Solving Geometry Problems Using Algebraic Equations 3.3

Solving Geometry Problems with Algebra

Using Algebraic Equations to Find Triangle Angles

Algebra can be used to solve geometric problems, such as finding the measures of angles in a triangle when given relationships between the angles. This approach involves translating word problems into algebraic equations and solving for unknowns.

• Key Concept: The sum of the measures of the angles in any triangle is always 180 degrees.

• Variables: Assign variables to represent the unknown angles. For example, let x be the smallest angle, y the middle-sized angle, and z the largest angle.

Example Problem

Problem Statement: The smallest angle of a certain triangle measures 36 degrees less than the middle-sized angle. The largest angle measures 16 degrees more than twice the smallest angle. Find the measure of each angle.

• Step 1: Define Variables

  • Let x = smallest angle

  • Let y = middle-sized angle

  • Let z = largest angle

• Step 2: Write Equations Based on Relationships

  • The smallest angle is 36 degrees less than the middle-sized angle: $x = y - 36$

  • The largest angle is 16 degrees more than twice the smallest angle: $z = 16 + 2x$

  • The sum of the angles in a triangle is 180 degrees: $x + y + z = 180$

• Step 3: Substitute and Solve

  • Substitute $y = x + 36$ and $z = 16 + 2x$ into the sum equation: $x + (x + 36) + (16 + 2x) = 180$

  • Simplify: $x + x + 36 + 16 + 2x = 180$ $4x + 52 = 180$

  • Solve for $x$: $4x = 180 - 52$ $4x = 128$ $x = 32$

  • Find $y$: $y = x + 36 = 32 + 36 = 68$

  • Find $z$: $z = 16 + 2x = 16 + 2(32) = 16 + 64 = 80$

• Step 4: Check the Solution

  • Add the angles: $32 + 68 + 80 = 180$

  • The solution satisfies the triangle angle sum property.

Summary Table

Angle	Expression	Value (degrees)
Smallest	$x$	32
Middle-sized	$x + 36$	68
Largest	$16 + 2x$	80

Key Takeaways

• Translating word problems into algebraic equations is a powerful method for solving geometry problems.

• Always check that your solution satisfies all given conditions and the fundamental properties (such as the sum of triangle angles).

Additional info: This type of problem is common in College Algebra, especially in sections dealing with applications of linear equations and systems of equations.