Proof of the Circle Theorems worksheet
10 GCSE practice questions with answers, free to print. Every sheet is generated, so you can make a fresh one whenever you need it.
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5 marks
\(S\), \(T\) and \(U\) are points on a circle, centre \(O\).
\(ST\) is a diameter.
(a)Explain why \(OS = OU\).
(b)Explain why angle \(OUS = x\) and angle \(OUT = y\).
(c)Hence prove that angle \(SUT = 90^\circ\).
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5 marks
\(TA\) and \(TB\) are tangents to a circle with centre \(O\), touching the circle at \(A\) and at \(B\).
(a)State the size of angle \(OAT\), and give a reason for your answer.
(b)Prove that triangle \(OAT\) and triangle \(OBT\) are congruent.
(c)Hence explain why \(TA = TB\).
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6 marks
\(E\), \(F\) and \(G\) are points on a circle, centre \(O\).
The line \(GO\) is continued to meet the circle again at \(H\).
(a)Explain why angle \(OEG = x\).
(b)Find the size of angle \(EOH\), in terms of \(x\).
(c)Hence prove that angle \(EOF = 2 \times\) angle \(EGF\).
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4 marks
\(W\), \(X\), \(Y\) and \(Z\) are points on a circle, centre \(O\).
\(Y\) and \(Z\) are on the same arc.
(a)Write down \(z\) in terms of \(x\), and \(z\) in terms of \(y\).
Give a reason for your answers.
(b)Hence prove that angle \(WYX = \) angle \(WZX\).
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5 marks
\(W\), \(X\), \(Y\) and \(Z\) are points on a circle, centre \(O\).
(a)Write down the angle \(XOZ\) on the same side of \(XZ\) as \(Y\), in terms of \(x\).
Give a reason for your answer.
(b)Write down the other angle \(XOZ\), in terms of \(y\).
(c)Hence prove that \(x + y = 180^\circ\).
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5 marks
\(S\), \(T\) and \(U\) are points on a circle, centre \(O\).
\(ST\) is a diameter.
(a)Explain why \(OS = OU\).
(b)Explain why angle \(OUS = x\) and angle \(OUT = y\).
(c)Hence prove that angle \(SUT = 90^\circ\).
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5 marks
\(TA\) and \(TB\) are tangents to a circle with centre \(O\), touching the circle at \(A\) and at \(B\).
(a)State the size of angle \(OAT\), and give a reason for your answer.
(b)Prove that triangle \(OAT\) and triangle \(OBT\) are congruent.
(c)Hence explain why \(TA = TB\).
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6 marks
\(E\), \(F\) and \(G\) are points on a circle, centre \(O\).
The line \(GO\) is continued to meet the circle again at \(H\).
(a)Explain why angle \(OEG = x\).
(b)Find the size of angle \(EOH\), in terms of \(x\).
(c)Hence prove that angle \(EOF = 2 \times\) angle \(EGF\).
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4 marks
\(A\), \(B\), \(C\) and \(D\) are points on a circle, centre \(O\).
\(C\) and \(D\) are on the same arc.
(a)State \(z\) in terms of \(x\), and \(z\) in terms of \(y\).
Give a reason for your answers.
(b)Hence prove that angle \(ACB = \) angle \(ADB\).
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5 marks
\(W\), \(X\), \(Y\) and \(Z\) are points on a circle, centre \(O\).
(a)Write down the angle \(XOZ\) on the same side of \(XZ\) as \(Y\), in terms of \(x\).
Give a reason for your answer.
(b)Write down the other angle \(XOZ\), in terms of \(y\).
(c)Hence prove that \(x + y = 180^\circ\).
Answers
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(a) \(OS\) and \(OU\) are both radii of the circle, so they are equal.
(b) Triangles \(OSU\) and \(OTU\) are isosceles, and base angles of an isosceles triangle are equal.
(c) The angles of triangle \(SUT\) give \(x + y + (x + y) = 180^\circ\), so \(x + y = 90^\circ\), and angle \(SUT = x + y\). -
(a) \(90^\circ\). \(TA\) is a tangent and \(OA\) is the radius at the point of contact, so the angle between them is a right angle.
(b) RHS: both have a right angle, the hypotenuse \(OT\) is common, and \(OA = OB\) as radii.
(c) \(TA\) and \(TB\) are corresponding sides of congruent triangles, so they are equal. -
(a) \(OE\) and \(OG\) are radii, so triangle \(OEG\) is isosceles and its base angles are equal.
(b) Angle \(EOH = 2x\)
(c) Angle \(FOH = 2y\) in the same way, so angle \(EOF = 2x + 2y = 2(x + y)\), and angle \(EGF = x + y\). -
(a) \(z = 2x\) and \(z = 2y\). Both \(Y\) and the centre stand on arc \(WX\), and the angle at the centre is twice the angle at the circumference.
(b) \(2x = z = 2y\), so \(x = y\), which means angle \(WYX = \) angle \(WZX\). -
(a) \(2x\), because the angle at the centre is twice the angle at the circumference standing on the same arc \(XYZ\).
(b) The other angle \(XOZ\) is \(2y\), standing on arc \(XWZ\).
(c) The two angles at \(O\) make a full turn, so \(2x + 2y = 360^\circ\) and \(x + y = 180^\circ\). -
(a) \(OS\) and \(OU\) are both radii of the circle, so they are equal.
(b) Triangles \(OSU\) and \(OTU\) are isosceles, and base angles of an isosceles triangle are equal.
(c) The angles of triangle \(SUT\) give \(x + y + (x + y) = 180^\circ\), so \(x + y = 90^\circ\), and angle \(SUT = x + y\). -
(a) \(90^\circ\). \(TA\) is a tangent and \(OA\) is the radius at the point of contact, so the angle between them is a right angle.
(b) RHS: both have a right angle, the hypotenuse \(OT\) is common, and \(OA = OB\) as radii.
(c) \(TA\) and \(TB\) are corresponding sides of congruent triangles, so they are equal. -
(a) \(OE\) and \(OG\) are radii, so triangle \(OEG\) is isosceles and its base angles are equal.
(b) Angle \(EOH = 2x\)
(c) Angle \(FOH = 2y\) in the same way, so angle \(EOF = 2x + 2y = 2(x + y)\), and angle \(EGF = x + y\). -
(a) \(z = 2x\) and \(z = 2y\). Both \(C\) and the centre stand on arc \(AB\), and the angle at the centre is twice the angle at the circumference.
(b) \(2x = z = 2y\), so \(x = y\), which means angle \(ACB = \) angle \(ADB\). -
(a) \(2x\), because the angle at the centre is twice the angle at the circumference standing on the same arc \(XYZ\).
(b) The other angle \(XOZ\) is \(2y\), standing on arc \(XWZ\).
(c) The two angles at \(O\) make a full turn, so \(2x + 2y = 360^\circ\) and \(x + y = 180^\circ\).