PB and PE bisect angles APC and FPC respectively and A, P and F are collinear points. Sum of angles BPA and EPF is equal to:
Answers
Step-by-step explanation:
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To understand this better, let's consider the angle bisectors of APC and FPC. When two line segments are angle bisectors of a triangle, they divide the triangle into two smaller triangles with equal area. This means that the angle bisectors also bisect the angles of the triangle, meaning that the angles formed by the angle bisectors and the vertices of the original triangle are equal.
So in this case, since PB bisects angle APC and PE bisects angle FPC, we have BPA = CPA/2 and EPF = FPC/2.
Since the sum of the interior angles in a triangle must equal 180 degrees, we can write the following equation for triangle APC:
BPA + APC + CPA = 180 degrees
Substituting BPA = CPA/2 and simplifying, we get:
CPA/2 + APC + CPA/2 = 180 degrees
APC = 180 degrees - CPA
Similarly, for triangle FPC, we have:
EPF + FPC + CPF = 180 degrees
Substituting EPF = FPC/2 and simplifying, we get:
FPC/2 + FPC + CPF/2 = 180 degrees
FPC = 180 degrees - CPF
Finally, since BPA + EPF = (CPA/2) + (FPC/2), we get:
BPA + EPF = (180 degrees - CPA) / 2 + (180 degrees - CPF) / 2
BPA + EPF = 180 degrees - (CPA + CPF) / 2
Since CPA + CPF = 180 degrees, we can substitute and simplify to get:
BPA + EPF = 180 degrees - (180 degrees) / 2
BPA + EPF = 180 degrees - 90 degrees
BPA + EPF = 90 degrees
Therefore, the sum of angles BPA and EPF is equal to 90 degrees.
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