Rectangle ABCD has AB=6 and BC=3. Point M is chosen on side AB so that ∠ AMD=∠ CMD. What is the degree measure of ∠ AMD?

2011 AMC 12B Problem 10

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10.

Rectangle

ABCD

has

AB=6

and

BC=3.

Point

M

is chosen on side

AB

so that

\angle AMD=\angle CMD.

What is the degree measure of

\angle AMD?

15

30

45

60

75

Since

AB\parallel CD,

the angle

\angle CDM

equals

\angle AMD

That makes

\triangle CMD

isosceles with

CM=CD=6,

\triangle MBC

30

60

90

Because

AB\parallel CD,

we have

\angle CDM=\angle AMD.

Combined with

\angle AMD=\angle CMD,

this gives

\angle CDM=\angle CMD,

\triangle CMD

is isosceles with

CM=CD=6.

Then

\triangle MBC

is right-angled at

B

with hypotenuse

CM=6

and leg

BC=3,

so it is a

30

60

90^\circ

triangle with

\angle BMC=30^\circ.

Finally,

\angle AMD+\angle CMD

+\angle BMC=180^\circ,

2\angle AMD+30^\circ=180^\circ,

giving

\angle AMD=75^\circ.

Thus, the correct answer is E.