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In this problem, we are given a right triangle \( \triangle CDE \) with the following angle measures:

- \( \angle C = (4x - 5)^\circ \)
- \( \angle D = (3x - 10)^\circ \)
- \( \angle E \) is the right angle, so \( m\angle E = 90^\circ \).

Since the sum of angles in any triangle is \(180^\circ\), we can set up the equation:

\[
m\angle C + m\angle D + m\angle E = 180^\circ
\]

Substitute the values we know:

\[
(4x - 5) + (3x - 10) + 90 = 180
\]

Now, we can solve for \( x \):

1. Combine like terms:
   \[
   7x - 15 + 90 = 180
   \]
   \[
   7x + 75 = 180
   \]

2. Subtract 75 from both sides:
   \[
   7x = 105
   \]

3. Divide by 7:
   \[
   x = 15
   \]

Now, substitute \( x = 15 \) back into the expressions for \( m\angle C \) and \( m\angle D \):

- For \( m\angle C \):
  \[
  m\angle C = 4x - 5 = 4(15) - 5 = 60 - 5 = 55^\circ
  \]

- For \( m\angle D \):
  \[
  m\angle D = 3x - 10 = 3(15) - 10 = 45 - 10 = 35^\circ
  \]

Thus, the degree measures for each angle in the triangle are:

- \( m\angle C = 55^\circ \)
- \( m\angle D = 35^\circ \)
- \( m\angle E = 90^\circ \)

Would you like more details or have any questions?

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Here are 5 related questions:

1. What is the sum of angles in any triangle?
2. How do we identify a right angle in a triangle?
3. Can the value of \( x \) change if the triangle type changes?
4. How can we check if the calculated angle measures are correct?
5. How does setting up equations help in solving for unknowns in geometry?

**Tip:** Always verify your solutions by substituting back to check consistency with given conditions.

In the triangle below, with right angle ∠E, suppose that m∠C = (4x - 5)° and m∠D = (3x - 10)°. Find the degree measure of each angle in the triangle.

Learn how to find the degree measure of each angle in a right triangle, given m∠C = (4x - 5)° and m∠D = (3x - 10)°. This problem involves triangle angle sum and solving for unknowns using linear equations, suitable for students in Grades 8-10.

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