How Many Degrees Are in a Triangle? Answer – There are 3 angles in a triangle whose measures add up to 180 degrees.
Explanation:
To find the sum of the angles in a triangle, let’s consider the following triangle ABC whose three angles are unknown.
A line DE is drawn through vertex A such that it is parallel to BC. BC is also extended on both sides. Now, all the angles in the triangle as well as the ones formed are named.
∠x, ∠y, and ∠z are the interior angles of ∆ABC. ∠r and ∠s, which are outside ∆ABC, are its exterior angles.
The construction of the parallel lines DE and BC cut by AB resembles two parallel lines cut by a transversal. Thus, according to parallel line rules, angles ∠r and ∠y are alternate interior angles and are hence equal.
⇒ ∠y = ∠r
Similarly, the transversal AC cuts the parallel lines DE and BC. Therefore, ∠z and ∠s are also alternate interior angles and hence equal.
⇒ ∠z = ∠s
The two equations above can now be substituted in the expression for the sum of the interior angles given by ∠x + ∠y + ∠z:
⇒ ∠x + ∠y + ∠z = ∠x + ∠r + ∠s
From the second figure of ∆ABC, it is evident that DE is a straight line and hence equals 180°. Further, since ∠x, ∠r, and ∠s all lie on DE, ∠x + ∠r + ∠s = 180°.
On substituting ∠r = ∠y and ∠s = ∠z in the equation:
∠x + ∠y + ∠z = 180°
This shows that the sum of the angles in a triangle is 180°, which means that the degrees in a triangle are 180°.
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