Deepen understanding of circle theorems including angles in the same segment, cyclic quadrilaterals, and the angle in a semicircle.
In a cyclic quadrilateral, opposite angles are supplementary (sum to 180°).
A cyclic quadrilateral has one angle of 75°. Find the opposite angle.
180°−75°=105°
Two opposite angles of a cyclic quadrilateral are (2x) and (x+30). Find x.
2x+x+30=180; 3x=150; x=50
The angle in a semicircle is always 90°. Angles subtended by the same arc in the same segment are equal.
A triangle is inscribed in a semicircle with one side as the diameter. Find the angle opposite the diameter.
90°, since the angle in a semicircle is always a right angle
Two angles in the same segment subtend the same arc. One is 55°. Find the other.
55°, since angles in the same segment are equal
1. A cyclic quadrilateral has one angle of 100°. Find the opposite angle.
2. Two opposite angles of a cyclic quadrilateral are (3x) and (x+40). Find x.
3. A triangle is inscribed in a semicircle. What is the angle opposite the diameter?
4. Two angles in the same segment subtend the same arc. One is 42°. Find the other.
5. A cyclic quadrilateral has angles x, x+10, x+20, x+30. Find x using the property that opposite angles are supplementary.