ABCD is a convex quadrilateral satisfying AB=BC=CD, AD=DB and \angle BAD = 75^\circ. What is the measure of \angle BCD?
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To find the measure of ∠BCD, we can start by drawing a diagram of the given convex quadrilateral ABCD.
Since AB = BC = CD, we can mark the lengths of the sides of the quadrilateral accordingly.
Next, we know that AD = DB, which means that triangle ABD is an isosceles triangle. Since ∠BAD is given as 75°, we can find the measure of ∠ADB by dividing the remaining angle (180° - 75°) by 2, which is 52.5°.
Now, let's mark ∠ADB as 52.5° on the diagram and add the remaining angles ∠ABD and ∠BDA. We can see that ∠BDA is also 52.5° due to the isosceles triangle property.
Since the interior angles of a quadrilateral sum up to 360°, we can find the measure of ∠BCD by subtracting the sum of the other three angles (∠BDA, ∠ABD, and ∠BAD) from 360°.
∠BCD = 360° - (∠BAD + ∠ABD + ∠BDA)
∠BCD = 360° - (75° + 52.5° + 52.5°)
∠BCD = 360° - 180°
∠BCD = 180°
Therefore, the measure of ∠BCD is 180°.