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 the measure of the medium angle be represented by the variable $x$ degrees.

Express the largest angle in terms of $x$: since it measures 55° less than twice the medium angle, write it as $2x - 55$ degrees.

Express the smallest angle in terms of $x$: since it measures 25° less than the medium angle, write it as $x - 25$ degrees.

Use the fact that the sum of the measures of the angles in a triangle is 180°, and set up the equation: $x + (2x - 55) + (x - 25) = 180$.

Simplify the equation and solve for $x$ to find the measure of the medium angle, then substitute back to find the largest and smallest angles.

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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

This theorem states that the sum of the interior angles of any triangle is always 180 degrees. It is fundamental for solving problems involving unknown angles in triangles by setting up equations based on this total.

Setting Up Algebraic Equations from Word Problems

Translating verbal descriptions into algebraic expressions is essential. Here, the largest and smallest angles are expressed in terms of the medium angle, allowing the formation of equations to solve for unknown values.

Solving Linear Equations

Once the algebraic equation representing the sum of angles is formed, solving linear equations involves combining like terms and isolating variables to find the measure of the medium angle, which then helps find the other angles.

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Textbook Question

Solve each problem. See Examples 5 and 9. A cashier has a total of 30 bills, made up of ones, fives, and twenties. The number of twenties is 9 more than the number of ones. The total value of the money is \$351. How many of each denomination of bill are there?

Textbook Question

Solve each problem. See Examples 5 and 9. Solve the system of equations (4), (5), and (6) from Example 9.

$25 x + 40 y + 20 z = 2200$ (4)

$4 x + 2 y + 3 z = 280$ (5)

$3 x + 2 y + z = 180$ (6)

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Textbook Question

Solve each problem. See Examples 5 and 9. The sum of two numbers is 47, and the difference between the numbers is 1. Find the numbers.

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Textbook Question

Solve each problem. See Examples 5 and 9. At the Berger ranch, 6 goats and 5 sheep sell for \$305, while 2 goats and 9 sheep sell for \$285. Find the cost of a single goat and of a single sheep.