Circle theorem question?

Question

look at this picture: http://i104.photobucket.com/albums/m174/tomw91/circle.jpg

i need to prove that a=b using circle theorems.
please show me how - i'm stuck!
thanks

Answer 1

OK

Denote
angle PTQ = b
angle QPR = c
angle QTR = d


We know that c = d because they are Angles Subtended on the Same Arc.

We know that b+d is a right angle (90 degrees) because it is an Angle in a Semi-Circle (PR is the diameter of the circle)

We know that a+c is a right angle (90 degrees) because the tangent to the circle at the point P is perpendicular to the radius OP.

Hence b+d = a+c (because b+d = 90 and a+c =90)
But we said that c = d. Therefore
b+d = a+d
Subtract d
=> b=a

Hope this helps.

Answer 2

because RCT is a tangent to the circle and and the radius continuously meets a tangent at a suitable attitude. Therefor RCB + 36 could upload to ninety, meaning RCB = fifty 4. BADC is a cyclic quadrilateral because all 4 corners contact the circle. In this kind of structure, the different corners could upload to 100 and eighty, so D is one hundred forty four, so the attitude at ABC + one hundred forty four = 100 and eighty, hence ABC = 36. the attitude subtended on the circumference of the circle is 1/2 that of the attitude on the middle, so AOC = seventy 2. by technique of the commerce phase theorem, CAD = 23, an identical as DCT, so ACD = 13 as they could upload to 100 and eighty to form that triangle. also by technique of the commerce phase theorem, ACR = one hundred forty four, making OCA = fifty 4 because OCR is ninety. Now OAD + one hundred forty four + 13 + fifty 4 + seventy 2 could be 360 to make a quadrilateral, that means that OAD = seventy seven. wish this all helps

Answer 3

Pyramids.

Answer 4

a = b (tan chord therom)