two equal squares, ABCD and DEFG, have the vertex D in common. The angle between the two squares is 60 degrees: what is the angle ACF?
After making your sketch it should be very easy to see that triangle DCE is equilateral and angle ACD = 45°
so angle ACF = 45 = 60 or 105°
To find the angle ACF, we need to analyze the given information and the relationship between the two squares.
First, let's draw the two squares ABCD and DEFG with vertex D in common:
A------B
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D------C
E------F
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G------H
We know that the angle between the two squares is 60 degrees. Let's call the point where the diagonal AC intersects line EH as point I:
A------B
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| I |
D------C
E------F
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G------H
Since we have a square, we know that each angle within the square is 90 degrees. Thus, we can deduce that angle BAC is equal to 90 degrees.
Now, let's consider triangle BAI. We have:
1) Angle BAI
2) Angle IAB (which is 90 degrees)
To find angle BAI, we can use the fact that the sum of the angles in a triangle is 180 degrees. Therefore, angle BAI can be calculated as:
Angle BAI = 180 degrees - angle IAB - angle ABI
Since angle IAB is 90 degrees, we can simplify it further:
Angle BAI = 180 degrees - 90 degrees - angle ABI
= 90 degrees - angle ABI
Now, let's focus on triangle ABI. We have:
1) Angle ABI
2) Angle IAB (which is 90 degrees)
3) Angle BAI (which we just calculated as 90 degrees - angle ABI)
To find angle ABI, we can use the fact that the sum of the angles in a triangle is 180 degrees. Therefore, angle ABI can be calculated as:
Angle ABI = 180 degrees - angle IAB - angle BAI
= 180 degrees - 90 degrees - (90 degrees - angle ABI)
= 180 degrees - 90 degrees - 90 degrees + angle ABI
= angle ABI
From this calculation, we can conclude that:
Angle ABI = Angle ABI
Since the two angles are equal, we know that angle ABI is 60 degrees (as given in the question).
Now, let's go back to triangle BAI and substitute the value of angle ABI:
Angle BAI = 90 degrees - angle ABI
= 90 degrees - 60 degrees
= 30 degrees
Finally, let's find angle ACF. Since angle ACF is the same as angle BAI, we can conclude that:
Angle ACF = 30 degrees
Therefore, the angle ACF is 30 degrees.