Proving circle theorums

Question 1

How do you prove that angles subtended from the diameter are 90 degrees?

Answer 1

1. Draw out the diagram
2. Split the example triangle into two isosceles triangles by drawing a line from the centre of the circle to the triangle’s point (ie a radius)
3. Label the equivalent angles in each isosceles triangle (a,a and b,b)
4. Show that the angle in question is a + b
5. Using the large triangle, put the angles into an equation :
   a+b+a+b=180 / 2a+2b=180
   > a+b=90
   90= a right angle.

Question 2

How do you prove angles at the centre are twice angles at the circumference?

Answer 2

1. Add in a radius from the centre through the x angle to form two isosceles triangles
2. Label the equivalent angles a,a and b and c.c and d and the other angle as e
3. You know that a + a + b =180 and c +c +d = 180
4. Use this to write the formula to find out the central angles : b + d +e = 360
   which is also : (180-2a) + (180-2c) + e =360
   or 360 -2a -2c +e =360
   and e - 2a -2c = 0
5. this means that e = 2a + 2c or 2(a+c) and seeing as a+c is our point angle which we can label as f,
   this proves that e=2f

Question 3

How do you prove angles in the same segment are equal?

Answer 3

1. Label your triangle angles in question a and b
2. draw in a second triangle coming from the same chord but with it’s point at the centre and label that angle c
3. Based on the theorum ‘ angles at the circumference are half angles at the centre’, angle c must be double angle a, however it must also be double angle b

therefore angles a and b have to be equal

Question 4

How do you prove opposite angles in a quadrilateral add to 180?

Answer 4

1. Pick two opposing angles and label them a and b
2. Draw two radii coming from the unlabelled angles and meeting at the centre
3. Based on the theorum ‘angles at the centre are half angles at the circumferance’, you can label the angles created by the two new lines 2a and 2b as they fit.
4. As angles around a point add to 360, you can say that 2a+2b = 360 or a+b=180

Question 5

How do you prove that alternate angles with a triangle and tangent are equal?

Answer 5

1. Create an example where the triangle is formed form the diameter and label the angles in question a and b
2. where the radius hits a tangent the angle is always 90 degrees, this also means that angle b + x = 90
3. Also, angles subtended at the circumference from a diameter are 90 so that can be labelled 90 as well
4. Angles in a triangle are 180 so in the triangle in question, 90 + x + a= 180
   or x + a = 90
5. Now you know that x+a=90 and x+b= 90 and so a and b must be equal