Answer:
Explanation:
To prove that the sum of the angles of any triangle PQR is two right angles, we can use the fact that the sum of the angles of any polygon with n sides is (n-2) times the sum of its interior angles.
Since a triangle has three sides, it follows that the sum of its interior angles is (3-2) times the sum of its interior angles, which simplifies to:
Sum of interior angles of triangle PQR = 1 times sum of interior angles of triangle PQR.
Therefore, we can write:
Sum of interior angles of triangle PQR = angle P + angle Q + angle R
Now, draw a line segment from one vertex of the triangle, say vertex P, to a point on the opposite side, forming two angles with the original angles of the triangle as shown in the figure below.
Q
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P-------R
Since the line segment divides the triangle into two smaller triangles, we can apply the fact that the sum of the angles of any polygon with n sides is (n-2) times the sum of its interior angles to each of the smaller triangles.
Thus, in triangle PQR, we have:
angle P + angle Q + angle R = (angle P + angle X + angle Q) + (angle R + angle Y + angle X)
Simplifying, we have:
angle P + angle Q + angle R = angle P + angle Q + angle R + angle X + angle Y
Subtracting angle P + angle Q + angle R from both sides, we get:
angle X + angle Y = 180°
This means that the sum of the two angles formed by the line segment is 180 degrees, or they are a pair of supplementary angles.
Since the sum of the angles of the original triangle is equal to the sum of the angles of the two smaller triangles plus the sum of the two angles formed by the line segment, we have:
angle P + angle Q + angle R = (angle P + angle X + angle Q) + (angle R + angle Y + angle X) + angle X + angle Y
Simplifying, we get:
angle P + angle Q + angle R = 2(angle P + angle Q + angle R)
Therefore, the sum of the angles of any triangle PQR is two right angles, or 180 degrees.