The measure of the smallest angle of a triangle is one-third...

The measure of the smallest angle of a triangle is one-third the measure of the largest angle. The middle angle measures $30^{\circ}$ less than the largest angle. Find the measures of the angles of the triangle. (Hint: Recall that the sum of the measures of the angles of a triangle is $180^{\circ}$.)

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Key Concepts

Algebraic Representation of Angle Relationships

Many geometric problems involve relationships between angles that are best expressed using algebraic equations. By assigning variables to unknown angle measures and representing the given relationships as equations, one can systematically solve for the unknown values, linking algebra with geometric concepts.

Triangle Angle Sum Property

The triangle angle sum property states that the measures of the three interior angles of a triangle add up to 180°. This principle is essential for establishing relationships between the angles when one or more of their measures are unknown, forming the basis for many geometric proofs and problem-solving techniques.

System of Equations

When a problem provides multiple relationships among angles, these can be translated into a system of equations. Solving such a system involves using methods like substitution or elimination to determine the numerical values of the unknown variables. This process is a fundamental skill that bridges algebra and geometry.

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