Given a circle with chord AB and diameter CD, prove that triangle ABD is isosceles if AB is perpendicular to CD.

Statements: Reasons:

1.AE=BE 1.Opposite the equal sides.

2.angle AED = Angle BED

SSS
congruent
triangle EAD = triangle EBD

To prove that triangle ABD is isosceles when chord AB is perpendicular to diameter CD, we can use the properties of perpendicular lines in a circle.

Let's start by identifying some key properties:
1. In a circle, a diameter that is perpendicular to a chord bisects the chord.
2. If a line is perpendicular to a chord, then it bisects the chord.

Now, let's prove that triangle ABD is isosceles:
1. Given that AB is perpendicular to CD, we know that CD is a diameter of the circle.
2. From Property 1, since CD is a diameter, it bisects AB. Therefore, AC = CB.
3. Since AC = CB, triangle ABC is isosceles.

Now, let's prove that ABD is isosceles:
1. Since AB is perpendicular to CD (given), AB bisects CD.
2. Let's denote the point of intersection of AB and CD as E.
3. By Property 2, AB bisects CD at point E, which means CE = DE.
4. Now, in triangle ABD, we have CE = DE (from step 3) and AC = CB (from triangle ABC).
5. By the transitive property of equality, we can conclude that AC = CE = DE = CB.
6. Therefore, triangle ABD is isosceles with AD = BD.

In conclusion, when chord AB in a circle is perpendicular to diameter CD, triangle ABD is isosceles with AD = BD.