To solve for the missing angle in the given figure, we need to analyze the relationships between the angles.

Step 1: Identify Given Angles

• The angle of $112^\circ$ is an exterior angle.
• The angle of $114^\circ$ is also an exterior angle.
• There's a right angle $90^\circ$.
• The angle of $55^\circ$ is given.
• The unknown angle is labeled $x$.

Step 2: Use the Exterior Angle Theorem

The exterior angle theorem states that the exterior angle is equal to the sum of the two opposite interior angles. However, here we need to approach this differently since the figure's angles involve a triangle and straight lines.

Step 3: Analyze the Triangle

In the triangle:

• The right angle is $90^\circ$.
• The angle opposite to the $55^\circ$ is unknown.

Step 4: Calculate the Missing Interior Angle

In a triangle, the sum of the interior angles is always $180^\circ$. Let's denote the angle opposite to $55^\circ$ as $y$.

$y = 180^\circ - 90^\circ - 55^\circ = 35^\circ$

Now, let's use the information of the straight line:

Step 5: Use Linear Pair Postulate

Since the angles on a straight line add up to $180^\circ$: $x = 180^\circ - 114^\circ - 35^\circ = 31^\circ$

So, the missing angle $x$ is 31 degrees.

Would you like any further details or have any other questions?

Related Questions:

1. What is the exterior angle theorem, and how is it applied?
2. How can you verify if the angles in a figure are correctly identified?
3. What is the sum of angles in any triangle, and how does it help in solving problems?
4. How do linear pair postulates assist in determining unknown angles?
5. Can the interior angles of a non-right triangle be calculated in the same way?

Tip:

Always ensure to check the sum of the angles in the triangle or along a straight line to validate your calculations.