Solve each problem. See Examples 5 and 9. The sum of the measures of the angles of any triangle is 180°. In a certain triangle, the largest angle measures 55° less than twice the medium angle, and the smallest angle measures 25° less than the medium angle. Find the measures of all three angles.
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Let's denote the medium angle as 'm'. According to the problem, the largest angle is '2m - 55' and the smallest angle is 'm - 25'.
Since the sum of the measures of the angles of any triangle is 180°, we can set up the following equation: m + (2m - 55) + (m - 25) = 180.
Simplify the equation by combining like terms. This will give you: 4m - 80 = 180.
Next, add 80 to both sides of the equation to isolate the term with 'm'. This will give you: 4m = 260.
Finally, divide both sides of the equation by 4 to solve for 'm'. Once you have the value of 'm', you can substitute it back into the expressions for the largest and smallest angles to find their measures.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Triangle Angle Sum Theorem
The Triangle Angle Sum Theorem states that the sum of the interior angles of a triangle is always 180 degrees. This fundamental property allows us to set up equations based on the measures of the angles in a triangle, which is essential for solving problems related to triangle geometry.
Algebraic expressions involve variables and constants combined using mathematical operations. In this problem, we need to express the measures of the angles in terms of a variable (the medium angle) to create equations that can be solved. Understanding how to manipulate these expressions is crucial for finding the angle measures.
A system of equations consists of two or more equations that share variables. In this problem, we will create a system based on the relationships between the angles. Solving this system will allow us to find the values of the angles, demonstrating the interconnectedness of the equations derived from the triangle's properties.