To solve for the degree measures of each angle in the triangle $\triangle GHI$, we can follow these steps:

Step 1: Set up the known angles

1. The triangle $GHI$ has a right angle at $\angle G$, so $m \angle G = 90^\circ$.
2. The problem provides expressions for the other two angles:
   • $m \angle H = (4x + 21)^\circ$
   • $m \angle I = (2x + 9)^\circ$

Step 2: Use the Triangle Sum Property

In any triangle, the sum of the angles is $180^\circ$. Therefore: $m \angle G + m \angle H + m \angle I = 180^\circ$ Substitute the values: $90 + (4x + 21) + (2x + 9) = 180$

Step 3: Simplify and Solve for $x$

Combine like terms: $90 + 4x + 21 + 2x + 9 = 180$ $6x + 120 = 180$ Subtract 120 from both sides: $6x = 60$ Divide by 6: $x = 10$

Step 4: Substitute $x = 10$ back into the expressions for $m \angle H$ and $m \angle I$

1. $m \angle H = 4x + 21 = 4(10) + 21 = 40 + 21 = 61^\circ$
2. $m \angle I = 2x + 9 = 2(10) + 9 = 20 + 9 = 29^\circ$

Step 5: Write the final answer

• $m \angle G = 90^\circ$
• $m \angle H = 61^\circ$
• $m \angle I = 29^\circ$

Would you like further clarification on any step?

Here are some additional questions for exploration:

1. What is the Triangle Sum Property, and why does it apply here?
2. How would the solution change if one angle was obtuse instead of acute?
3. Could you solve for $x$ if the given angle measures were different?
4. What are the implications if $m \angle G$ wasn’t $90^\circ$?
5. Can you prove that this is a right triangle by other geometric properties?

Tip: Always verify your final angle measures by ensuring they sum to $180^\circ$.