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To solve this problem, we first need to break down the given information and understand the geometry of the triangle.

### Step-by-step analysis:

- In triangle \( \triangle ABC \), we know that:
  - \( \angle C \) is a right angle (\( 90^\circ \)).
  - \( CD \) is the altitude to side \( AB \), which means \( CD \) is perpendicular to \( AB \) and divides \( \triangle ABC \) into two smaller right triangles \( \triangle CBD \) and \( \triangle CAD \).
  - \( \angle A = 65^\circ \), as given.
  
From this, let's start calculating the unknown angles in triangles \( \triangle CBD \) and \( \triangle CAD \).

### Triangle \( \triangle ABC \):
Since \( \triangle ABC \) is a right triangle, we can calculate \( \angle B \):
\[
\angle A + \angle B + \angle C = 180^\circ
\]
Substituting \( \angle A = 65^\circ \) and \( \angle C = 90^\circ \):
\[
65^\circ + \angle B + 90^\circ = 180^\circ
\]
\[
\angle B = 180^\circ - 155^\circ = 25^\circ
\]
So, \( \angle B = 25^\circ \).

### Triangle \( \triangle CBD \) (Right triangle):
- \( \angle DCB = 90^\circ \) because \( CD \) is perpendicular to \( AB \).
- \( \angle B \) in \( \triangle ABC \) is shared with \( \triangle CBD \), so \( \angle DBC = 25^\circ \).

Now, we can calculate \( \angle CDB \):
\[
\angle DBC + \angle DCB + \angle CDB = 180^\circ
\]
\[
25^\circ + 90^\circ + \angle CDB = 180^\circ
\]
\[
\angle CDB = 180^\circ - 115^\circ = 65^\circ
\]

### Triangle \( \triangle CAD \) (Right triangle):
- \( \angle ACD = 90^\circ \) because \( CD \) is perpendicular to \( AB \).
- \( \angle A \) in \( \triangle ABC \) is shared with \( \triangle CAD \), so \( \angle DAC = 65^\circ \).

Now, we can calculate \( \angle ADC \):
\[
\angle DAC + \angle ACD + \angle ADC = 180^\circ
\]
\[
65^\circ + 90^\circ + \angle ADC = 180^\circ
\]
\[
\angle ADC = 180^\circ - 155^\circ = 25^\circ
\]

### Final Angles:
- \( m \angle DBC = 25^\circ \)
- \( m \angle DCB = 90^\circ \)
- \( m \angle CDB = 65^\circ \)
- \( m \angle ACD = 90^\circ \)
- \( m \angle ADC = 25^\circ \)

Would you like a further breakdown of any step or details on specific parts? Here are some related questions to deepen understanding:

1. Why is \( CD \) perpendicular to \( AB \)?
2. How do the angles in a right triangle add up to \( 180^\circ \)?
3. What is the significance of \( \angle C = 90^\circ \) in the context of the altitude?
4. How are the angles in \( \triangle ABC \) related to the smaller triangles \( \triangle CBD \) and \( \triangle CAD \)?
5. Can we use trigonometric ratios to find the lengths of the sides in these triangles?

**Tip:** Always check if a triangle is a right triangle when one angle is \( 90^\circ \), as this simplifies calculations using the sum of angles.

In triangle ABC, ∠C is a right angle and CD is the altitude to AB. Find the angles in triangles CBD and CAD if m∠A = 65°.

Solve a geometry problem where ∠C is a right angle and CD is the altitude to AB. Given m∠A = 65°, learn how to find the angles in triangles CBD and CAD using the angle sum property of triangles and right triangle properties. Suitable for students in Grades 9-12.

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