Example 2
Given BOC = 62 ° and DCA = 44°, find BAC and ABD.
BOC = BAC (both subtended by arc BC)
C
BAC =62 °
DCA = ABD (both subtended by arc DA)
ABD = 44 °
Exercise 11
Find the angles marked with letters. A line passes through the centre
only when point O is shown.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
B
Circle theorems
11.
12.
14.
15.
13.
16.
C
ABCD is a cyclic quadrilateral. The corners touch the circle.
c} The opposite angles in a cyclic quadrilateral add up to 180 °
(the angles are supplementary).
,,.._
,.._
A+ C = 180 °
B+ f> = 180 °
Proof:
Draw radii OA and OC.
Let ADC = x and ABC= y.
AOC obtuse= 2x
(angle at the centre)
AOC reflex= 2y
(angle at the centre)
2x + 2y = 360 (angles at a point)
°
x + y = 180 ° as required
d) The angle in a semicircle is a right angle.
In the diagram, AB is a diameter.
ACB = 90 ° .
Geometry