Sum and Difference of Angles: Adding / subtracting angles

To solve this problem, we must identify the configuration shown in the diagram, which involves a right triangle. The key to resolving this problem is recognizing the geometric properties of the triangle presented:

1. In a right triangle, the sum of the two non-right angles must equal $90^\circ$. This is a fundamental property of right triangles where one angle is $90^\circ$.

2. Given that the problem involves angles $\alpha$ and $\beta$ positioned as they are in the right triangle's context, we observe that the angle at $O$, formed by the two arms making the right angle, is $90^\circ$. Note: the vertex $O$ is presented as the intersection of the vertical and horizontal directions.

3. Thus, $\alpha$ and $\beta$ are the acute angles of a right triangle:

4. Since the sum of the angles in any triangle must equal $180^\circ$, and one of these angles is the right angle, the remaining two must sum to $90^\circ$.

Therefore, the size of the angle $\alpha + \beta$ is precisely $90^\circ$.

Thus, the solution to this problem is $\alpha + \beta = 90$ degrees.

To find the size of angle $\alpha$, we proceed as follows:

• Step 1: Establish the relationship for the angles as supplementary: $\alpha + (2x + 30) = 180^\circ$.
• Step 2: Substitute $\alpha = x + 30$ into the equation:

Substituting, we get:

$(x + 30) + (2x + 30) = 180$

Combine like terms:

$3x + 60 = 180$

Step 3: Solve for $x$:
Subtract 60 from both sides:

$3x = 120$

Divide both sides by 3:

$x = 40$

• Step 4: Substitute $x = 40$ back into $\alpha = x + 30$ to find $\alpha$:

$\alpha = 40 + 30 = 70$

Thus, the size of the angle $\alpha$ is 70.