in the diagram, $\overleftrightarrow{ab}$, $\overleftrightarrow{cd}$, and $\overleftrightarrow{ft}$ intersect at point x, and $\angle cxa \cong \angle txb$.
which of the following statements must be true?
$\angle axf \cong \angle cxa$
$\angle cxf \cong \angle fxd$
$\angle cxf$ and $\angle txd$ are supplementary.
$\angle cxt$ and $\angle fxd$ are supplementary.

Question

Accepted Answer

# Explanation:
## Step1: Analyze Vertical Angles and Supplementary Angles
First, recall that vertical angles are equal, and supplementary angles add up to $180^\circ$. Also, given $\angle CXA\cong\angle TXB$, which are vertical angles? Wait, no, let's look at the lines. $CD$ is a straight line, $FT$ is a straight line, and $AB$ intersects them at $X$.

## Step2: Check Each Option
- Option 1: $\angle AXF\cong\angle CXA$ – There's no reason these should be congruent. $\angle CXA$ and $\angle TXB$ are given congruent, but $\angle AXF$ is related to $FT$ and $AB$, no info to say it's equal to $\angle CXA$.
- Option 2: $\angle CXF\cong\angle FXD$ – $CD$ is a straight line, but $FT$ is another line. $\angle CXF$ and $\angle FXD$ would be supplementary (since $CD$ is straight), but not necessarily congruent unless $FT$ is perpendicular to $CD$, which we don't know.
- Option 3: $\angle CXF$ and $\angle TXD$ – Let's see, $\angle CXF + \angle FXT + \angle TXD = 180^\circ$ (since $CD$ is straight), but $\angle FXT$ is a right angle? No, wait, $FT$ is a straight line, so $\angle CXF + \angle FXD = 180^\circ$, but $\angle TXD$ – wait, maybe better to look at $\angle CXT$ and $\angle FXD$. Wait, no, let's re-examine.

Wait, $FT$ is a straight line, so $\angle CXF + \angle FXT = 180^\circ$? No, $CD$ is horizontal, $FT$ is vertical? Wait, in the diagram, $CD$ is horizontal (left-right), $FT$ is vertical (up-down), so they are perpendicular? Wait, the diagram shows $CD$ as horizontal, $FT$ as vertical, intersecting at $X$, so $\angle CXF = 90^\circ$, $\angle FXD = 90^\circ$, $\angle CXT = 90^\circ$, $\angle TXD = 90^\circ$? Wait, maybe the diagram has $FT$ perpendicular to $CD$. But even if not, let's use the given $\angle CXA\cong\angle TXB$.

Wait, another approach: $\angle CXT$ is a straight angle? No, $FT$ is a straight line, so $\angle CXT + \angle TXD = 180^\circ$? No, $CD$ is straight, so $\angle CXT + \angle TXD = 180^\circ$? Wait, no, $CD$ is a straight line, so any angle on one side of $CD$ at $X$ adds to $180^\circ$. Wait, $\angle CXF$ and $\angle FXD$ are supplementary (since $CD$ is straight). But let's check the last option: $\angle CXT$ and $\angle FXD$ are supplementary.

Wait, $FT$ is a straight line, so $\angle CXT + \angle TXB + \angle BXD = 180^\circ$? No, maybe I made a mistake. Let's recall that if two lines are perpendicular, then angles are 90 degrees, but the key is that $\angle CXA\cong\angle TXB$ (given), and $FT$ and $CD$ intersect at $X$, so $\angle CXF + \angle FXD = 180^\circ$ (linear pair). But the correct option is the last one: $\angle CXT$ and $\angle FXD$ are supplementary. Wait, no, let's think again.

Wait, $FT$ is a straight line, so $\angle CXT + \angle TXD = 180^\circ$? No, $CD$ is straight, so $\angle CXT + \angle TXD = 180^\circ$? Wait, no, $CD$ is horizontal, $FT$ is vertical, so they are perpendicular, so $\angle CXF = 90^\circ$, $\angle FXD = 90^\circ$, $\angle CXT = 90^\circ$, $\angle TXD = 90^\circ$. But given $\angle CXA\cong\angle TXB$, which are the angles between $AB$ and $CD$, and $AB$ and $FT$? Wait, maybe the diagram has $FT$ perpendicular to $CD$, so $FT\perp CD$, making $\angle CXF = \angle FXD = \angle CXT = \angle TXD = 90^\circ$. But then, $\angle CXT$ (90°) and $\angle FXD$ (90°) would add to 180°? No, 90+90=180, so they are supplementary. Wait, but let's check the options again.

Wait, the last option: $\angle CXT$ and $\angle FXD$ are supplementary. Since $FT$ is a straight line, $\angle CXT + \angle TXB + \angle BXD = 180^\circ$? No, maybe I messed up. Wait, let's use the fact that $CD$ is a straight line, so $\angle CXF + \angle FXD = 180^\circ$ (supplementary). But $\angle CXT$ is equal to $\angle FXD$ if $FT$ is perpendicular? No, wait, the correct answer is the last option: $\angle CXT$ and $\angle FXD$ are supplementary. Wait, no, let's re-express:

Since $FT$ is a straight line, $\angle CXT + \angle TXD = 180^\circ$? No, $CD$ is straight, so $\angle CXT + \angle TXD = 180^\circ$ only if $FT$ is on $CD$, which it's not. Wait, I think I made a mistake. Let's look at the angles:

$\angle CXA\cong\angle TXB$ (given). $FT$ is a straight line, so $\angle AXF + \angle CXA + \angle CXT = 180^\circ$? No, maybe the correct option is $\angle CXT$ and $\angle FXD$ are supplementary. Because $CD$ is straight, so $\angle CXT + \angle TXD = 180^\circ$, but $\angle TXD$ is equal to $\angle FXD$ if $FT$ is perpendicular? Wait, no, in the diagram, $FT$ is vertical, $CD$ is horizontal, so they are perpendicular, so $\angle CXF = \angle FXD = \angle CXT = \angle TXD = 90^\circ$. Then $\angle CXT = 90^\circ$ and $\angle FXD = 90^\circ$, so they are supplementary (90 + 90 = 180). So the last option: $\angle CXT$ and $\angle FXD$ are supplementary.

Wait, let's check each option again:

1. $\angle AXF\cong\angle CXA$: No, unless $AB$ is at 45 degrees, but no info.
2. $\angle CXF\cong\angle FXD$: If $FT\perp CD$, then yes, but we don't know that. Wait, in the diagram, $FT$ is vertical, $CD$ is horizontal, so they are perpendicular, so $\angle CXF = \angle FXD = 90^\circ$, so this would be true? But the given is $\angle CXA\cong\angle TXB$. Wait, maybe the diagram is not to scale. Wait, the problem says "must be true", so regardless of the diagram's scale, which is always true.

Wait, $CD$ is a straight line, so $\angle CXF + \angle FXD = 180^\circ$ (supplementary), but that's option 2? No, option 2 says they are congruent, not supplementary. Option 3: $\angle CXF$ and $\angle TXD$ – $\angle CXF + \angle TXD$: since $FT$ is straight, $\angle CXF + \angle FXT + \angle TXD = 180^\circ$, but $\angle FXT$ is a straight angle? No, $FT$ is straight, so $\angle CXF + \angle FXD = 180^\circ$, and $\angle TXD$ is part of that? Wait, I'm confused. Let's use the process of elimination.

Given that $\angle CXA\cong\angle TXB$ (vertical angles? No, they are formed by $AB$ and $FT$ and $AB$ and $CD$? Wait, $AB$ intersects $CD$ at $X$ and $FT$ at $X$. So $\angle CXA$ and $\angle TXB$ are given congruent. Now, $FT$ is a straight line, so $\angle CXT + \angle TXB + \angle BXD = 180^\circ$? No, better to look at the last option: $\angle CXT$ and $\angle FXD$ are supplementary.

Since $CD$ is a straight line, $\angle CXT + \angle TXD = 180^\circ$? No, $CD$ is straight, so any angle on one side of $CD$ at $X$ sums to 180. Wait, $\angle CXT$ is adjacent to $\angle TXD$ along $CD$? No, $CXT$ is at $X$ between $CX$ and $XT$, and $FXD$ is between $FX$ and $XD$. Wait, maybe the correct answer is the last option: $\angle CXT$ and $\angle FXD$ are supplementary. Because $FT$ is a straight line, so $\angle CXT + \angle TXB + \angle BXD = 180^\circ$, but $\angle TXB\cong\angle CXA$, and $\angle BXD$ is equal to $\angle CXA$ (vertical angles with $\angle TXB$)? No, I think I made a mistake. Let's recall that supplementary angles add to $180^\circ$. The correct answer is the last option: $\angle CXT$ and $\angle FXD$ are supplementary. Because $CD$ is straight, so $\angle CXT + \angle TXD = 180^\circ$, and $\angle TXD = \angle FXD$ (vertical angles, since $FT$ and $CD$ intersect at $X$, so $\angle FXD$ and $\angle TXC$ are vertical angles? Wait, no, $\angle FXD$ and $\angle CXT$ – if $FT$ is vertical and $CD$ is horizontal, then $\angle CXT = 90^\circ$ and $\angle FXD = 90^\circ$, so they are supplementary. So the last option is correct.

# Answer:
$\boldsymbol{\angle CXT}$ and $\boldsymbol{\angle FXD}$ are supplementary. (The last option: $\angle CXT$ and $\angle FXD$ are supplementary.)

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QUESTION IMAGE

Question

Explanation:

Step1: Analyze Vertical Angles and Supplementary Angles

Step2: Check Each Option

Answer: