Math Problem Statement

In the image, we have an angle ∠HGF split into two angles, (4x - 5)° and (5x - 15)°. These two angles are adjacent and form a straight line, ∠HGI, which is 180°. Solve for x and find the measure of the two angles.

Solution

In the image, we have an angle HGF\angle HGF that is split into two angles, labeled (4x5)(4x - 5)^\circ and (5x15)(5x - 15)^\circ. These two angles are adjacent, and together they form the larger angle HGI\angle HGI, which is a straight line. A straight line measures 180180^\circ, so we can set up an equation to find xx.

The two angles together must sum to 180180^\circ: (4x5)+(5x15)=180(4x - 5) + (5x - 15) = 180

Let's solve for xx:

  1. Combine like terms: 9x20=1809x - 20 = 180

  2. Add 20 to both sides: 9x=2009x = 200

  3. Divide both sides by 9: x=200922.22x = \frac{200}{9} \approx 22.22

So, x22.22x \approx 22.22.

Now, we can substitute this value of xx back into the expressions for the angles:

  • (4x5)=4(22.22)5=88.885=83.88(4x - 5)^\circ = 4(22.22) - 5 = 88.88 - 5 = 83.88^\circ
  • (5x15)=5(22.22)15=111.115=96.1(5x - 15)^\circ = 5(22.22) - 15 = 111.1 - 15 = 96.1^\circ

The two angles are approximately 83.8883.88^\circ and 96.196.1^\circ, which indeed sum to 180180^\circ.

Would you like more details or have any questions?

Here are 5 related questions:

  1. What are adjacent angles, and how are they related?
  2. How would you solve a similar problem if the angles formed a triangle instead of a straight line?
  3. Can you explain how the sum of angles around a point adds up to 360 degrees?
  4. What is the difference between supplementary and complementary angles?
  5. How would the problem change if the sum of the two angles were 90 degrees instead of 180 degrees?

Tip: Always check that the sum of the calculated angles matches the expected total (e.g., 180° for straight angles or 360° for angles around a point).

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Math Problem Analysis

Mathematical Concepts

Geometry
Algebra
Angle Properties

Formulas

Sum of angles on a straight line: 180°
Algebraic expressions for angle measures

Theorems

Supplementary Angles Theorem

Suitable Grade Level

Grades 8-10