The problem in the image involves finding the value of $x$ and determining the measures of each labeled angle in a quadrilateral. Let's break it down:

Step 1: Identifying the angles

• One of the angles is labeled $x^\circ$.
• Another angle is labeled $(x - 30)^\circ$.
• There is a right angle, which measures $90^\circ$.
• The fourth angle is also unknown, but we will use the fact that the sum of all interior angles in a quadrilateral equals $360^\circ$.

Step 2: Applying the sum of interior angles

The sum of the four angles in any quadrilateral is $360^\circ$. Therefore, we can write the equation: $x + (x - 30) + 90 + 90 = 360$ Simplify the equation: $x + (x - 30) + 180 = 360$ $2x - 30 + 180 = 360$ $2x + 150 = 360$ $2x = 360 - 150$ $2x = 210$ $x = 105$

Step 3: Finding the measures of each angle

Now that we know $x = 105^\circ$, we can find the other angles:

• The first angle is $x = 105^\circ$.
• The second angle is $x - 30 = 105^\circ - 30^\circ = 75^\circ$.
• The third and fourth angles are both $90^\circ$ (right angles).

So, the angles are:

• $105^\circ$
• $75^\circ$
• $90^\circ$
• $90^\circ$

Would you like further details on how this applies or have any other questions?

Relative questions:

1. What is the sum of interior angles in a quadrilateral?
2. How do you solve for an unknown angle when given multiple angles in a figure?
3. What are the properties of a right angle?
4. Can the same approach be used for triangles? How would it differ?
5. How does knowing the properties of a quadrilateral help in solving geometry problems?

Tip:

Always remember that the sum of the interior angles of any quadrilateral is $360^\circ$, which is key in solving for unknown angles.