Question

I am having trouble solving this Hinge Theorem Geometry proof. I understand the Hinge Theorem, but how is m<1 congruent to m<2?

Answer 1

Please see below.

Explanation:

We can use cosine formula for triangles.

We have `RQ=ST`

also `m/_1>m/_2` as `/_1` is exterior angle and is hence sum of `/_2` and `m/_QTS`

and as both angles are less than `180^@`, we have `cos(/_1)< cos(/_2)`

Now using cosine formula in `DeltaQRS`, we have

`RS^2=RQ^2+QS^2-2RQxxQSxxcos(/_1)` ..................(A)

and in `DeltaQST`, we have

`QT^2=ST^2+QS^2-2STxxQSxxcos(/_2)` ..................(B)

Compare RHS of (A) and (B),

we have `RQ=ST`, but as `cos(/_1)< cos(/_2)`

`RS^2 > QT^2`

i.e. `RS>QT`