Answer:
True
Step-by-step explanation:
The perpendicular bisector of the opposite side to the vertex bisects the angle at the vertex into two equal parts and also bisects the triangle into two equal parts.
Let A be the angle at the vertex, then assume that the angle is an isosceles triangle with base angles B.
We need to show that A = 180 - 2B for an isoceles triangle
The perpendicular bisector bisects A into two so the new angle in the vertex one half of the bisected triangle is A/2.
Since this half triangle is a right-angled triangle, the third angle in it is 90.
So, A/2 + B + 90 = 180 (Sum of angles in a triangle)
subtracting 90 from both sides, we have
A/2 + B + 90 - 90 = 180 - 90
A/2 + B = 90
subtracting B from both sides, we have
A/2 + B = 90
A/2 = 90 - B
multiplying through by 2, we have
A = 2(90 - B)
A = 180 - 2B
Since A = 180 - 2B, then our triangle is an isosceles triangle.
