Let's start by drawing a diagram:
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B
/ \
/ \
m / \ n
/ \
/_ _ _ _ _\
A c C
We know that the median m is half the length of ac, so we can label ac as 2m.
We also know that the median m makes an angle of 32 degrees with side ac. Since the median m also bisects bc, we know that angles Cbm and Abm are congruent, so each must measure (180 - 32)/2 = 74 degrees.
Now we can use the fact that the angles of a triangle add up to 180 degrees to solve for angle bac:
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angle BAC = 180 - angle ABC - angle ACB
We can find angle ABC as follows:
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angle ABC = angle ABm + angle Cbm
Since Abm is a right angle and Cbm is 74 degrees, we have:
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angle ABC = 90 + 74 = 164
Similarly, we can find angle ACB as follows:
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angle ACB = angle ACm + angle CBm
Since ACm is a right angle and CBm is 74 degrees, we have:
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angle ACB = 90 + 74 = 164
Now we can plug in these values to find angle BAC:
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angle BAC = 180 - angle ABC - angle ACB
= 180 - 164 - 164
= 180 - 328
= -148
This answer doesn't make sense since angles are measured between 0 and 180 degrees. So we made a mistake somewhere. Let's check our work.
We found that ABC is 164 degrees, which means that angle BCA must be (180 - 164)/2 = 8 degrees. But this is a problem since angle BCA should be greater than angle ACB, which we found to be 164 degrees. So we made a mistake when we assumed that angles Cbm and Abm are congruent.
Let's try again:
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B
/ \
/ \
m / \ n
/ \
/_ _ _ _ _\
A c C
Since the median m bisects bc, we know that angles Cbm and Abn are congruent, so each must measure (180 - 32)/2 = 74 degrees.
Now we can find angle BAC as follows:
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angle BAC = angle BAm + angle CAM
Since BAm and CAM are congruent, we have:
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angle BAC = 2 * angle BAm
We know that angle BAm is equal to 90 - 32/2 = 74 degrees, so:
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angle BAC = 2 * 74
= 148
Therefore, the measure of angle bac is 148 degrees.