Parts of a Triangle: Determine the size of the given angles

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.

To solve this problem, let's identify the angle relationships based on the given information:

Step 1: From the diagram, angle $\beta$ is stated to be $65^\circ$. It is important to recognize that $\alpha$ and $\beta$ are positioned such that they form a pair of vertical angles.

Step 2: According to the vertical angle theorem, vertical angles are congruent. This means that if $\beta = 65^\circ$, then $\alpha$ must also equal $65^\circ$ because they are vertical angles created by intersecting lines.

Therefore, the size of angle $\alpha$ is $\mathbf{65^\circ}$.