Question 1189508
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Let x be the measure of angle DCE. 


Because Angle DCE = Angle ACB, this means ACB is also x.


Extend out segment CE so it's a longer line. Make sure this longer line intersects AB. Label this intersection point G.


Note in the diagram that angles DCE and ACG are vertical angles. This means angle ACG is also x.


We're told that angle BCF is a right angle (aka 90 degrees). If we knew what angle FCG was, then we could solve for x.


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Angle FBC is 23 degrees in the diagram.


Focus on right triangle FBC.
Interior angle B is 23 degrees.
The other acute angle of this right triangle must be 90-23 = 67 degrees. 
This is angle F of triangle FBC.
More broadly, this is angle CFA.


Now focus your attention on triangle CFG. 
This is a right triangle due to CE ⊥ AB, ie the segments are perpendicular.


For triangle CFG, we found F = 67 earlier. This means C = 90-F = 90-67 = 23 degrees


Angle FCG = angle ABC = 23 degrees


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We now have enough info to solve for x.
Note that the angles FCG, GCA, and ACB are 23, x and x respectively. They combine to the angle BCF = 90.


So,
(angleFCG) + (angleGCA) + (angleACB) = angle BCF
(23) + (x) + (x) = 90
2x+23 = 90
2x = 90-23
2x = 67
x = 67/2
x = 33.5


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For the last part, we'll focus entirely on triangle ABC


We're given that B = 23 from the diagram.
We just found that C = 33.5 which was the measure of x.


A+B+C = 180
A+23+33.5 = 180
A+56.5 = 180
A = 180-56.5
A = <font color=red>123.5</font> which is the measure of angle BAD.


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Final Answer:
<font color=red>123.5 degrees</font>
This value is exact.
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