Example 1 As shown in Figure 1, the lengths of the sides $AB$, $BC$, $CD$, and $DA$ of quadrilateral $ABCD$ are 1, 9, 8, and 6, respectively. For the following statements:
(1) Quadrilateral $ABCD$ is circumscribed around a circle;
(2) Quadrilateral $ABCD$ is not inscribed in a circle;
(3) The diagonals are not perpendicular to each other;
(4) $\angle ADC \geqslant 90^{\circ}$;
(5) $\triangle BCD$ is an isosceles triangle.
The correct understanding is ( ').
(A)(1) true, (2) false, (4) true
(B)(3) true, (4) false, (5) true
(C)(3) true, (4) false, (5) false
(D)(2) false, (3) false, (4) true
Solution 1: If $\angle A D C=90^{\circ}$, then $A C=10$; if $\angle A D C>90^{\circ}$, then $A C>10$. From (4), we get $A C \geqslant 10$.

But in $\triangle A B C$, $A C A C^{2}$.
Therefore, $\angle A D C$ is an acute angle, which means proposition (4) is false, thus negating options (A) and (D). The options (B) and (C) are clearly judged by proposition (5), so we should proceed to proposition (5).
From $B D<A B+A D=7<C D<B C$, we know that $\triangle B C D$ is not an isosceles triangle, thus negating option (B).
Given that there is only one correct option, we should choose (C).