a)

Draw the radius OU first.

SVU = 180° - STU (opposite angles of a cyclic quadrilateral add up to 180°. SVUT is the cyclic quadrilateral in this case)

OVU = SVU (common angle)

OV = OU (both are radius of same circle)


So triangle OVU is isosceles.

OVU = OUV (base angles of isosceles triangle)

SOU = OVU + OUV (exterior angle = sum of 2 interior opposite angles)

= 2 OVU (since OUV = OVU)



b)


OUP and OSP = 90°
(tangent perpendicular to radius)


SPU = 360° - OUP - OSP
(sum of angles in a quadrilateral add up to 360°, the quadrilateral in this case is SOUP)



c)

TSP = TWS
(angles in alternate segments

VST = VSP - TSP (adjacent angles of the same angle VSP)


VSP = OSP = 90° (common angle)

so VST = OSP - TSP


d)

VUT = 180° - VST

(Opposite angles of a cyclic quadrilateral add up to 180°. SVUT is the cyclic quadrilateral in this case)

J: please note that 'opposite angles, cyclic quadrilateral' is now, under syllabus 4048 ' opposite angles of the same segment are supplementary'

As Eric has pointed out, its now:
Angles in opposite segments are supplementary

Thanks for the heads up

Thanks for all the explanations!!!

Most welcome, J and Cheryl.

Just to clarify, would writing ' opposite angles of a cyclic quadrilateral are supplementary/add up to 180°' be penalised?

Interesting to know this. Didn’t realise they have changed it.

I highly doubt they should penalise it. The thing is that the quadrilateral has to be cyclic, in the sense that the four vertices must lie on the circumference of the triangle. So I believe either one should be ok.

It’s currently phrased as

“angles in opposite angles are supplementary”.

For quite a while already I was wondering why this phrase has not been used before, considering that “angles in the same segment are equal” already suggests that drawing a fourth point on the other side of the chord instead would make the angles automatically in opposite segments.

Hi Eric, you made a spelling error - have posted the pertinent page from syllabus 4048 as an answer, for reference. Thanks for the feedback.

J and Eric: Honestly cannot say with certainty whether older, factually correct reasons will be accepted by Cambridge examiners, but why take the risk? I advise my students to follow the phrasing used within the syllabus, whether math or science.

The latest syllabus is available for download as a PDF from www.seab.gov.sg
SEAB = Singapore Examinations and Assessment Board, which oversees all National Exams from PSLE to the A-Levels.

True that. Better to stay safe