英文三角形數學題(15分)

(a)

Since angle EBC is 90 degree, therefore, BC = EC cos angle ECB
Similarly, angle EBA is 90 degree, therefore, BD = AD cos angle BDA

Given BD=BC and AD=EC
EC cos angle ECB = BC = BD = AD cos angle BDA
cos angle ECB = cos angle BDA
angle ECB = angle BDA

AB = AD sin angle BDA = BD sin ECB = BE
therefore all angles and length of triange ABD and triangle EBC are the same

(b)
since AB=BE, therefore angle BEA = 45 degree
From triangle ADE, total interior angle = 180 degree

angle DAE + angle ADE + angle AED = 180 degree

since angle BEA = 45 degree, angle AED = 135 degree
therefore angle DAE + angle ADE = 45 degree, or

From above, we know that angle ECB = angle ADE
therefore angle ECB = 45 degree - angle DAE