Finding Right Triangle Angles When One Is Expressed Relative to Another
Apply the geometry problem-solving strategy when the angles of a right triangle are defined relative to one another, requiring an algebraic expression for each before setting up the equation.
Problem: The measure of one angle of a right triangle is degrees more than the measure of the smallest angle. Find the measures of all three angles.
- Read the problem.
- Identify what you are looking for: the measures of all three angles.
- Name: Choose a variable to represent the smallest angle. Let be the measure of the first (smallest) angle. Since the second angle is degrees more, its measure is . The third angle is a right angle, meaning its measure is . Draw the figure and label it with these expressions.
- Translate: Write the appropriate formula (the triangle angle sum property) and substitute the expressions:
- Solve the equation. Combine like terms: . Subtract from both sides: . Divide by : . Substitute to find the other angles: the second angle is , and the third angle is .
- Check the solutions: Does ? Yes, .
- Answer: The three angles measure , , and .
Unlike problems where each angle is an explicit given number, this type of problem requires defining one angle in terms of another. Establishing all unknown angles through a single variable creates a two-step linear equation.
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