# Unlocking the Secrets of Triangle Angle Sum

Triangles, with their three sides and three angles, have intrigued mathematicians for centuries. Whether you're studying the majestic pyramids of Egypt or a slice of pie, triangles are everywhere! One fundamental property of triangles is the sum of their interior angles. Ever wondered why the angles inside a triangle always add up to a specific value? Dive into this post to unlock the mysteries of the triangle angle sum!

## Step-by-step Guide: Triangle Angle Sum

**The Triangle Angle Sum Theorem:**

The Triangle Angle Sum Theorem states that the sum of the interior angles of any triangle is always \(180^\circ\).

**Proof:**

- Take a triangle \(ABC\).
- Draw a line \(DE\) parallel to side \(AC\) through vertex \(B\).
- Angle \(A\) is congruent to angle \(DBA\) because they are corresponding angles (due to the parallel lines).
- Angle \(C\) is congruent to angle \(BCD\) for the same reason.

Adding the angles of triangle \(ABC\):

\(\angle A + \angle B + \angle C = \angle DBA + \angle ABC + \angle BCD\)

Given that a straight line has an angle of \(180^\circ\), we have:

\(\angle DBA + \angle ABC + \angle BCD = 180^\circ\)

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

### Examples

**Example 1:**

Given a triangle \(PQR\) with angles \(P = 60^\circ\), \(Q = 50^\circ\), find angle \(R\).

**Solution:**

Using the Triangle Angle Sum theorem:

\(\angle P + \angle Q + \angle R = 180^\circ\)

Substitute the given values:

\(60^\circ + 50^\circ + \angle R = 180^\circ\)

Adding the angles together:

\(\angle R = 180^\circ – 110^\circ\)

\(\angle R = 70^\circ\)

**Example 2:**

In triangle \(XYZ\), if angle \(X\) is twice angle \(Y\) and angle \(Z\) is \(40^\circ\), find the angles \(X\) and (Y).

**Solution:**

Let \( \angle Y = a^\circ \)

Therefore, \( \angle X = 2a^\circ \)

Using the Triangle Angle Sum theorem:

\(a^\circ + 2a^\circ + 40^\circ = 180^\circ\)

Combine like terms:

\(3a^\circ + 40^\circ = 180^\circ\)

Subtract 40 from both sides:

\(3a^\circ = 140^\circ\)

Divide both sides by \(3\):

\(a^\circ = \frac{140}{3}\)

\(a^\circ = 46.67^\circ\)

So, \( \angle Y = 46.67^\circ \) and \( \angle X = 93.34^\circ\).

### Practice Questions:

- If two angles of a triangle are \(70^\circ\) and \(45^\circ\), find the third angle.
- In triangle \(DEF\), angle \(D\) is half the size of angle \(E\) and angle \(F\) is \(60^\circ\). Calculate the angles \(D\) and \(E\).
- A triangle has angles in the ratio 2:3:5. What are the angles?

**Answers:**

- \(65^\circ\)
- \(D = 40^\circ\), \(E = 80^\circ\)
- \(40^\circ\), \(60^\circ\), \(80^\circ\)

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