ABCD is a convex quadrilateral satisfying AB=BC=CD,AD=DB and ∠BAD=75∘. What is the measure of ∠BCD?
Since AD=DB, triangle ABD is isosceles, so
∠DAB = ∠DBA = 75° and ∠ADB = 30°
Now draw AC, and again we have isosceles triangles ABC and BCD.
∠BCD = 75°
150
To find the measure of ∠BCD, we can use the given information and the properties of a convex quadrilateral.
Let's start by drawing the convex quadrilateral ABCD. Since AB=BC=CD, we know that all four sides have the same length.
Now, since AD=DB, we can conclude that triangle ABD is an isosceles triangle. This means that angles ∠ABD and ∠BAD are congruent.
We are given that ∠BAD = 75∘, so ∠ABD is also 75∘.
Now, we can use the fact that the sum of the angles of a quadrilateral is 360∘. Since we already know two angles (∠ABD and ∠BAD), we can find the measure of ∠BCD by subtracting the sum of the known angles from 360∘.
Sum of known angles = ∠ABD + ∠BAD = 75∘ + 75∘ = 150∘
∠BCD = 360∘ - Sum of known angles = 360∘ - 150∘ = 210∘
Therefore, the measure of ∠BCD is 210∘.