ABCD is a convex quadrilateral satisfying AB=BC=CD, AD=DB and \angle BAD = 75^\circ. What is the measure of \angle BCD?
In order to find the measure of angle BCD, we can start by drawing a diagram and labeling relevant information.
We are given that ABCD is a convex quadrilateral with AB = BC = CD, AD = DB, and angle BAD = 75 degrees.
Let's label the angles in the quadrilateral as follows:
Angle BAD = 75 degrees (given)
Angle ABC = x degrees (to be determined)
Angle BCD = y degrees (to be determined)
Angle CDA = 75 degrees (since angle BAD = angle CDA)
Angle ABD = 180 - Angle BAD - Angle ABC = 180 - 75 - x degrees
Angle BDA = 180 - Angle ABD - Angle BAD = 180 - (180 - 75 - x) - 75 = x degrees
Since AD = DB, we have a isosceles triangle ABD. Therefore, angle BAD = angle ABD. So, we can conclude that angle DBA = 75 degrees.
Since angles BDA and DBA add up to 180 degrees, we have:
x + x + 75 = 180
2x + 75 = 180
2x = 180 - 75
2x = 105
x = 52.5
Now, we can find the measure of angle BCD.
In triangle BCD, we know that angles BCD and CBD are equal since BC = CD.
So, angle BCD = angle CBD = y degrees.
Since angles BCD, ABD, and BDA add up to 180 degrees, we have:
y + (180 - 75 - x) + x = 180
y + 180 - 75 - 52.5 = 180
y + 52.5 = 180 - 180 + 75
y + 52.5 = 75
y = 75 - 52.5
y = 22.5
Therefore, the measure of angle BCD is 22.5 degrees.
To find the measure of angle BCD, we can start by using the given information and applying the properties of a convex quadrilateral.
Given:
AB = BC = CD
AD = DB
∠BAD = 75°
Since AB = BC = CD, we can conclude that angles ABC and BCD are congruent. Let's denote the measure of angle ABC (which is also the measure of angle BCD) as x.
Since ABCD is a convex quadrilateral, the sum of the angles in a quadrilateral is 360°. We can use this information to set up an equation:
∠BAD + ∠ABC + ∠BCD + ∠CDA = 360°
Substituting the given values:
75° + 75° + x + 75° = 360°
Simplifying the equation:
225° + x = 360°
Subtracting 225° from both sides:
x = 360° - 225°
x = 135°
Therefore, the measure of ∠BCD is 135°.