Math Problem Statement

In ΔFGH, m∠F = (5x - 16)°, m∠G = (2x - 13)°, and m∠H = (8x + 14)°. Find m∠H.

Solution

Let's solve this step by step!

Problem:

We are given the angles of FGH\triangle FGH:
mF=(5x16),mG=(2x13),mH=(8x+14).m\angle F = (5x - 16)^\circ, \, m\angle G = (2x - 13)^\circ, \, m\angle H = (8x + 14)^\circ.

The sum of the interior angles of any triangle is 180180^\circ.
So, mF+mG+mH=180.m\angle F + m\angle G + m\angle H = 180^\circ.

Step 1: Substitute the given expressions for the angles

(5x16)+(2x13)+(8x+14)=180.(5x - 16) + (2x - 13) + (8x + 14) = 180.

Step 2: Simplify the equation

Combine like terms: 5x+2x+8x1613+14=180.5x + 2x + 8x - 16 - 13 + 14 = 180. 15x15=180.15x - 15 = 180.

Step 3: Solve for xx

Add 15 to both sides: 15x=195.15x = 195. Divide both sides by 15: x=13.x = 13.

Step 4: Find mHm\angle H

Substitute x=13x = 13 into mH=8x+14m\angle H = 8x + 14: mH=8(13)+14=104+14=118.m\angle H = 8(13) + 14 = 104 + 14 = 118^\circ.

Final Answer:

mH=118.m\angle H = 118^\circ.

Would you like further explanation or a breakdown of any specific step?


Here are 5 related questions to expand your understanding:

  1. What is the measure of mFm\angle F and mGm\angle G using x=13x = 13?
  2. How would you check your solution to ensure it satisfies the triangle's angle sum?
  3. Can you explain why the sum of interior angles in a triangle is always 180180^\circ?
  4. How would the problem change if one angle were an exterior angle?
  5. What if xx were negative—would it still produce valid triangle angles?

Tip: Always recheck your substitution to confirm accurate values for variables and expressions!

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Math Problem Analysis

Mathematical Concepts

Algebra
Geometry
Triangle Angle Sum Property

Formulas

Sum of angles in a triangle: m∠F + m∠G + m∠H = 180°

Theorems

Triangle Angle Sum Theorem

Suitable Grade Level

Grades 8-10