Circles

Opposite angles sum to 180°, All four vertices lie on a circle, It can be inscribed in a circle

A cyclic quadrilateral has all four vertices on a circle, which means it can be inscribed in a circle. The key property is that opposite angles sum to 180° (Theorem 9.10). Adjacent angles are not necessarily equal, and diagonals are not necessarily equal unless it's a special case like rectangle.

90°

An angle inscribed in a semicircle is always a right angle (90°). This follows from Theorem 9.7: the arc of semicircle subtends 180° at center, so angle at circumference = 180° ÷ 2 = 90°. This is a fundamental property of circles.

5 cm

The perpendicular from center bisects the chord, so half of AB = 3 cm. In the right triangle formed by radius, perpendicular distance, and half-chord: radius² = 4² + 3² = 16 + 9 = 25. Therefore, radius = √25 = 5 cm.

70°

According to Theorem 9.8, angles in the same segment of a circle are equal. Since both ∠ABC and ∠ADC are inscribed angles subtending the same arc BC from points on the same side, they must be equal. Therefore, ∠ADC = ∠ABC = 70°.