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In this problem, we are given three angles related to each other in the figure, and we need to solve for \( x \).

We are provided with the following information:
- \( m \angle HGS = 3x + 5 \)
- \( m \angle SGF = 21x + 3 \)
- \( m \angle HGF = 176^\circ \)

Since the points \( H \), \( G \), and \( F \) lie on a straight line, the angles \( \angle HGS \), \( \angle SGF \), and \( \angle HGF \) are related as follows:

\[
m \angle HGS + m \angle SGF = m \angle HGF
\]

Substitute the given expressions for the angles into this equation:

\[
(3x + 5) + (21x + 3) = 176
\]

Now, let's solve for \( x \). 

\[
3x + 5 + 21x + 3 = 176
\]
\[
24x + 8 = 176
\]
\[
24x = 176 - 8
\]
\[
24x = 168
\]
\[
x = \frac{168}{24} = 7
\]

So, \( x = 7 \).

### Would you like to see further details or clarifications?

Here are 5 related questions:
1. What are supplementary angles, and how do they relate to this problem?
2. Can we find the actual measures of \( \angle HGS \) and \( \angle SGF \) now that we know \( x \)?
3. What would the equation look like if the sum of the angles was different?
4. What are linear pairs of angles, and how do they apply here?
5. How can we use similar techniques to solve problems involving angles in polygons?

**Tip:** Always check that the sum of angles in a line equals \( 180^\circ \) when working with supplementary or linear angles!

Find x if m∠HGS = 3x + 5, m∠SGF = 21x + 3, and m∠HGF = 176°.

Find the value of x using the relationships between angles and algebraic expressions. This problem involves solving for x in the equation where m∠HGS = 3x + 5, m∠SGF = 21x + 3, and the total angle m∠HGF = 176°. Suitable for Grades 8-10.

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