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.