there is a geometric figure with angles 115°, 85°, 155° and a right angle, and we need to find the unknown angle (marked as ?).

Question

Accepted Answer

# Explanation:
## Step1: Find adjacent supplementary angles
First, find the supplementary angles to $115^\circ$ and $155^\circ$. For $115^\circ$, the supplementary angle is $180^\circ - 115^\circ = 65^\circ$. For $155^\circ$, the supplementary angle is $180^\circ - 155^\circ = 25^\circ$.
## Step2: Use angle sum in quadrilateral
A quadrilateral's interior angles sum to $360^\circ$, and there's a right angle ($90^\circ$) and an $85^\circ$ angle. Let the unknown angle (let's call it $x$) be the angle we need to find related to the right angle for the final angle $?$. Wait, actually, let's re - evaluate. The figure seems to be a combination of angles around a point or in a polygon. Wait, the sum of angles around a point is $360^\circ$, but there's a right angle ($90^\circ$), $85^\circ$, the supplementary angles $65^\circ$ and $25^\circ$, and the angle $?$. Wait, no, let's do it properly.

Wait, the first triangle - like part: the angle adjacent to $115^\circ$ is $180 - 115=65^\circ$. The angle adjacent to $155^\circ$ is $180 - 155 = 25^\circ$. Now, in the quadrilateral (or the angle sum around the right - angled vertex), we know that the sum of angles in a quadrilateral is $360^\circ$, but there's a right angle ($90^\circ$), an $85^\circ$ angle, the $65^\circ$ angle, the $25^\circ$ angle, and the angle $?$ and the right angle? Wait, no, let's look at the right angle. Let's assume that we have a set of angles where we can use the fact that the sum of angles in a polygon or around a point. Wait, maybe a better approach: the sum of all angles around the point (or in the figure) should account for the right angle, $85^\circ$, the two supplementary angles, and the angle $?$.

Wait, let's start over. The angle next to $115^\circ$ (linear pair) is $180 - 115 = 65^\circ$. The angle next to $155^\circ$ (linear pair) is $180 - 155=25^\circ$. Now, we have a right angle ($90^\circ$), an $85^\circ$ angle, the $65^\circ$ angle, the $25^\circ$ angle, and the angle $?$, and we know that the sum of angles in a quadrilateral (if it's a quadrilateral with a right angle) or the sum of angles around a point? Wait, no, the sum of interior angles of a quadrilateral is $360^\circ$. Let's consider the quadrilateral with angles $85^\circ$, $90^\circ$, the angle adjacent to $115^\circ$ ($65^\circ$), the angle adjacent to $155^\circ$ ($25^\circ$), and the angle $?$? No, that can't be. Wait, maybe the figure is such that we have a right angle, $85^\circ$, $65^\circ$, $25^\circ$, and we need to find the remaining angle (which is $?$) such that when we consider the sum of angles in a polygon or the linear pairs.

Wait, another approach: The sum of angles in a triangle is $180^\circ$, but here we have a right angle. Wait, let's use the fact that the sum of all angles involved should be such that:

We know that $85^\circ+65^\circ + 90^\circ+25^\circ+?= 360^\circ$? No, $85 + 65=150$, $150+90 = 240$, $240 + 25=265$, $360 - 265 = 95$? No, that's not right. Wait, maybe the angle $?$ is related to the right angle. Wait, no, let's look at the linear pair with the right angle. Wait, the correct way:

The angle adjacent to $115^\circ$ is $180 - 115=65^\circ$

The angle adjacent to $155^\circ$ is $180 - 155 = 25^\circ$

Now, in the figure, we have a right angle ($90^\circ$), an $85^\circ$ angle, the $65^\circ$ angle, the $25^\circ$ angle, and the angle $?$. Since the sum of angles in a quadrilateral is $360^\circ$, but we also have to consider that the right angle is part of it. Wait, no, maybe the figure is a pentagon? No, let's think of the angles around the point where the right angle is. The sum of angles around a point is $360^\circ$. So we have: $85^\circ+65^\circ + 90^\circ+25^\circ+?= 360^\circ$

Wait, $85 + 65=150$, $150+90 = 240$, $240+25 = 265$, then $360 - 265=95$? No, that's not correct. Wait, maybe I made a mistake in identifying the angles. Let's look at the right angle. The angle we need to find (let's call it $x$) and the right angle: Wait, no, the right angle is $90^\circ$, and we have $85^\circ$, the angle from the $115^\circ$ (65°), the angle from the $155^\circ$ (25°), and $x$. Wait, maybe the sum of these angles should be equal to $360^\circ$ (around a point), so:

$85 + 65+90 + 25+x=360$

$85 + 65=150$, $150 + 90=240$, $240+25 = 265$

$x=360 - 265=95$? No, that can't be. Wait, maybe the figure is a triangle - like with a right angle. Wait, no, let's try a different approach. The sum of the interior angles of a triangle is $180^\circ$. Wait, maybe the angle we need is related to the right angle and the other angles. Wait, let's see:

The angle next to $115^\circ$ is $65^\circ$, the angle next to $155^\circ$ is $25^\circ$. In the left - hand triangle, we have angles $85^\circ$, $65^\circ$, and the third angle of the triangle: $180-(85 + 65)=180 - 150 = 30^\circ$. Then, in the right - hand part, we have the right angle ($90^\circ$), the $25^\circ$ angle, and the angle related to the $30^\circ$ angle. Wait, maybe the angle $?$ is $90-(30 + 25)$? No, $30+25 = 55$, $90 - 55 = 35$? No, that's not right.

Wait, I think I messed up. Let's start over. The key is that the sum of angles around a point is $360^\circ$, and we have a right angle ($90^\circ$), an $85^\circ$ angle, a $115^\circ$ - supplementary angle ($65^\circ$), a $155^\circ$ - supplementary angle ($25^\circ$), and the angle $?$. Wait, no, the right angle is $90^\circ$, and the angle $?$ is a right angle? No, the figure has a right - angled symbol, so one angle is $90^\circ$. Let's list all the angles around the common vertex:

- Angle from the $85^\circ$ side: $85^\circ$
- Angle from the $115^\circ$ side (linear pair): $180 - 115=65^\circ$
- Right angle: $90^\circ$
- Angle from the $155^\circ$ side (linear pair): $180 - 155 = 25^\circ$
- Angle $?$

The sum of these angles should be $360^\circ$ (since they are around a point), so:

$85+65 + 90+25+?=360$

$85 + 65=150$, $150+90 = 240$, $240 + 25=265$

$?=360 - 265 = 95$? No, that's not correct. Wait, maybe the right angle is not part of the sum? No, the right angle is in the figure. Wait, maybe the figure is a quadrilateral with angles $85^\circ$, $90^\circ$, the angle from $115^\circ$ (65°), the angle from $155^\circ$ (25°), and the angle $?$. But a quadrilateral has 4 angles, so maybe I included an extra angle. Let's correct: A quadrilateral has 4 interior angles. So if we have angles $85^\circ$, $90^\circ$, $65^\circ$, and $?$, and the $25^\circ$ is from another part? No, I'm getting confused.

Wait, let's use the fact that the sum of angles in a triangle is $180^\circ$. The left - hand triangle: angles are $85^\circ$, the angle adjacent to $115^\circ$ (65°), so the third angle is $180-(85 + 65)=30^\circ$. Now, the right - hand part: we have a right angle ($90^\circ$), the angle adjacent to $155^\circ$ (25°), and the angle $?$, and the $30^\circ$ angle. Wait, maybe the sum of $30^\circ+25^\circ+? = 90^\circ$ (since it's a right angle). So $?=90-(30 + 25)=90 - 55 = 35^\circ$. Wait, that makes sense. Let's check:

The angle from the left triangle: $180 - 85 - 65=30^\circ$

The angle from the right side: $25^\circ$

The right angle is $90^\circ$, so the remaining angle (?) is $90-(30 + 25)=35^\circ$

Yes, that works. So the steps are:

## Step1: Find the angle in the left triangle
The angle adjacent to $115^\circ$ is a linear pair, so $180 - 115 = 65^\circ$. In the left triangle with angles $85^\circ$ and $65^\circ$, the third angle $A$ is $180-(85 + 65)=30^\circ$.

## Step2: Find the angle adjacent to $155^\circ$
The angle adjacent to $155^\circ$ is a linear pair, so $180 - 155 = 25^\circ$.

## Step3: Find the unknown angle (?)
Since there is a right angle ($90^\circ$), and the sum of the angles $30^\circ$, $25^\circ$, and $?$ should equal $90^\circ$ (because they are within the right - angled region), we have $?=90-(30 + 25)=35^\circ$.

# Answer:
$35^\circ$

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QUESTION IMAGE

Question

Explanation:

Step1: Find adjacent supplementary angles

Step2: Use angle sum in quadrilateral

Answer: