Question

question 5
use the image to determine the false statement.
image of angle diagram with points a, b, c, d, e, f
options:

• ∠acd is an obtuse angle
• ∠fce is an acute angle
• ∠ecd and ∠fce only are both acute angles
• ∠acf is a right angle

Explanation:

Step1: Analyze $\angle ACD$

$\angle ACB$ is a straight angle ($180^\circ$), and $\angle ECB$ has a right angle ($90^\circ$) and $\angle ECD = 45^\circ$, so $\angle ACD=180^\circ - \angle DCB$. Since $\angle DCB = 45^\circ$ (as $\angle ECD = 45^\circ$ and $\angle ECB = 90^\circ$? Wait, no, $\angle ECB$ is a right angle? Wait, the diagram shows $\angle ACE$ is a right angle (since $AC$ and $CE$ are perpendicular? Wait, the right angle at $C$ between $AC$ and $CE$? Wait, the diagram has a right angle symbol at $C$ between $AC$ and $CE$? Wait, the points: $A---C---B$ is a straight line, $CE$ is perpendicular to $AC$? Wait, the right angle is between $AC$ and $CE$, so $\angle ACE = 90^\circ$. Then $\angle ECD = 45^\circ$, so $\angle ACD=\angle ACE+\angle ECD = 90^\circ + 45^\circ=135^\circ$, which is obtuse (greater than $90^\circ$, less than $180^\circ$), so $\angle ACD$ is obtuse. So first statement is true.

Step2: Analyze $\angle FCE$

$\angle FCE$: since $\angle ACE = 90^\circ$, and $\angle FCE$ is part of it (between $FC$ and $CE$), and since $\angle ECD = 45^\circ$, and $FC$ is between $AC$ and $CE$ (since the right angle is at $AC$ and $CE$), so $\angle FCE$ is less than $90^\circ$ (because $\angle ACE = 90^\circ$ and $FC$ is inside $\angle ACE$? Wait, no, the right angle is between $AC$ and $CE$, so $\angle ACE = 90^\circ$. Then $FC$ is another ray between $AC$ and $CE$, so $\angle FCE$ is acute (less than $90^\circ$), so second statement is true.

Step3: Analyze "$\angle ECD$ and $\angle FCE$ only are both acute angles"

$\angle ECD = 45^\circ$ (acute), $\angle FCE$ is acute (as above), but also, what about $\angle DCB$? Wait, $\angle DCB$: since $\angle ECB$ is a right angle? Wait, no, $\angle ECB$: $CE$ is perpendicular to $AC$, so $\angle ACE = 90^\circ$, so $\angle ECB = 90^\circ$ (since $AC$ and $CB$ are straight, so $\angle ACB = 180^\circ$, so $\angle ECB = 180^\circ - 90^\circ = 90^\circ$? Wait, no, $\angle ECD = 45^\circ$, so $\angle DCB = \angle ECB - \angle ECD = 90^\circ - 45^\circ = 45^\circ$, which is acute. Wait, so $\angle DCB$ is also acute. So the statement says only $\angle ECD$ and $\angle FCE$ are acute, but $\angle DCB$ is also acute (45 degrees), so this statement is false.

Step4: Analyze $\angle ACF$

$\angle ACF$: since there's a right angle symbol at $C$ between $AC$ and $CE$, and $FC$ is such that $\angle ACF$: wait, the right angle is between $AC$ and $CE$, so if $FC$ is such that $\angle ACF$ is a right angle? Wait, no, the right angle is between $AC$ and $CE$, so $\angle ACE = 90^\circ$. If $FC$ is equal to $EC$? Wait, no, the diagram shows $\angle FCE$ and $\angle ECD$: since $\angle ECD = 45^\circ$, and if $FC$ is symmetric to $DC$ with respect to $CE$, then $\angle FCE = 45^\circ$, so $\angle ACF = \angle ACE - \angle FCE = 90^\circ - 45^\circ = 45^\circ$? Wait, no, that can't be. Wait, maybe the right angle is between $AC$ and $CF$? Wait, the diagram has a right angle symbol at $C$ between $AC$ and $CF$? Wait, the original diagram: $A---C---B$ straight line, $F$ and $D$ above $C$, $CE$ is a vertical line (perpendicular to $AC$), with a right angle at $C$ between $AC$ and $CE$. Then $FC$ is a ray from $C$ to $F$, $DC$ to $D$. The right angle symbol is between $AC$ and $CE$, so $\angle ACE = 90^\circ$. Then $\angle ACF$: if $FC$ is such that $\angle ACF$ is a right angle? Wait, no, the option says $\angle ACF$ is a right angle. Wait, maybe the right angle is between $AC$ and $CF$, so $\angle ACF = 90^\circ$, which would be true if $CF$ is perpendicular to $AC$, same as $CE$. But in the diagram, maybe $FC$ and $EC$ are symmetric, so $\angle ACF = 90^\circ$, so that statement is true.

So the false statement is "$\angle ECD$ and $\angle FCE$ only are both acute angles".

Answer:

$\angle ECD$ and $\angle FCE$ only are both acute angles