Let's solve this geometry problem about a triangle ABC where we are given a specific relationship between its angles. We are told that 2 times angle A is equal to 3 times angle B, which is also equal to 6 times angle C. Our goal is to find the sum of angle A and angle C.

Understanding the Given Relationship

The problem provides the relationship: \(2 \angle A = 3 \angle B = 6 \angle C\).

This means all three expressions are equal to some common value. Let's call this common value \(k\).

\(2 \angle A = k\)

\(3 \angle B = k\)

\(6 \angle C = k\)

Expressing Angles in Terms of a Common Variable

From the relationships above, we can express each angle in terms of \(k\):

• \(\angle A = \frac{k}{2}\)
• \(\angle B = \frac{k}{3}\)
• \(\angle C = \frac{k}{6}\)

Using the Angle Sum Property of a Triangle

We know that the sum of the interior angles in any triangle is always 180 degrees. For triangle ABC, this means:

\(\angle A + \angle B + \angle C = 180^\circ\)

Solving for the Common Variable \(k\)

Now, substitute the expressions for \(\angle A\), \(\angle B\), and \(\angle C\) in terms of \(k\) into the angle sum equation:

\(\frac{k}{2} + \frac{k}{3} + \frac{k}{6} = 180^\circ\)

To solve for \(k\), find a common denominator for the fractions, which is 6.

\(\frac{3k}{6} + \frac{2k}{6} + \frac{k}{6} = 180^\circ\)

Combine the terms on the left side:

\(\frac{3k + 2k + k}{6} = 180^\circ\)

\(\frac{6k}{6} = 180^\circ\)

\(k = 180^\circ\)

Calculating the Angles

Now that we have the value of \(k\), we can find the measure of each angle:

• \(\angle A = \frac{k}{2} = \frac{180^\circ}{2} = 90^\circ\)
• \(\angle B = \frac{k}{3} = \frac{180^\circ}{3} = 60^\circ\)
• \(\angle C = \frac{k}{6} = \frac{180^\circ}{6} = 30^\circ\)

Verifying the Angles

Let's check if these angles satisfy the given relationship and the angle sum property:

• Check given relationship:
  • \(2 \angle A = 2 \times 90^\circ = 180^\circ\)
  • \(3 \angle B = 3 \times 60^\circ = 180^\circ\)
  • \(6 \angle C = 6 \times 30^\circ = 180^\circ\)
  The relationship \(2 \angle A = 3 \angle B = 6 \angle C\) is satisfied as all are equal to \(180^\circ\).
• Check angle sum property:
  • \(\angle A + \angle B + \angle C = 90^\circ + 60^\circ + 30^\circ = 180^\circ\)
  The angle sum property is satisfied.

Finding ∠A + ∠C

The question asks for the value of \(\angle A + \angle C\).

\(\angle A + \angle C = 90^\circ + 30^\circ = 120^\circ\)

Thus, the sum of angle A and angle C is \(120^\circ\).

Revision Table: Triangle Angles

Additional Information: Types of Triangles

Based on their angles, triangles can be classified:

• Acute Triangle: All three angles are less than \(90^\circ\).
• Right Triangle: One angle is exactly \(90^\circ\).
• Obtuse Triangle: One angle is greater than \(90^\circ\).

In this problem, we found \(\angle A = 90^\circ\), \(\angle B = 60^\circ\), and \(\angle C = 30^\circ\). Since one angle is \(90^\circ\), the triangle ABC is a right triangle.