angle 1+m \angle 3+m \angle 4=93^{\circ} \\ & m \angle 2+m \angle 4=62^{\circ} \\ & m \angle 2=m \angle 3 \\ & m \angle 1=m \angle 2 \end{aligned}$$

Fill in the blank. Then draw a sketch that helps illustrate your answer. If two planes intersect, then their intersection is ______.

State whether any conclusions can be made using the true statement, given that x=3. If $4 x \geq 12$, then y=6 x.

Solve for each variable. Explain your reasoning. Two lines intersect each other and following are the pairs of vertically opposite angles, (4w+10) $^{\circ}$, (2x-30) $^{\circ}$ 2(x+25) $^{\circ}$ , 13W $^{\circ}$

State whether any conclusions can be made using the true statement, given that x=3. If 2 x+3<4 x<5 x, then $y \leq x$.

Suppose M is the midpoint of $\overline{A B}, P$ is the midpoint of $\overline{A M}$, and Q is the midpoint of $\overline{P M}$. If $a$ and $b$ are the coordinates of points $A$ and $B$ on a number line, find the coordinates of $P$ and $Q$ in terms of $a$ and $b$.

Give a counterexample that demonstrates that the converse of the statement is false. If a polygon is a square, then it has four sides.

Give an argument for the statement, including a reason for each step. If B lies between A and C and AB=3 and BC=8, then AC=11.

Fill in the blank. Then draw a sketch that helps illustrate your answer. If two points lie in a plane, then the ______ containing them lies in the plane.

Give an argument for the statement, including a reason for each step. If $\angle 1$ and $\angle 2$ are right angles, then they are supplementary.

Fill in the blank. Then draw a sketch that helps illustrate your answer. Through any _____ points there exists exactly one line.

Suppose point T is the midpoint of $\overline{R S}$ and point $W$ is the midpoint of $\overline{R T}$. If $\overline{X Y} \cong \overline{R T}$ and $\overline{T S}$ has a length of $z$, write the length of the segment in terms of z. $\overline{W T}$

State whether any conclusions can be made using the true statement, given that x=3. If x>2 x-10, then x=y.