Two angles of a triangle are 50 and 130. What is the third?

Step 1: Recall the Angle Sum Property of a Triangle

The Angle Sum Property of a Triangle states that the sum of the interior angles of any triangle is always equal to $180^\circ$. This is a fundamental property of Euclidean geometry.

Step 2: Substitute the given angles

We are given two angles of the triangle as $50^\circ$ and $130^\circ$. Let's substitute these values into the Angle Sum Property equation, where $C$ represents the unknown third angle.

Step 3: Calculate the sum of the known angles

First, we add the two known angles together: $50^\circ + 130^\circ = 180^\circ$. This simplifies our equation, making it easier to solve for the third angle.

Step 4: Solve for the third angle

To find the third angle $C$, we subtract the sum of the known angles ($180^\circ$) from the total sum of angles in a triangle ($180^\circ$). This gives us the measure of the missing angle.

Step 5: Interpret the result

A triangle cannot have an angle of $0^\circ$. This result indicates that a triangle with angles $50^\circ$ and $130^\circ$ cannot exist. The sum of two angles is already $180^\circ$, which means the third angle would have to be $0^\circ$, making it a degenerate triangle (a straight line).

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