Two angles of a triangle are \( 45^\circ \) and \( 85^\circ \). What is the measure of the third angle?

["Understanding Triangle Angles: Finding the Missing Angle in a 45°–85°–? Triangle", "When learning geometry, one of the most fundamental concepts is identifying the angles of a triangle. A triangle always has three interior angles that add up to exactly ( 180^\circ ). This principle forms the basis for solving various angle-related problems—especially when two angles are known.", "In this article, we’ll explore a classic triangle problem: given two angles of ( 45^\circ ) and ( 85^\circ ), how do we find the measure of the third angle?", "### The Angle Sum Property of Triangles", "The key rule to remember is that the sum of all interior angles in any triangle equals ( 180^\circ ). This is often expressed in formulas as:", "[\n\ ext{Angle}_A + \ ext{Angle}_B + \ ext{Angle}_C = 180^\circ\n]", "When two angles are known, finding the third is simple. Subtract the sum of the known angles from ( 180^\circ ).", "### Step-by-Step Calculation", "Let the two known angles be:", "[\n\angle A = 45^\circ \quad \ ext{and} \quad \angle B = 85^\circ\n]", "Now substitute into the angle sum equation:", "[\n45^\circ + 85^\circ + \angle C = 180^\circ\n]", "Add the known angles:", "[\n130^\circ + \angle C = 180^\circ\n]", "Now solve for the third angle ( \angle C ):", "[\n\angle C = 180^\circ - 130^\circ = 50^\circ\n]", "### Conclusion: The Third Angle Measures ( 50^\circ )", "Thus, in a triangle with angles ( 45^\circ ) and ( 85^\circ ), the measure of the third angle is ( 50^\circ ), making the three angles ( 45^\circ, 85^\circ, ) and ( 50^\circ ), neatly summing to ( 180^\circ ).", "### Why This Matters", "Understanding how to find a missing angle is crucial in geometry for solving problems involving unknown values in various contexts—such as real-world applications, trigonometry, and engineering design. Mastering this basic principle builds a strong foundation for more advanced topics.", "---", "Key Takeaways:\n- The sum of angles in a triangle is always ( 180^\circ ).\n- To find the unknown angle, subtract the sum of known angles from ( 180^\circ ).\n- For angles ( 45^\circ ) and ( 85^\circ ), the third angle is ( 50^\circ ).", "Start with these essential steps, and you’ll confidently tackle any triangle angle puzzle!"]