Week 3 Challenge Q17

[vibrantrobin]

I was reading the explanation for this question and I was confused about how we know that angle BFI is equal to angle FEH?

Thank you,
Lucy

[quacker88]

Great question, @vibrantrobin !
So, the solution uses two theorems. For \(\angle FEH\), it uses the Inscribed Angle Theorem.

As long as the vertex of the angle (in this diagram, it's point \(B\)) is on the circle, the degree measure of the angle is equal to half of the intercepted arc. Using this, in your problem, \(\angle FEH=\frac12 \widehat{FGH}\) (this isn't the correct symbol for arc but it's all I could find).

For \(\angle BFI\), it uses another theorem. It's a special case of when a tangent and a secant interesect ON the circle.

Whatever arc the two lines cut out, the angle is half of the arc measure. Using this, in your problem, \(\angle BFH = \angle BFI = \frac12 \widehat{FGH} \).

Since they are both equal to \(\frac12 \widehat{FGH}\), they are equal!

There are a lot of theorems that involve secants and tangents, like these three for example:

You should look more into it if you're interested! Hope this helps