SOLUTION: In the diagram below, Angle DCE=Angle ACB, Angle BCF is a right angle, and CE ⊥ AB. Find the measure of Angle BAD Diagram: https://imgur.com/a/enff97E

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Question 1189508: In the diagram below, Angle DCE=Angle ACB, Angle BCF is a right angle, and CE ⊥ AB. Find the measure of Angle BAD

Diagram: https://imgur.com/a/enff97E
Found 2 solutions by math_tutor2020, Edwin McCravy:
Answer by math_tutor2020(3817)   (Show Source): You can put this solution on YOUR website!

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.

-----------------------------------------------------

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 = 123.5 which is the measure of angle BAD.

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Final Answer:
123.5 degrees
This value is exact.
Answer by Edwin McCravy(20055)   (Show Source): You can put this solution on YOUR website!


Since ∠BCF is a right angle, ΔBCF is a right triangle and
m∠FCG = 90o - 23o = 67o.

The fact that CE ⊥ AB, means that an extension of CE down to an 
extension of AB, the extensions will intersect at a right angle. So
we extend EC down to FA, which is a left extension of AB. Let G be 
the point where the extension of EC intersepts FA.



Now that we extended EC to G we see that ∠DCE ≅ ∠ACG because
they are vertical angles. We also see that 

Since ∠CGF is a right angle, ΔCGF is a right triangle and
m∠GCF = 90o - 67o = 23o.

Let the common measure of ∠DCE, ∠ACG, and ∠ACB be xo



Since Angle BCF is a right angle,









Since the three angles of ΔBAC must have sum 180o,

m∠BAD + m∠ABC + m∠BCA = 180o

m∠BAD + 23o + 33.5o = 180o

m∠BAD + 56.5o = 180o

m∠BAD = 123.5o

Edwin
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