Unlocking the Secrets of 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:

1. Take a triangle \(ABC\).
2. Draw a line \(DE\) parallel to side \(AC\) through vertex \(B\).
3. Angle \(A\) is congruent to angle \(DBA\) because they are corresponding angles (due to the parallel lines).
4. 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:

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

Answers:

1. \(65^\circ\)
2. \(D = 40^\circ\), \(E = 80^\circ\)
3. \(40^\circ\), \(60^\circ\), \(80^\circ\)