The measures of the angles in quadrilateral BDFE are as follows: m∠BDF = 100, m∠EFC = 100, m∠EBG = 260 - (m∠CBD + m∠FDG), and m∠BFE = 110, satisfying the Polygon Interior Angle-Sum Theorem for a quadrilateral.
Here are the steps to find the measures of the angles of quadrilateral BDFE:
Find the measures of the angles of ACBD and AFDG.
To find the measures of the angles of ACBD, we can use the fact that the sum of the angles of a triangle is 180 degrees. Therefore, we have:
m∠ACB + m∠CBD + m∠BDA = 180
m∠ACB + m∠CBD + 50 = 180
m∠ACB + m∠CBD = 130
Similarly, to find the measures of the angles of AFDG, we can use the same fact and get:
m∠AFD + m∠FDG + m∠GDA = 180
m∠AFD + m∠FDG + 50 = 180
m∠AFD + m∠FDG = 130
Use the Angle Addition Postulate to find m∠BDF.
The Angle Addition Postulate states that if a point lies on the interior of an angle, then the measure of the angle is equal to the sum of the measures of the two adjacent angles formed by the point. Therefore, we have:
m∠BDF = m∠BDA + m∠FDA
m∠BDF = 50 + 50
m∠BDF = 100
Use the Angle Addition Postulate to find m∠EFC.
Similarly, we can use the Angle Addition Postulate to find m∠EFC as follows:
m∠EFC = m∠FDC + m∠CDE
m∠EFC = 50 + 50
m∠EFC = 100
Use the Angle Addition Postulate to find m∠EBG.
We can also use the Angle Addition Postulate to find m∠EBG as follows:
m∠EBG = m∠EBD + m∠DBG
m∠EBG = m∠CBD + m∠FDG
m∠EBG = 130 - m∠CBD + 130 - m∠FDG
m∠EBG = 260 - (m∠CBD + m∠FDG)
Use the Polygon Interior Angle-Sum Theorem to find m∠BFE.
The Polygon Interior Angle-Sum Theorem states that the sum of the measures of the interior angles of a convex n-gon is (n-2)180 degrees. Therefore, for a quadrilateral, we have:
m∠BFE + m∠EFC + m∠CFD + m∠BDF = (4-2)180
m∠BFE + m∠EFC + m∠CFD + m∠BDF = 360
m∠BFE + 100 + 50 + 100 = 360
m∠BFE = 110