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.