Equations Using the Exterior Angle Theorem--Example 6

Topic

Equations

Description

This example demonstrates a more complex application of the Exterior Angle Theorem in solving triangle-related equations. In this scenario, we have a triangle with one known interior angle of 28°, an unknown interior angle x, and an exterior angle expressed as 3x. The Exterior Angle Theorem states that an exterior angle of a triangle is equal to the sum of the two non-adjacent interior angles. Here, the equation is set up as 3x = 28° + x. This algebraic representation requires more sophisticated problem-solving skills. To solve for x, we first subtract x from both sides, resulting in 2x = 28°. Then, we divide both sides by 2 to get x = 14°. We can verify that the exterior angle 3x is indeed 42°, which satisfies the theorem. This problem showcases how the theorem can be applied in situations where the exterior angle is expressed as a multiple of a variable. The methodology for solving such equations involves setting up the equation based on the theorem, then using algebraic manipulation to isolate the unknown variable. This process combines geometric reasoning with more advanced algebraic skills, challenging students to think critically about the relationships between angles in a triangle. By working through this type of problem, students develop a deeper understanding of how variables can represent both interior and exterior angles, preparing them for more complex geometric and algebraic challenges in higher-level mathematics.

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