Finding Angle ACD in a Right-Angled Triangle

This problem involves a right-angled triangle ▵ABC and a perpendicular line segment CD from the right angle vertex C to the hypotenuse AB. We are given one acute angle of the triangle and need to find the measure of an angle formed by the perpendicular.

Understanding the Geometry of the Triangle

We are given:

• ▵ABC is right angled at C, which means ∠ACB = \(90^\circ\).
• CD is perpendicular to AB, which means CD ⊥ AB. This implies ∠CDA = \(90^\circ\) and ∠CDB = \(90^\circ\).
• The measure of angle A is ∠A = \(55^\circ\).

We need to find the measure of ∠ACD.

Step-by-Step Solution to Find ∠ACD

We can solve this problem by using the angle sum property of triangles.

Method 1: Using Triangle ACD

Consider the right-angled triangle ▵ACD.

• We know ∠CDA = \(90^\circ\) (since CD ⊥ AB).
• We are given ∠A = \(55^\circ\).
• The sum of angles in any triangle is \(180^\circ\). So, in ▵ACD, we have:

\[\angle A + \angle ACD + \angle CDA = 180^\circ\]

Substitute the known values:

\[55^\circ + \angle ACD + 90^\circ = 180^\circ\]

Combine the known angles:

\[\angle ACD + 145^\circ = 180^\circ\]

Subtract \(145^\circ\) from both sides to find ∠ACD:

\[\angle ACD = 180^\circ - 145^\circ\]

\[\angle ACD = 35^\circ\]

Method 2: Using Triangle ABC and Triangle BCD (Alternative Approach)

First, find ∠B in the right-angled ▵ABC.

• In ▵ABC, ∠A + ∠B + ∠ACB = \(180^\circ\).
• Substitute the known values: \(55^\circ + \angle B + 90^\circ = 180^\circ\).
• So, ∠B = \(180^\circ - 90^\circ - 55^\circ = 35^\circ\).

Now, consider the right-angled triangle ▵BCD.

• We know ∠CDB = \(90^\circ\).
• We just found ∠B = \(35^\circ\).
• In ▵BCD, ∠B + ∠BCD + ∠CDB = \(180^\circ\).
• Substitute the known values: \(35^\circ + \angle BCD + 90^\circ = 180^\circ\).
• So, ∠BCD = \(180^\circ - 90^\circ - 35^\circ = 55^\circ\).

Finally, note that ∠ACB is the sum of ∠ACD and ∠BCD.

• ∠ACB = ∠ACD + ∠BCD
• We know ∠ACB = \(90^\circ\) and ∠BCD = \(55^\circ\).
• So, \(90^\circ = \angle ACD + 55^\circ\).
• Subtract \(55^\circ\) from both sides: ∠ACD = \(90^\circ - 55^\circ = 35^\circ\).

Both methods yield the same result.

The measure of angle ACD is \(35^\circ\).

Conclusion on Finding Angle ACD

Based on the angle sum property of triangle ACD, with ∠A = \(55^\circ\) and ∠CDA = \(90^\circ\), the measure of ∠ACD is \(35^\circ\).

Revision Table: Right Triangle Geometry

Additional Information: Properties of Altitude from Right Angle

When an altitude is drawn from the right angle vertex C to the hypotenuse AB in ▵ABC:

• The altitude CD divides ▵ABC into two smaller triangles, ▵ACD and ▵BCD.
• These two smaller triangles are similar to the original triangle ▵ABC and also similar to each other.
  • ▵ACD ~ ▵ABC
  • ▵BCD ~ ▵ABC
  • ▵ACD ~ ▵BCD
• This similarity leads to important geometric mean relationships:
  • CD is the geometric mean of AD and DB: \(CD^2 = AD \times DB\)
  • AC is the geometric mean of AD and AB: \(AC^2 = AD \times AB\)
  • BC is the geometric mean of DB and AB: \(BC^2 = DB \times AB\)
• In terms of angles:
  • ∠ACD = ∠B
  • ∠BCD = ∠A

Using the angle property ∠ACD = ∠B from the additional information, we can verify our result. Since ∠A = \(55^\circ\) and ∠ACB = \(90^\circ\) in ▵ABC, ∠B = \(180^\circ - 90^\circ - 55^\circ = 35^\circ\). Therefore, ∠ACD must also be \(35^\circ\).