Because BD=DC
So the triangle BDC is an isosceles triangle
So angle BDC= angle DCB
So angle ABC= angle ACB
So AB=AC
Because the angle between the two sides of triangle ABD and triangle ADC is the same,
So triangle ABD=ACD
Advertising extension line to BC, focus e
Because angle BAE= angle CAE, AB = AC.
So triangle AEB=AEC, angle AEB= angle AEC.
Angle AEB+ angle AEC= 180 degrees.
So AEB angle =90 degrees.
So vertical
The second question:
According to Pythagorean theorem
Triangle ABF is a right triangle.
Because: AB=8, BC=AD= 10.
So: AF= 10, BF=6.
So: CF=BC-BF= 10-6=4.
Because angle BAC+ angle AFB=90 degrees
Angle AFE=90 degrees
So angle EFC= angle BAF
Because CF=4
So ef = 5 and ce = 3.
Third question
When angle 1= angle 3= angle 2
AB=AD
Triangle ABD is an isosceles triangle.
Angle ABD= angle ADB
Angle CAB= angle EAD
Angle ADE= 180 degrees-angle 3- angle ADB= 180 degrees-angle 1- angle ABD= angle ABD.
So the two angles in the triangle ABC and the triangle AED are the same.
Similarly, these three angles are the same as triangles with similar phases.
Because AB=AD, three angles and one side of two triangles are the same.
So BC=DE
The fourth question
Because the angle 1 equals the angle 2, the angle ADO= the angle AEO=90 degrees.
So AD=AE
Because angle AEB= angle ADC=90 degrees
So AB=AC
Because BD=AB-AD, CE=AC-AE.
So BD=CE
The fifth question
Make an auxiliary line, d is perpendicular to AB, and focus e.
Because the angle BD=AD, the triangle ABD is an isosceles triangle.
Because AB=6CM
So AE=BE=3CM.
Because angle DBC= angle DBA
Angle ACB= Angle DEB=90 degrees.
So two triangles are similar triangles.
Because the sides BD are the same, BC=BE=3cm.