A Math

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E Math (IGCSE)

H1 Math

H2 Math

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N(A) A Math

N(A) E Math

PSLE

PSLE Foundation Mathematics

<p>In the diagram, \(TA\) and \(TB\) are tangents from \(T\) to the circle with centre \(O\) at \(A\) and \(B\), respectively. \(AC\) is a diameter of the circle. Prove that</p>
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<ol class="i-list"><li data-list-item-id="item-i" data-list-item-path="item-i"><div class="list-item-paragraph">angle \(AOB = 2 \times\) angle \(ACB\),</div></li><li data-list-item-id="item-ii" data-list-item-path="item-ii"><div class="list-item-paragraph">\(CB\) is parallel to \(OT\),</div></li><li data-list-item-id="item-iii" data-list-item-path="item-iii"><div class="list-item-paragraph">triangles \(OBC\) and \(TAB\) are similar, and</div></li><li data-list-item-id="item-iv" data-list-item-path="item-iv"><div class="list-item-paragraph">quadrilateral \(AOBT\) is a cyclic quadrilateral, i.e., there exists a circle that passes through \(A\), \(O\), \(B\) and \(T\).</div></li></ol>

In the diagram, \(TA\) and \(TB\) are tangents from \(T\) to the circle with centre \(O\) at \(A\) and \(B\), respectively. \(AC\) is a diameter of the circle. Prove that

triangles \(OBC\) and \(TAB\) are similar, and

quadrilateral \(AOBT\) is a cyclic quadrilateral, i.e., there exists a circle that passes through \(A\), \(O\), \(B\) and \(T\).

Angles at the Centre and Circumference

Alternate Segment Theorem

Tangents Perpendicular to Radii

Similarity

Proving Concyclicity

Angles Formed by Two Parallel Lines and a Transversal

Equal Tangents from an External Point

Proofs in Plane Geometry

<p>In the diagram, \(O\) is the centre of the larger circle. \(AOB\), \(AKC\) and \(BKD\) are straight lines. Prove that</p>
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<ol class="i-list"><li data-list-item-id="item-i" data-list-item-path="item-i"><div class="list-item-paragraph">\(DK = CK\) ,</div></li><li data-list-item-id="item-ii" data-list-item-path="item-ii"><div class="list-item-paragraph">\(AB\) is parallel to \(CD\) , and</div></li><li data-list-item-id="item-iii" data-list-item-path="item-iii"><div class="list-item-paragraph">\(DKB\) bisects angle \(ODC\) .</div></li></ol>
<p>As shown, \(A,B,C,D\) lie on the larger circle and \(A,O,D,K\) lie on the smaller circle.</p>

In the diagram, \(O\) is the centre of the larger circle. \(AOB\), \(AKC\) and \(BKD\) are straight lines. Prove that

As shown, \(A,B,C,D\) lie on the larger circle and \(A,O,D,K\) lie on the smaller circle.

Properties of Special Triangles

Angles in the Same Segment

<p>In the diagram, \(O\) is the centre of the smaller circle. \(ABC\) and \(AED\) are straight lines. Prove that</p>
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<ol class="i-list"><li data-list-item-id="item-i" data-list-item-path="item-i"><div class="list-item-paragraph">\(AC = AD\), and</div></li><li data-list-item-id="item-ii" data-list-item-path="item-ii"><div class="list-item-paragraph">\(BE\) is parallel to \(CD\).</div></li></ol>
<p>As shown in the diagram, \(A,B,O,D\) lie on one circle, and \(B,C,D,E\) lie on the circle with centre \(O\).</p>

In the diagram, \(O\) is the centre of the smaller circle. \(ABC\) and \(AED\) are straight lines. Prove that

As shown in the diagram, \(A,B,O,D\) lie on one circle, and \(B,C,D,E\) lie on the circle with centre \(O\).

Angles in Cyclic Quadrilaterals

<p>In the diagram, \(TAR\) is the tangent to the circle at \(A\). \(DCA\) and \(DOB\) are straight lines. It is given that angle \(BAR = 3 \times \text{angle}\hspace{0.5em} OCB\), prove that</p>
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<ol class="i-list"><li data-list-item-id="item-i" data-list-item-path="item-i"><div class="list-item-paragraph">angle \(ODC = 2 \times \text{angle}\hspace{0.5em} OBC\),</div></li><li data-list-item-id="item-ii" data-list-item-path="item-ii"><div class="list-item-paragraph">triangle \(COD\) is an isosceles triangle, and</div></li><li data-list-item-id="item-iii" data-list-item-path="item-iii"><div class="list-item-paragraph">angle \(DCO = \text{angle}\hspace{0.5em} CAT + 90^\circ\).</div></li></ol>

In the diagram, \(TAR\) is the tangent to the circle at \(A\). \(DCA\) and \(DOB\) are straight lines. It is given that angle \(BAR = 3 \times \text{angle } OCB\), prove that

angle \(ODC = 2 \times \text{angle } OBC\),

triangle \(COD\) is an isosceles triangle, and

angle \(DCO = \text{angle } CAT + 90^\circ\).

<p>In the diagram, the smaller circle has centre \(O\) and intersects the larger circle at \(B\) and \(C\). The point \(O\) also lies on the larger circle. \(ABD\) and \(ACE\) are straight lines, where&nbsp;\(A\) is on the smaller circle and \(D, E\) are on the larger circle. \(F\) is the point of intersection of \(BE\) and \(CD\). Prove that</p>
<div class="graph-block" data-graph-config="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" data-width="219" data-height="248" data-proportional style="width:219px;height:248px;max-width:100%;display:flow-root;margin:0 auto;--graph-width:219px;--graph-aspect:219 / 248;"></div>
<ol class="i-list"><li data-list-item-id="item-i" data-list-item-path="item-i"><div class="list-item-paragraph">angle \(AEB = 180^\circ - 2 \times\) angle \(BAC\) ,</div></li><li data-list-item-id="item-ii" data-list-item-path="item-ii"><div class="list-item-paragraph">triangle \(ABE\) is an isosceles triangle, and</div></li><li data-list-item-id="item-iii" data-list-item-path="item-iii"><div class="list-item-paragraph">angle \(DFE = 360^\circ - 3 \times\) angle \(BAC\) .</div></li></ol>

In the diagram, the smaller circle has centre \(O\) and intersects the larger circle at \(B\) and \(C\). The point \(O\) also lies on the larger circle. \(ABD\) and \(ACE\) are straight lines, where \(A\) is on the smaller circle and \(D, E\) are on the larger circle. \(F\) is the point of intersection of \(BE\) and \(CD\). Prove that

angle \(AEB = 180^\circ - 2 \times\) angle \(BAC\) ,

triangle \(ABE\) is an isosceles triangle, and

angle \(DFE = 360^\circ - 3 \times\) angle \(BAC\) .

<p>In the figure, \(O\) is the centre of the circle \(BCDE\) and \(ABC\) and \(AED\) are straight lines, prove that</p>
<div class="content-block content-block--columns"><div class="question-columns question-columns--2" data-layout="2"><div class="question-column" style="flex: 1; align-self: flex-start;"><ol class="i-list"><li data-list-item-id="item-1780398684393-mvhu24dmu" data-list-item-path="item-1780398684393-mvhu24dmu"><div class="list-item-paragraph">\(AD=AC\),</div></li><li data-list-item-id="item-1780398684841-fkn4cdoq8" data-list-item-path="item-1780398684841-fkn4cdoq8"><div class="list-item-paragraph">Given that&nbsp;\(AB=AE\), show that&nbsp;\(BE\) is parallel to \(CD\).</div></li></ol></div><div class="question-column" style="flex: 1; align-self: flex-start;"><div class="content-block"><figure class="question-figure alignright" style="text-align: right;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1768111313403-sdv4ekep96g.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjgxMTEzMTM0MDMtc2R2NGVrZXA5NmcucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MjcyNTY3MiwiZXhwIjoxNzkyNzI1OTcyfQ.zmp3A3vWAPFMSijl2gKQJ_SqKDin2CoirdKJc25GZ-0" style="max-width: min(220px, 100%); height: auto; display: block;" class="alignright custom-width" /></figure></div></div></div></div>
<p>As shown in the diagram, \(A,B,O,D\) lie on one circle, and \(B,C,D,E\) lie on the circle with centre \(O\).</p>

In the figure, \(O\) is the centre of the circle \(BCDE\) and \(ABC\) and \(AED\) are straight lines, prove that

<p>In the figure, \(TA\) and \(TB\) are tangents to the circle, centre \(O\), from \(T\).</p>
<div class="content-block content-block--columns"><div class="question-columns question-columns--2-1" data-layout="2-1"><div class="question-column" style="flex: 2; align-self: flex-start;"><ol class="i-list"><li><div class="list-item-paragraph">Given that&nbsp;\(AOC\) is a diameter of the circle, state a reason why \(\angle AOB=2\angle ACB\).</div></li></ol><p>Hence, prove that</p><ol class="i-list" start="2"><li><div class="list-item-paragraph">\(CB\) is parallel to \(OT\),</div></li><li><div class="list-item-paragraph">A circle with diameter \(OT\) can be drawn to pass through \(A\) and \(B\),</div></li><li><div class="list-item-paragraph">\(\Delta OBC\) is similar to \(\Delta TAB\).</div></li></ol></div><div class="question-column" style="flex: 1; align-self: flex-start;"><div class="content-block"><figure class="question-figure alignright" style="text-align: right;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1768110672714-c5voub3uhv7.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjgxMTA2NzI3MTQtYzV2b3ViM3VodjcucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MjcyNTY3MiwiZXhwIjoxNzkyNzI1OTcyfQ.Uilk80Idx2fR-54TfK5gCaADsXvgFSJyzpNktt7sn60" style="max-width: min(200px, 100%); height: auto; display: block;" class="alignright custom-width" /></figure></div></div></div></div>

In the figure, \(TA\) and \(TB\) are tangents to the circle, centre \(O\), from \(T\).

<p>In the figure, \(O\) is the centre of the circle \(ABCD\) and it intersects the circle \(AOCE\) at \(A\) and \(C\). Given that&nbsp;\(ECB\) and \(ADE\) are straight lines, prove that</p>
<div class="content-block content-block--columns"><div class="question-columns question-columns--2" data-layout="2"><div class="question-column" style="flex: 1; align-self: flex-start;"><ol class="i-list"><li><div class="list-item-paragraph">\(DE = CE\),</div></li><li><div class="list-item-paragraph">\(\angle CDE = \angle BAD\).</div></li></ol></div><div class="question-column" style="flex: 1; align-self: flex-start;"><div class="content-block"><figure class="question-figure alignright" style="text-align: right;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1768107234018-pwb2bh37fue.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjgxMDcyMzQwMTgtcHdiMmJoMzdmdWUucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MjcyNTY3MiwiZXhwIjoxNzkyNzI1OTcyfQ.d0dWtczYYIGn7eeC2ZIoJeh5bwcUn6D8SU8W7O3Ut2I" style="max-width: min(210px, 100%); height: auto; display: block;" class="alignright custom-width" /></figure></div></div></div></div>

In the figure, \(O\) is the centre of the circle \(ABCD\) and it intersects the circle \(AOCE\) at \(A\) and \(C\). Given that \(ECB\) and \(ADE\) are straight lines, prove that

<p>In the figure, \(O\) is the centre of the circle \(ABCD\). The smaller circle \(AOKD\) intersects the larger circle at \(A\) and \(D\). Given that&nbsp;\(AOB\), \(AKC\) and \(BKD\) are straight lines, prove that</p>
<div class="content-block content-block--columns"><div class="question-columns question-columns--2" data-layout="2"><div class="question-column" style="flex: 1; align-self: flex-start;"><ol class="i-list"><li><div class="list-item-paragraph">\(DK = CK\),</div></li><li><div class="list-item-paragraph">\(BKD\) bisects \(\angle ODC\).</div></li></ol></div><div class="question-column" style="flex: 1; align-self: flex-start;"><div class="content-block"><figure class="question-figure alignright" style="text-align: right;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1768107139966-543ba3xiig6.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjgxMDcxMzk5NjYtNTQzYmEzeGlpZzYucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MjcyNTY3MiwiZXhwIjoxNzkyNzI1OTcyfQ.gcl9AquMw8W2dC1vjVb4NK3Nq0EF4j7JKde_p6Hu7X8" style="max-width: min(170px, 100%); height: auto; display: block;" class="alignright custom-width" /></figure></div></div></div></div>

In the figure, \(O\) is the centre of the circle \(ABCD\). The smaller circle \(AOKD\) intersects the larger circle at \(A\) and \(D\). Given that \(AOB\), \(AKC\) and \(BKD\) are straight lines, prove that

<p>In the figure, \(AE\) is a tangent to the circle whose centre is \(O\) and \(ABC\) is a straight line.<br>\(\angle BAE = 38{}^\circ\), \(\angle BOE = 72{}^\circ\) and \(\angle DCE = 40{}^\circ\).<br>Find, stating your reasons clearly</p>
<div class="content-block content-block--columns"><div class="question-columns question-columns--2" data-layout="2"><div class="question-column" style="flex: 1; align-self: flex-start;"><ol class="abc-list"><li><div class="list-item-paragraph">\(\angle ECB\),</div></li><li><div class="list-item-paragraph">\(\angle OEC\),</div></li><li><div class="list-item-paragraph">\(\angle CED\),</div></li><li><div class="list-item-paragraph">\(\angle OBA\).</div></li></ol></div><div class="question-column" style="flex: 1; align-self: flex-start;"><div class="content-block"><figure class="question-figure alignright" style="text-align: right;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1767602304815-a9qazn86x5.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc2MDIzMDQ4MTUtYTlxYXpuODZ4NS5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzkyNzI1NjcyLCJleHAiOjE3OTI3MjU5NzJ9.753aAiDWX3EKWVrJv3FXeWTvrP9dG7B_JZjzrK0h0T8" style="max-width: min(250px, 100%); height: auto; display: block;" class="alignright custom-width" /></figure></div></div></div></div>

In the figure, \(AE\) is a tangent to the circle whose centre is \(O\) and \(ABC\) is a straight line.
\(\angle BAE = 38{}^\circ\), \(\angle BOE = 72{}^\circ\) and \(\angle DCE = 40{}^\circ\).
Find, stating your reasons clearly

Determining Angular Values from Diameters, Chords, and Arcs

Properties of Circle

<p>In the diagram, \(A\), \(B\), \(C\) and \(D\) are points on the circumference of the circle centre \(O\).<br>\(ABE\) and \(DCE\) are straight lines and \(ST\) is a tangent to the circle at \(A\).<br>\(ST\) is parallel to \(DE\), \(\angle CEB=36{}^\circ\) and \(\angle DAB=83{}^\circ\).</p>
<div class="content-block content-block--columns"><div class="question-columns question-columns--2" data-layout="2"><div class="question-column" style="flex: 1; align-self: flex-start;"><ol class="abc-list"><li><div class="list-item-paragraph">Calculate</div><ol class="i-list"><li><div class="list-item-paragraph">angle \(OAB\),</div></li><li><div class="list-item-paragraph">angle \(ABC\),</div></li><li><div class="list-item-paragraph">angle \(AOD\).</div></li></ol></li><li><div class="list-item-paragraph">Show that&nbsp;\(ACD\) is an isosceles triangle.</div></li></ol></div><div class="question-column" style="flex: 1; align-self: flex-start;"><div class="content-block"><figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1767601914948-dxhsqkds96b.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc2MDE5MTQ5NDgtZHhoc3FrZHM5NmIucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MjcyNTY3MiwiZXhwIjoxNzkyNzI1OTcyfQ.QICSpGqvovhXU5C5AWrRCd5dHE4_3yVJbZ26Ar3I6Pw" style="max-width: min(400px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div></div></div></div>

In the diagram, \(A\), \(B\), \(C\) and \(D\) are points on the circumference of the circle centre \(O\).
\(ABE\) and \(DCE\) are straight lines and \(ST\) is a tangent to the circle at \(A\).
\(ST\) is parallel to \(DE\), \(\angle CEB=36{}^\circ\) and \(\angle DAB=83{}^\circ\).

Investigating Geometric Proofs through Circle Angle Invariants

<div class="content-block content-block--columns"><div class="question-columns question-columns--2" data-layout="2"><div class="question-column" style="flex: 1; align-self: flex-start;"><p>In the diagram, \(O\) is the centre of the circle.<br>\(PQ\) is parallel to \(SR\), \(\angle PRS=42{}^\circ\) and \(\angle OSR=28{}^\circ\).<br>Calculate</p><ol class="abc-list"><li><div class="list-item-paragraph">\(\angle POS\),</div></li><li><div class="list-item-paragraph">\(\angle PQR\),</div></li><li><div class="list-item-paragraph">\(\angle OPR\).</div></li></ol></div><div class="question-column" style="flex: 1; align-self: flex-start;"><div class="content-block"><figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1767601424278-uidoomrf8j.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc2MDE0MjQyNzgtdWlkb29tcmY4ai5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzkyNzI1NjcyLCJleHAiOjE3OTI3MjU5NzJ9.P-bAB7dsP5m6coSZ2UE9JCCeFhLmf-MzcjbL28_sBMo" style="max-width: min(170px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div></div></div></div>

Theoretical Foundations of Subtended Angles and Arcs

<p>Find the value of all the unknowns in each diagram, stating your reasons clearly.</p>
<div class="content-block content-block--columns"><div class="question-columns question-columns--2" data-layout="2"><div class="question-column" style="flex: 1; align-self: flex-start;"><ol class="abc-list"><li><div class="content-block list-item-image" style="text-align: left;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1767600941766-ve646enj0k.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc2MDA5NDE3NjYtdmU2NDZlbmoway5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzkyNzI1NjcyLCJleHAiOjE3OTI3MjU5NzJ9.xX2071F1OemnJ8Uxq3DC3L8u4o0Q8Ouaf4_-71Jr0xw" alt="List item image" class="alignleft custom-width" style="max-width: min(170px, 100%); height: auto;" /></div></li></ol></div><div class="question-column" style="flex: 1; align-self: flex-start;"><ol class="abc-list" start="2"><li><div class="content-block list-item-image" style="text-align: left;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1767600949603-hw00mfenavm.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc2MDA5NDk2MDMtaHcwMG1mZW5hdm0ucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MjcyNTY3MiwiZXhwIjoxNzkyNzI1OTcyfQ.LJBSyXYpw21y3UrDVIed-aSTHi0XzrCeNfhZeScW41U" alt="List item image" class="alignleft custom-width" style="max-width: min(170px, 100%); height: auto;" /></div></li></ol></div></div></div>

Find the value of all the unknowns in each diagram, stating your reasons clearly.

Angles in a Semicircle and the Converse

Geometric Invariants within Circular Segments

<p>Find the value of all the unknowns in each diagram, stating your reasons clearly.</p>
<div class="content-block content-block--columns"><div class="question-columns question-columns--2" data-layout="2"><div class="question-column" style="flex: 1; align-self: flex-start;"><ol class="abc-list"><li><div class="content-block list-item-image" style="text-align: left;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1767600680009-piavagfuvo.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc2MDA2ODAwMDktcGlhdmFnZnV2by5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzkyNzI1NjcyLCJleHAiOjE3OTI3MjU5NzJ9._Vh2CvXnsqdSE1I8HeEUNdhEZjCckRNcy4667HB3jZM" alt="List item image" class="alignleft custom-width" style="max-width: min(180px, 100%); height: auto;" /></div></li></ol></div><div class="question-column" style="flex: 1; align-self: flex-start;"><ol class="abc-list" start="2"><li><div class="content-block list-item-image" style="text-align: left;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1767600690575-tcaxv2nr0w.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc2MDA2OTA1NzUtdGNheHYybnIwdy5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzkyNzI1NjcyLCJleHAiOjE3OTI3MjU5NzJ9.M3U4sd7e7bs5LzESKaNV8LW72ASBZ08dNU7P9CRuK4E" alt="List item image" class="alignleft custom-width" style="max-width: min(170px, 100%); height: auto;" /></div></li></ol></div></div></div>

Angles in the Same Segment, at the Centre and in a Semicircle

<p>Find the value of all the unknowns in each diagram, stating your reasons clearly.</p>
<div class="content-block content-block--columns"><div class="question-columns question-columns--2" data-layout="2"><div class="question-column" style="flex: 1; align-self: flex-start;"><ol class="abc-list"><li><div class="content-block list-item-image" style="text-align: left;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1767600504325-bgri1il66cv.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc2MDA1MDQzMjUtYmdyaTFpbDY2Y3YucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MjcyNTY3MiwiZXhwIjoxNzkyNzI1OTcyfQ.HSpJw4qCafhVK5xyWQt6rVxnlZ-gYNTD4kfTwN8d0c8" alt="List item image" class="alignleft custom-width" style="max-width: min(170px, 100%); height: auto;" /></div></li></ol></div><div class="question-column" style="flex: 1; align-self: flex-start;"><ol class="abc-list" start="2"><li><div class="content-block list-item-image" style="text-align: left;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1767600510983-bhahkm6c3h.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc2MDA1MTA5ODMtYmhhaGttNmMzaC5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzkyNzI1NjcyLCJleHAiOjE3OTI3MjU5NzJ9.BCT9sKPbFv6QLgtwv4ccCEeLeVBx3t6yqgcJx3crXDQ" alt="List item image" class="alignleft custom-width" style="max-width: min(150px, 100%); height: auto;" /></div></li></ol></div></div></div>

Circle Geometry: Angles in a Semi-Circle and Thales' Theorem

<p>Find the value of all the unknowns in each diagram, stating your reasons clearly.</p>
<div class="content-block content-block--columns"><div class="question-columns question-columns--2" data-layout="2"><div class="question-column" style="flex: 1; align-self: flex-start;"><ol class="abc-list"><li><div class="content-block list-item-image" style="text-align: left;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1767600341778-7hkan2duf8b.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc2MDAzNDE3NzgtN2hrYW4yZHVmOGIucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MjcyNTY3MiwiZXhwIjoxNzkyNzI1OTcyfQ.Qxd5i4cYqve2Oeb2Bgu9E-QF8zrV0-oJfW0XCp7mev8" alt="List item image" class="alignleft custom-width" style="max-width: min(150px, 100%); height: auto;" /></div></li></ol></div><div class="question-column" style="flex: 1; align-self: flex-start;"><ol class="abc-list" start="2"><li><div class="content-block list-item-image" style="text-align: left;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1767600345531-ovc6f9h606r.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc2MDAzNDU1MzEtb3ZjNmY5aDYwNnIucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MjcyNTY3MiwiZXhwIjoxNzkyNzI1OTcyfQ.4yeffZj_hKQEKV0Xu_3quh8LkM9VHXz9C0TllQr2zcc" alt="List item image" class="alignleft custom-width" style="max-width: min(150px, 100%); height: auto;" /></div></li></ol></div></div></div>

Angle Properties of Circles: Fundamental Theorems and Applications

<p>Find the value of all the unknowns in each diagram, stating your reasons clearly.</p>
<div class="content-block content-block--columns"><div class="question-columns question-columns--2" data-layout="2"><div class="question-column" style="flex: 1; align-self: flex-start;"><ol class="abc-list"><li><div class="content-block list-item-image" style="text-align: left;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1767600129448-deon2wf3nu.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc2MDAxMjk0NDgtZGVvbjJ3ZjNudS5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzkyNzI1NjcyLCJleHAiOjE3OTI3MjU5NzJ9.lyDGoOU7L6Y1TsHJHjC4bxbRxLgKS9C6g3jFsMamqIo" alt="List item image" class="alignleft custom-width" style="max-width: min(150px, 100%); height: auto;" /></div></li></ol></div><div class="question-column" style="flex: 1; align-self: flex-start;"><ol class="abc-list" start="2"><li><div class="content-block list-item-image" style="text-align: left;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1767600135362-32pkiuqg9dr.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc2MDAxMzUzNjItMzJwa2l1cWc5ZHIucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MjcyNTY3MiwiZXhwIjoxNzkyNzI1OTcyfQ.5bamS9zAQNM5J9f9fuyttvv1X1NG1B2_aIF0JDwtbC8" alt="List item image" class="alignleft custom-width" style="max-width: min(150px, 100%); height: auto;" /></div></li></ol></div></div></div>

Angles Subtended by an Arc at the Centre and Circumference

<div class="content-block content-block--columns"><div class="question-columns question-columns--2" data-layout="2"><div class="question-column" style="flex: 1; align-self: flex-start;"><p>In the diagram, the points \(A\), \(B\), \(C\), \(D\) lie on a circle centre \(O\).<br>Given that&nbsp;\(\angle BOD=100{}^\circ\) and \(\angle ADO=20{}^\circ\), calculate</p><ol class="abc-list"><li><div class="list-item-paragraph">\(\angle BAD\),</div></li><li><div class="list-item-paragraph">\(\angle BCD\),</div></li><li><div class="list-item-paragraph">\(\angle OBD\),</div></li><li><div class="list-item-paragraph">\(\angle ABO\).</div></li></ol></div><div class="question-column" style="flex: 1; align-self: flex-start;"><div class="content-block"><figure class="question-figure alignright" style="text-align: right;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1767599940165-33598q5qj38.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc1OTk5NDAxNjUtMzM1OThxNXFqMzgucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MjcyNTY3MiwiZXhwIjoxNzkyNzI1OTcyfQ.SrECtSPQjWKm6-3N8iV_xLi2MgmQjKpmz381A2STMjc" style="max-width: min(180px, 100%); height: auto; display: block;" class="alignright custom-width" /></figure></div></div></div></div>

Angle Properties of Circles: Theorems and Deductions

<p>Find the value of all the unknowns in each diagram, stating your reasons clearly.</p>
<div class="content-block content-block--columns"><div class="question-columns question-columns--2" data-layout="2"><div class="question-column" style="flex: 1; align-self: flex-start;"><ol class="abc-list"><li><div class="content-block list-item-image" style="text-align: left;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1767598975386-t1axg1j608n.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc1OTg5NzUzODYtdDFheGcxajYwOG4ucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MjcyNTY3MiwiZXhwIjoxNzkyNzI1OTcyfQ.fBR7sLCAvB2mPCebOR-fuHmtvbSY5k0I8W7O51cRP_0" alt="List item image" class="alignleft custom-width" style="max-width: min(150px, 100%); height: auto;" /></div></li></ol></div><div class="question-column" style="flex: 1; align-self: flex-start;"><ol class="abc-list" start="2"><li><div class="content-block list-item-image" style="text-align: left;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1767598980853-yk2wumep3wi.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc1OTg5ODA4NTMteWsyd3VtZXAzd2kucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MjcyNTY3MiwiZXhwIjoxNzkyNzI1OTcyfQ.PKT7wEtp0-NufsKy18lVqQiXfusam3JtQPKalRwL3bg" alt="List item image" class="alignleft custom-width" style="max-width: min(150px, 100%); height: auto;" /></div></li></ol></div></div></div>

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