prove: the bisector...?

Question

The bisector of the angle opposite the base of an isosceles triangle bisects the base and is perpendicular to the base.

Answer 1

Let ABC be an isoscekes triangle.Let AD,the bisector of To prove that AD=DC and AD is perpendicular on BC
Proof: In triangles ABD and ACD,
AB=AC [equal sides of isosceles triangle]
Side AD is common
Therefore,the two triangles are congruent[SAS congruency]
Therefore,BD=CD,and
or.AD is perpendicular on BC

Answer 2

Hi,

Remember that definition of an isoceles triangle (ABC) is that angle ABC = angle ACB, and side AB = side AC.

Now construct segment AD that bisects angle BAC:

Note that the sum of angles in a triangle are 180degrees.

Now since we've bisected the top angle (BAC), and the total sum of angles in the new triangle (ABD) are 180.

Because all triangles have 180degrees
180 = (angle ABD) + (angle BDA) + (angle DAB)
since we bisected the original triangle
90 = (angle ABD) + 1/2 (angle ABC)
which is the same as
90 = (angle ABD) + (angle DAB)
Then again the new triangle must have 180 degrees
180 = (angle ABD) + (angle DAB) + (angle BDA)
then
(angle BDA) = 90 degrees, and Triangles ABD, and ACD are congruent, making segments BD, and DC congruent.

QED.

Hope that helps, sorry about the messy notation. It's tough to write geometry with a keyboard.
Matt

Answer 3

Let the triangle be ABC wher AB is the base and C is the vertex. Draw the bisector of angle c and let it meet the base at point D.

Since triangle is isosceles, AC = BC
angle A = angle B [base angles of iisosceles triangle are =]
CD bisects angle C [Given]
Therefore angle ACD = angle BCD [definition of bisector]
Therefore triangle ACD congruent triangle BCD [ASA]
Therefore AD = BD [CPCTC]
Therefore CD bisects AB [definition of bisector]

Answer 4

Draw the triangle and the bisector to touch the base.
The sides are equal. Hence the opposite angles are also equal.
The angles on the sides of the bisector are equal. Two angles and the included sides being equal, the two triangles are identical.So the base is bisected.
Since the triangles are identical the angles are 90 degrees. Q.E.D.

Answer 5

its quite simple if u know the basics of geometry and the explanation is quite simple with a figure.
simply follow the site:

http://www.sonoma.edu/users/w/wilsonst/Courses/Math_150/Theorems/Isosceles/I3.html