One possible rigid transformation that could be done to prove congruency is a rotation about
point A. Since Angles CFB and DFE are vertical angles we know that they are congruent to each
other, because they are congruent to each other, we can use point f as a rotation point because it
is shared and congruent. By rotating triangle CFB 180 degrees about point f, we will be able to
see if the two figures are congruent or not.
Proof
Reasoning
Measure of angle M is congruent to angle X
Given
Measure of angle N is congruent to angle Y
Given
Line segment YO is congruent to line
segment NZ
Given
Line segment OZ is congruent to Line seg.
OZ
Reflexive prop.
Line seg. YZ is congruent to line seg. ON
Substitution prop.
Triangles MNO and XYZ are congruent
AAS
Since we already know that angle X and Z are congruent, line segments XY and ZY are
congruent and that Angle Y is congruent to itself using the reflexive property. In order to make
both triangles congruent angle B and A should be congruent, which would make both triangles
congruent under AAS.
Angle Z in the picture is being bisected by line segment ZX. since two of the sides on the
triangle are congruent, the triangle is isosceles. That means that the angle bisector also bisects
the base of the triangle. This makes line segment ZX the perpendicular bisector of line segment
WY.
We already know that angle CXY and BXY are congruent, as well as angles CAX and BAX are
congruent, and that line segments AC and AB are congruent. We want to prove that angles XCY
and XBY are congruent. Line segment AY is an angle bisector of the angle CAB, since triangle
CAB is equilateral and we know that if there is an angle bisector in an isosceles triangle, it will
bisect the base of it. Therefore, point Y is the midpoint of line segment CB, which makes line
segment CY and YB congruent. Since the line segment AY bisects line segment CB, two right
angles will form on the other side of the line. We can also see that line segment XY is congruent
to Xy by the reflexive property. From here either using SAS or ASA we can then determine
triangles CXY and BXY are congruent. Because we know they are congruent and that all angles
and sides are congruent, using CPCTC we can then determine that angles XCY and XBY are
congruent.