A Math

A Math (IGCSE)

E Math

H1 Math

H2 Math

IB AA HL Math

IB AA SL Math

IB AI HL Math

IB AI SL Math

IP Math

A polygon has n sides. When the number of sides is doubled, its interior angle increases by 20. Find n.

A polygon has $n$ sides. When the number of sides is doubled, its interior angle increases by $20^\circ$. Find $n$.

Polygons

2025 SJI PRELIMS P1 Q5

<p>In the figure below, the surface of a modern football is made of <strong>20</strong> white hexagons and $x$ black pentagons, where each pentagon is surrounded by <strong>5</strong> hexagons and each hexagon is surrounded by <strong>3</strong> pentagons.</p>
<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/1767769381727-jovzf50mpm.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc3NjkzODE3Mjctam92emY1MG1wbS5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzg5NzcxMDI4LCJleHAiOjE3ODk3NzEzMjh9.EWmiZnRCKTFDglBwRW5NlZ9PBeR5UYsqBOFYvEzVJ6g" style="max-width: min(300px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<ol class="i-list"><li><div class="list-item-paragraph">Find the value of $x$.</div></li><li><div class="list-item-content"><span class="list-item-text">Find the number of vertices on a football, i.e., the number of points where the angles of black pentagons meet with the angles of white hexagons.</span></div></li></ol>

In the figure below, the surface of a modern football is made of **20** white hexagons and $x$ black pentagons, where each pentagon is surrounded by **5** hexagons and each hexagon is surrounded by **3** pentagons.

Find the number of vertices on a football, i.e., the number of points where the angles of black pentagons meet with the angles of white hexagons.

<p>Given the figure below, find the sum of angles <em>a, b, c, d, e, f, g, h, i</em> and <em>j </em>.</p>
<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/1767769252271-07c6gn85593l.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc3NjkyNTIyNzEtMDdjNmduODU1OTNsLnBuZyIsInNjb3BlIjoiZG93bmxvYWQiLCJpYXQiOjE3ODk3NzEwMjgsImV4cCI6MTc4OTc3MTMyOH0.GTz_H2R-6grxS_BhQrgbsgcPDhMaNK35VkGqdswGmPA" style="max-width: min(300px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>

Given the figure below, find the sum of angles *a, b, c, d, e, f, g, h, i* and *j *.

<p>In the figure below, <strong>ABCDE</strong> is a regular pentagon, <strong>ABMN</strong> and <strong>AEQP</strong> are squares.</p>
<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/1767769077295-ftyyzy06ow4.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc3NjkwNzcyOTUtZnR5eXp5MDZvdzQucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc4OTc3MTAyOCwiZXhwIjoxNzg5NzcxMzI4fQ.VntCN6BzaO1LgES6jkyLcLZDJ2RFpD_HHxMkoXxZYiA" style="max-width: min(350px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Find the size of angle <strong>BAE</strong>.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find the size of angle <strong>CDF</strong>.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find the size of angle <strong>BFE</strong>.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find the size of angle <strong>NAP</strong>.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find the size of angle <strong>ABF</strong>.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find the size of angle <strong>QEF</strong>.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Hence determine the type of quadrilateral <strong>ABFE</strong>, stating your reasons clearly.</span></div></li></ol>

In the figure below, **ABCDE** is a regular pentagon, **ABMN** and **AEQP** are squares.

Hence determine the type of quadrilateral **ABFE**, stating your reasons clearly.

<p>In the figure below, $ABCD$ and $ACEF$ are rhombuses, where $AE$ intersects $CD$ and $CF$ at $M$ and $N$, respectively. Given that angle $ACB = 26^\circ$, find the sizes of:</p>
<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/1767768855941-woh8nihn5r.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc3Njg4NTU5NDEtd29oOG5paG41ci5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzg5NzcxMDI4LCJleHAiOjE3ODk3NzEzMjh9.jAwt7m9n5kr3MoTThniOrjWMtMKMOjaroKTkq4myCY8" style="max-width: min(400px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">angle $CEN$,</span></div></li><li><div class="list-item-content"><span class="list-item-text">angle $CDF$,</span></div></li><li><div class="list-item-content"><span class="list-item-text">angle $DMN$,</span></div></li><li><div class="list-item-content"><span class="list-item-text">angle $MCN$.</span></div></li><li><div class="list-item-paragraph">Suppose now, given that angle $ACB \neq 26^\circ$ and triangle $CMN$ is a right-angled isosceles triangle, where $CN = MN$.</div><div class="list-item-paragraph">Find the new size of angle $ACB$.</div></li></ol>

In the figure below, $ABCD$ and $ACEF$ are rhombuses, where $AE$ intersects $CD$ and $CF$ at $M$ and $N$, respectively. Given that angle $ACB = 26^\circ$, find the sizes of:

Triangles

Quadrilaterals

Triangles, Quadrilaterals

<p>In the figure below, $ABCD$ is a rhombus. Produce $CB$ to $E$ such that $BD = BE$. Given that angle $BCD = 60°$, find the sizes of</p>
<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/1767768565074-t29rr082xul.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc3Njg1NjUwNzQtdDI5cnIwODJ4dWwucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc4OTc3MTAyOCwiZXhwIjoxNzg5NzcxMzI4fQ.LTwJIpLZKxUOuR3sjwtngy5f73TkdYF_LpSp705IuwM" style="max-width: min(300px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">angle $ADB$,</span></div></li><li><div class="list-item-content"><span class="list-item-text">angle $ABC$,</span></div></li><li><div class="list-item-content"><span class="list-item-text">angle $BED$,</span></div></li><li><div class="list-item-content"><span class="list-item-text">angle $BFE$.</span></div></li><li><div class="list-item-paragraph">Determine the type of triangles $ABD$ and $BCD$.</div></li><li><div class="list-item-paragraph">Construct the line $AE$. Determine the type of quadrilateral $AEBD$, stating your reasons clearly.</div></li></ol>

In the figure below, $ABCD$ is a rhombus. Produce $CB$ to $E$ such that $BD = BE$. Given that angle $BCD = 60°$, find the sizes of

<p>In the figure below, $ABC$ is a triangle enclosed by straight lines $MN$, $PQ$ and $RS$, where angles such as angle $CAN$, angle $ABR$ and angle $ACS$ are known as exterior angles of triangle $ABC$.</p>
<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/1767768078907-mxh93o674t.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc3NjgwNzg5MDctbXhoOTNvNjc0dC5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzg5NzcxMDI4LCJleHAiOjE3ODk3NzEzMjh9.y0ZMFqgsIS8iJgcT_fPCUk8fMInbvR_danMCfhz1upo" style="max-width: min(400px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Using this figure, show that angle $CAN = $ angle $ABC + $ angle $ACB$, i.e., the size of an exterior angle at a vertex of a triangle equals the sum of the sizes of the interior angles at the other two vertices of the triangle (remote interior angles), which is known as the Exterior Angle Theorem.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Apply the Exterior Angle Theorem on the other two exterior angles, angle $ABR$ and angle $ACS$, by stating the relationship between the exterior angles and corresponding interior angles.</span></div></li></ol>

In the figure below, $ABC$ is a triangle enclosed by straight lines $MN$, $PQ$ and $RS$, where angles such as angle $CAN$, angle $ABR$ and angle $ACS$ are known as exterior angles of triangle $ABC$.

Using this figure, show that angle $CAN = $ angle $ABC + $ angle $ACB$, i.e., the size of an exterior angle at a vertex of a triangle equals the sum of the sizes of the interior angles at the other two vertices of the triangle (remote interior angles), which is known as the Exterior Angle Theorem.

Apply the Exterior Angle Theorem on the other two exterior angles, angle $ABR$ and angle $ACS$, by stating the relationship between the exterior angles and corresponding interior angles.

<p>In the figure below, a number of congruent (identical) regular $k$-gons are placed in a ring to enclose a regular decagon, of which $ABCDE$ is a part.</p>
<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/1767686515751-x4oglx5md6.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc2ODY1MTU3NTEteDRvZ2x4NW1kNi5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzg5NzcxMDI4LCJleHAiOjE3ODk3NzEzMjh9.RsyZXX26OntdyQyiJC5LfpHUSGmlXaLF7kzEHyiDd38" style="max-width: min(350px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Find the size of angle $ABC$ and the value of $k$.</span></div></li><li><div class="list-item-content"><span class="list-item-text">If a number of congruent regular $n$-gons are placed in a ring to enclose a regular $m$-gon, find the relationship between $m$ and $n$ by expressing $m$ in terms of $n$.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Hence, or otherwise, explain whether a nonagon can be enclosed by placing a number of congruent regular polygons in a ring.</span></div></li></ol>

In the figure below, a number of congruent (identical) regular $k$-gons are placed in a ring to enclose a regular decagon, of which $ABCDE$ is a part.

Find the size of angle $ABC$ and the value of $k$.

If a number of congruent regular $n$-gons are placed in a ring to enclose a regular $m$-gon, find the relationship between $m$ and $n$ by expressing $m$ in terms of $n$.

Hence, or otherwise, explain whether a nonagon can be enclosed by placing a number of congruent regular polygons in a ring.

 Polygons

<p>In the figure below, $ABCDE$ is part of a regular dodecagon ($12$-gon). $P$ is on the extension of $ED$. Find the sizes of</p>
<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/1767686079507-z4z0kyywjhq.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc2ODYwNzk1MDctejR6MGt5eXdqaHEucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc4OTc3MTAyOCwiZXhwIjoxNzg5NzcxMzI4fQ.PWihpHY_p6WK8iyurC561IFqvjUb8GlDo8q10nZ8cUo" style="max-width: min(350px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">angle $ABC$,</span></div></li><li><div class="list-item-content"><span class="list-item-text">angle $CED$,</span></div></li><li><div class="list-item-content"><span class="list-item-text">angle $BCE$,</span></div></li><li><div class="list-item-content"><span class="list-item-text">angle $CDP$.</span></div></li><li><div class="list-item-paragraph">Hence find the sum of interior angles and the sum of exterior angles of the regular dodecagon, respectively.</div></li></ol>

In the figure below, $ABCDE$ is part of a regular dodecagon ($12$-gon). $P$ is on the extension of $ED$. Find the sizes of

<p>In the figure below, $XABCY$ is part of a regular polygon, $ABGFED$ is a regular hexagon and $BCHG$ is a square, where $AB, BC$ and $BG$ are common sides. Find the sizes of</p>
<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/1767685957380-kznp43tgaaa.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc2ODU5NTczODAta3pucDQzdGdhYWEucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc4OTc3MTAyOCwiZXhwIjoxNzg5NzcxMzI4fQ.RwDzd60u88s8mFsJUAK3CygLKAhqA32X551s2qJl5BA" style="max-width: min(300px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">angle $ABG$,</span></div></li><li><div class="list-item-content"><span class="list-item-text">angle $ABC$,</span></div></li><li><div class="list-item-content"><span class="list-item-text">angle $DAX$,</span></div></li><li><div class="list-item-content"><span class="list-item-text">angle $HCY$.</span></div></li><li><div class="list-item-paragraph">Hence find the number of sides and the sum of exterior angles of the polygon, respectively.</div></li></ol>

In the figure below, $XABCY$ is part of a regular polygon, $ABGFED$ is a regular hexagon and $BCHG$ is a square, where $AB, BC$ and $BG$ are common sides. Find the sizes of

<p>In the figure below, $ABCDEF$ and $AGHIJB$ are regular hexagons, where $AB$ is the common side. Find the sizes of</p>
<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/1767685818197-wbz20kd5tzg.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc2ODU4MTgxOTctd2J6MjBrZDV0emcucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc4OTc3MTAyOCwiZXhwIjoxNzg5NzcxMzI4fQ.NH6i6p1nU1LvNo4khMnelXiIVe4xK9G_d3aQLozKE7I" style="max-width: min(300px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">angle $BCD$,</span></div></li><li><div class="list-item-content"><span class="list-item-text">angle $AGB$,</span></div></li><li><div class="list-item-content"><span class="list-item-text">angle $AFD$,</span></div></li><li><div class="list-item-content"><span class="list-item-text">angle $BDF$.</span></div></li><li><div class="list-item-paragraph">Hence determine the type of quadrilateral $BGFD$, stating your reasons clearly.</div></li></ol>

In the figure below, $ABCDEF$ and $AGHIJB$ are regular hexagons, where $AB$ is the common side. Find the sizes of

<p>In the figure below, $ABCD$ is a kite, where the diagonals $AC, BD$ intersect at $O$. $E$ is a point on $BD$, such that $AECD$ is a rhombus. Given that angle $EAO = 33^\circ$, angle $BCF = 87^\circ$, where $F$ is a point on the extension of $DC$, find the sizes of</p>
<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/1767685675672-jf2j3h56sdi.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc2ODU2NzU2NzItamYyajNoNTZzZGkucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc4OTc3MTAyOCwiZXhwIjoxNzg5NzcxMzI4fQ.-OlVrwDOrwFCr38BdhIiy0teDZcz6pfLrC0wIdHSYDo" style="max-width: min(300px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">angle $ADO$,</span></div></li><li><div class="list-item-content"><span class="list-item-text">angle $AEC$,</span></div></li><li><div class="list-item-content"><span class="list-item-text">angle $DCE$,</span></div></li><li><div class="list-item-content"><span class="list-item-text">angle $BCE$.</span></div></li><li><div class="list-item-paragraph">Hence determine the type of triangle $ABC$, stating your reasons clearly.</div></li></ol>

In the figure below, $ABCD$ is a kite, where the diagonals $AC, BD$ intersect at $O$. $E$ is a point on $BD$, such that $AECD$ is a rhombus. Given that angle $EAO = 33^\circ$, angle $BCF = 87^\circ$, where $F$ is a point on the extension of $DC$, find the sizes of

<p>In the figure below, $ABCD$ is a rhombus, where diagonals $AC$ and $BD$ intersect at $O$. $EOF$ is a straight line segment that passes through $O$ and intersects sides $AB, CD$ at $E$ and $F$, respectively. Given that angle $BCO = 24^\circ$, angle $BEO = 36^\circ$, find the sizes of</p>
<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/1767685553927-74cnl96hhqh.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc2ODU1NTM5MjctNzRjbmw5NmhocWgucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc4OTc3MTAyOCwiZXhwIjoxNzg5NzcxMzI4fQ.LyYrwLqbYH9jpRHpYRnnkeLDufQj9lNE1pbo_7gPraA" style="max-width: min(400px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">angle $ADO$,</span></div></li><li><div class="list-item-content"><span class="list-item-text">angle $CFO$,</span></div></li><li><div class="list-item-content"><span class="list-item-text">angle $DOE$,</span></div></li><li><div class="list-item-content"><span class="list-item-text">angle $COE$.</span></div></li></ol>

In the figure below, $ABCD$ is a rhombus, where diagonals $AC$ and $BD$ intersect at $O$. $EOF$ is a straight line segment that passes through $O$ and intersects sides $AB, CD$ at $E$ and $F$, respectively. Given that angle $BCO = 24^\circ$, angle $BEO = 36^\circ$, find the sizes of

<p>In the figure below, $ABCD$ is a trapezium, where $AB$ is parallel to $CD$ and $AD = CD$. Produce $CB$ such that $AB = BE$. Given that angle $ADC = 94^\circ$, angle $ABC = 58^\circ$, answer the following questions:</p>
<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/1767685404303-v1no5pwdqsl.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc2ODU0MDQzMDMtdjFubzVwd2Rxc2wucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc4OTc3MTAyOCwiZXhwIjoxNzg5NzcxMzI4fQ.X3FU-QHvL-vn3P13a_u_eF8cuaNXsAYRls1NLx4od24" style="max-width: min(400px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<ol class="abc-list"><li><div class="list-item-paragraph">Show that $AC$ is the bisector of angle $BAD$.</div></li></ol>
<p>Find the sizes of</p>
<ol class="abc-list" start="2"><li><div class="list-item-content"><span class="list-item-text">angle $AEB$,</span></div></li><li><div class="list-item-content"><span class="list-item-text">angle $DAE$,</span></div></li><li><div class="list-item-content"><span class="list-item-text">angle $BCD$,</span></div></li><li><div class="list-item-content"><span class="list-item-text">angle $ACB$.</span></div></li></ol>

In the figure below, $ABCD$ is a trapezium, where $AB$ is parallel to $CD$ and $AD = CD$. Produce $CB$ such that $AB = BE$. Given that angle $ADC = 94^\circ$, angle $ABC = 58^\circ$, answer the following questions:

<p>In the figure below, $ABCD$ is a right-angled trapezium, where $AD$ is parallel to $BC$ and $AB$ is perpendicular to $BC$. $E$ and $F$ are on sides $AD$ and $CD$, respectively. Given that angle $AEB = 40^\circ$, angle $CBF = 16^\circ$ and angle $BCF = 54^\circ$, find the sizes of</p>
<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/1767685224809-zut1018a7l.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc2ODUyMjQ4MDktenV0MTAxOGE3bC5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzg5NzcxMDI4LCJleHAiOjE3ODk3NzEzMjh9.H1WAL_RIZevTmwr7RT2vntN72bwY5Q8WPCSjPTA3YyE" style="max-width: min(400px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">angle $ABC$,</span></div></li><li><div class="list-item-content"><span class="list-item-text">angle $EDF$,</span></div></li><li><div class="list-item-content"><span class="list-item-text">angle $EBF$,</span></div></li><li><div class="list-item-content"><span class="list-item-text">angle $BFD$.</span></div></li></ol>

In the figure below, $ABCD$ is a right-angled trapezium, where $AD$ is parallel to $BC$ and $AB$ is perpendicular to $BC$. $E$ and $F$ are on sides $AD$ and $CD$, respectively. Given that angle $AEB = 40^\circ$, angle $CBF = 16^\circ$ and angle $BCF = 54^\circ$, find the sizes of

Triangles and Quadrilaterals

<p>In the figure below, $ABCD$ is a square. $E$ and $F$ are on sides $AD$ and $CD$, respectively, such that $BEF$ is an equilateral triangle. Fold triangle $BEF$ along $BE$ to obtain quadrilateral $EGBF$. Given that angle $DEF = 45^\circ$, find the sizes of</p>
<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/1767684971309-myq7445o3jk.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc2ODQ5NzEzMDktbXlxNzQ0NW8zamsucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc4OTc3MTAyOCwiZXhwIjoxNzg5NzcxMzI4fQ.fHnebOAVEA7FwZBGeiZ5LPfgL4O0ffzlklEAl0tQOsw" style="max-width: min(400px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">angle $BFC$,</span></div></li><li><div class="list-item-content"><span class="list-item-text">angle $ABE$,</span></div></li><li><div class="list-item-content"><span class="list-item-text">angle $FEG$,</span></div></li><li><div class="list-item-content"><span class="list-item-text">angle $ABG$.</span></div></li><li><div class="list-item-paragraph">Hence determine the type(s) of triangles $DEF$ and $EGB$, and the type of quadrilateral $EGBF$, stating your reasons clearly.</div></li></ol>

In the figure below, $ABCD$ is a square. $E$ and $F$ are on sides $AD$ and $CD$, respectively, such that $BEF$ is an equilateral triangle. Fold triangle $BEF$ along $BE$ to obtain quadrilateral $EGBF$. Given that angle $DEF = 45^\circ$, find the sizes of

<p>In the figure below, $ABCD$ is a rectangle. $O$ the point of intersection of diagonals $AC$ and $BD$. Produce $BC$ to $E$, such that $DB = DE$. Given that angle $DAO = 40^\circ$, find the sizes of</p>
<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/1767684700447-jg08fxyi5vh.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc2ODQ3MDA0NDctamcwOGZ4eWk1dmgucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc4OTc3MTAyOCwiZXhwIjoxNzg5NzcxMzI4fQ.a3SN8VMzgHfRdW_Jc3AtSJknzN4dY16125e4_13Prrs" style="max-width: min(400px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">angle $BCO$,</span></div></li><li><div class="list-item-content"><span class="list-item-text">angle $BOC$,</span></div></li><li><div class="list-item-content"><span class="list-item-text">angle $DOC$,</span></div></li><li><div class="list-item-content"><span class="list-item-text">angle $CED$.</span></div></li><li><div class="list-item-paragraph">Hence, determine the type of quadrilateral $ACED$, stating your reasons clearly.</div></li></ol>

In the figure below, $ABCD$ is a rectangle. $O$ the point of intersection of diagonals $AC$ and $BD$. Produce $BC$ to $E$, such that $DB = DE$. Given that angle $DAO = 40^\circ$, find the sizes of

 Triangles and Quadrilaterals

<p>In the figure below, $ABCD$ is a parallelogram, where $AB$ is parallel to $CD$ and $AD$ is parallel to $BC$. $E$ is on side $CD$ and $F$ is on $BE$ such that $AE = EF$. Given that angle $AEF = 48^\circ$, angle $BAF = 18^\circ$ and angle $ADE = 62^\circ$, find the sizes of</p>
<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/1767684544661-qv51kfduusf.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc2ODQ1NDQ2NjEtcXY1MWtmZHV1c2YucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc4OTc3MTAyOCwiZXhwIjoxNzg5NzcxMzI4fQ.d64wCIbIH38FyNfwweNsvmxFvW8Sj45v7dcUfNzy4Z8" style="max-width: min(400px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">angle $FBA$,</span></div></li><li><div class="list-item-content"><span class="list-item-text">angle $CBE$,</span></div></li><li><div class="list-item-content"><span class="list-item-text">angle $DAE$,</span></div></li><li><div class="list-item-content"><span class="list-item-text">angle $CEB$.</span></div></li><li><div class="list-item-paragraph">Hence, determine the type of triangle $ABE$, stating your reason clearly.</div></li></ol>

In the figure below, $ABCD$ is a parallelogram, where $AB$ is parallel to $CD$ and $AD$ is parallel to $BC$. $E$ is on side $CD$ and $F$ is on $BE$ such that $AE = EF$. Given that angle $AEF = 48^\circ$, angle $BAF = 18^\circ$ and angle $ADE = 62^\circ$, find the sizes of

<p>In the figure below, $ABCD$ is a parallelogram, where $AB$ is parallel to $CD$ and $AD$ is parallel to $BC$. $E$ is on side $AD$. Produce $BE$ and $CD$, intersecting at $F$. Given that angle $BAE = 106^\circ$ and angle $BED = 143^\circ$, find the sizes of</p>
<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/1767684314526-uhov961v4h.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc2ODQzMTQ1MjYtdWhvdjk2MXY0aC5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzg5NzcxMDI4LCJleHAiOjE3ODk3NzEzMjh9.mZ0RQb0wNMHv0Jn3CNQ_MMLBcguo-EeNzkaPsFewa9M" style="max-width: min(400px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">reflex angle $CBE$,</span></div></li><li><div class="list-item-content"><span class="list-item-text">angle $ABE$,</span></div></li><li><div class="list-item-content"><span class="list-item-text">angle $EFD$,</span></div></li><li><div class="list-item-content"><span class="list-item-text">angle $CDE$.</span></div></li><li><div class="list-item-paragraph">Hence determine the type(s) of triangles $ABE$, $DEF$ and $CBF$, stating your reasons clearly.</div></li></ol>

In the figure below, $ABCD$ is a parallelogram, where $AB$ is parallel to $CD$ and $AD$ is parallel to $BC$. $E$ is on side $AD$. Produce $BE$ and $CD$, intersecting at $F$. Given that angle $BAE = 106^\circ$ and angle $BED = 143^\circ$, find the sizes of

<p>In the figure below, the triangle $ABD$
 is a right-angled triangle, where angle $BAD=90^{\circ}$
. $BE$
 is the angle bisector of angle $ABD$
 and $C$
 is on line segment $BD$
. Given that $BE=CE$
, angle $BEA=52^{\circ}$
, find the sizes of</p>
<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/1767684072932-qht8hl0xd8.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc2ODQwNzI5MzItcWh0OGhsMHhkOC5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzg5NzcxMDI4LCJleHAiOjE3ODk3NzEzMjh9.2tnlkW93CDZqrQTKKm31viP48NQ3y_qlDyJjAgzItXI" style="max-width: min(400px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<ol class="abc-list"><li><div class="list-item-paragraph">reflex angle $EBC$,</div></li><li><div class="list-item-paragraph"> angle $ABC$,</div></li><li><div class="list-item-paragraph"> angle $BEC$,</div></li><li><div class="list-item-paragraph"> angle $CED$,</div></li></ol>

In the figure below, the triangle $ABD$
 is a right-angled triangle, where angle $BAD=90^{\circ}$
. $BE$
 is the angle bisector of angle $ABD$
 and $C$
 is on line segment $BD$
. Given that $BE=CE$
, angle $BEA=52^{\circ}$
, find the sizes of

Math Question Bank