# 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:

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:
$$60^\circ + 50^\circ + \angle R = 180^\circ$$
$$\angle R = 180^\circ – 110^\circ$$
$$\angle R = 70^\circ$$

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?

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

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