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Thus it is proved that the measure of the inscribed angle is half that of the intercepted arc.

Theorem: If two inscribed angles intercept the same arc or arcs of equal measure then the inscribed angles have equal measure.


Figure 7.7

In figure 7.7 Ð CAB and Ð CDB intercept the same arc CXB.

Prove that m Ð CAB = Ð CDB.

From the previous theorem it is known that

m Ð CAB = m (arc CXB) and also

m Ð CDB = m (arc CXB)

\ m Ð CAB = m Ð CDB

Therefore if two inscribed angles intercept the same arc or arcs of equal measure the two inscribed angles are equal in measure.

Theorem: If the inscribed angle intercepts a semicircle the inscribed angle measures 900.


Figure 8.8

The inscribed angle Ð ACB intercepts a semicircle arc AXB (figure 8.8). We have to prove that m Ð ACB = 900.

m Ð ACB = m (arc AXB)

= (1800)

= 900

Therefore if an inscribed angle intercepts a semicircle the inscribed angle is a right angle.

Example 1

a) In the above figure name the central angle of arc AB.

b) In the above figure what is the measure of arc AB.

c) Name the major arc in the above figure.

Solution:

a) Ð AOB

b) 800. The measure of an arc is the measure of its central angle.

c) Arc AXB

Example 2

a) In the above figure name the inscribed angle and the intercepted arc.

b) What is m (arc PQ)

Solution:

a) inscribed angle - Ð PRQ

intercepted arc - arc PQ

b) 600. The measure of an intercepted arc is twice the measure of its inscribed angle.

Example 3

Ð PAQ and Ð PBQ intercept the same arc PQ what is the m Ð PBQ and m (arc PQ) ?

Solution:

m Ð PBQ = 400 If two inscribed angles intercept the same arc their measures are equal m (arc PQ) = 800 as m (arc) = 2m (inscribed angle).

Index

7.1 Introduction
7.2 Lines of circle
7.3 Arcs
7.4 Inscribed angels
7.5 Some properties od tangents, secants and chords
7.6 Chords and their arcs
7.7 Segments of chords secants and tangents
7.8 Lengths of arcs and area of sectors

Chapter 8

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