Circle-Theorem-Proofs Wjec-Eduqas Higher

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Exam code:C300

Circle theorem proofs

How do I prove circle theorems using radii to form isosceles triangles?

This type of proof can be used to prove the following circle theorems
- The angle in a semicircle is always 90°
- The angle at the centre is twice the angle at the circumference
- Angles in the same segment are equal
- Opposite angles in a cyclic quadrilateral add up to 180°

How do I prove that the angle in a semicircle is 90°?

This circle theorem states that an angle subtended at the circumference of a semicircle is always a right angle
- Although it is a special case of the angle at the centre and circumference circle theorem, it can be proved without using any other circle theorems
STEP 1
Draw a radius from the centre of the circle to the angle subtended at the circumference
- This will form two isosceles triangles
STEP 2
Label the two angles formed at the angle subtended at the circumference $x$ and $y$ .
- The angle you are trying to prove is 90° is now $x space plus space y$
STEP 3
Label the remaining angles in each of the isosceles triangles with algebraic expressions in terms of $x$ and $y$
- Angles at the base of an isosceles triangle are equal
  - Therefore the two remaining angles at the circumference are also $x$ and $y$
- Angles in a triangle add up to 180°
  - Therefore each angle at the centre will be labelled with the expression $180 space minus space 2 x$ and $180 space minus space 2 y$

STEP 4
The angles at the centre lie on a diameter, which is a straight line, therefore <img alt=”180 space minus space 2 x space plus space 180 space minus space 2 y space equals space 180″ data-mathml='<math ><semantics><mrow><mn>180</mn><mo> </mo><mo>-</mo><mo> </mo><mn>2</mn><mi>x</mi><mo> </mo><mo>+</mo><mo> </mo><mn>180</mn><mo> </mo><mo>-</mo><mo> </mo><mn>2</mn><mi>y</mi><mo> </mo><mo>=</mo><mo> </mo><mn>180</mn></mrow><annotation encoding=”application/vnd.wiris.mtweb-params+json”>{“language”:”en”,”fontFamily”:”Times New Roman”,”fontSize”:”18″,”autoformat”:true}</annotation></semantics></math>’ height=”22″ role=”math” src=”data:image/svg+xml;charset=utf8,%3Csvg%20xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F2000%2Fsvg%22%20xmlns%3Awrs%3D%22http%3A%2F%2Fwww.wiris.com%2Fxml%2Fmathml-extension%22%20height%3D%2222%22%20width%3D%22220%22%20wrs%3Abaseline%3D%2216%22%3E%3C!–MathML%3A%20%3Cmath%20xmlns%3D%22http%3A%2F%2Fwww.w3.org%2F1998%2FMath%2FMathML%22%3E%3Cmn%3E180%3C%2Fmn%3E%3Cmo%3E%26%23xA0%3B%3C%2Fmo%3E%3Cmo%3E-%3C%2Fmo%3E%3Cmo%3E%26%23xA0%3B%3C%2Fmo%3E%3Cmn%3E2%3C%2Fmn%3E%3Cmi%3Ex%3C%2Fmi%3E%3Cmo%3E%26%23xA0%3B%3C%2Fmo%3E%3Cmo%3E%2B%3C%2Fmo%3E%3Cmo%3E%26%23xA0%3B%3C%2Fmo%3E%3Cmn%3E180%3C%2Fmn%3E%3Cmo%3E%26%23xA0%3B%3C%2Fmo%3E%3Cmo%3E-%3C%2Fmo%3E%3Cmo%3E%26%23xA0%3B%3C%2Fmo%3E%3Cmn%3E2%3C%2Fmn%3E%3Cmi%3Ey%3C%2Fmi%3E%3Cmo%3E%26%23xA0%3B%3C%2Fmo%3E%3Cmo%3E%3D%3C%2Fmo%3E%3Cmo%3E%26%23xA0%3B%3C%2Fmo%3E%3Cmn%3E180%3C%2Fmn%3E%3C%2Fmath%3E–%3E%3Cdefs%3E%3Cstyle%20type%3D%22text%2Fcss%22%3E%40font-face%7Bfont-family%3A’math14f338326459958a9aae793361b’%3Bsrc%3Aurl(data%3Afont%2Ftruetype%3Bcharset%3Dutf-8%3Bbase64%2CAAEAAAAMAIAAAwBAT1MvMi7iBBMAAADMAAAATmNtYXDEvmKUAAABHAAAAERjdnQgDVUNBwAAAWAAAAA6Z2x5ZoPi2VsAAAGcAAABdWhlYWQQC2qxAAADFAAAADZoaGVhCGsXSAAAA0wAAAAkaG10eE2rRkcAAANwAAAAEGxvY2EAHTwYAAADgAAAABRtYXhwBT0FPgAAA5QAAAAgbmFtZaBxlY4AAAO0AAABn3Bvc3QB9wD6AAAFVAAAACBwcmVwa1uragAABXQAAAAUAAADSwGQAAUAAAQABAAAAAAABAAEAAAAAAAAAQEAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAACAgICAAAAAg1UADev96AAAD6ACWAAAAAAACAAEAAQAAABQAAwABAAAAFAAEADAAAAAIAAgAAgAAACsAPSIS%2F%2F8AAAArAD0iEv%2F%2F%2F9b%2Fxd3xAAEAAAAAAAAAAAAAAVQDLACAAQAAVgAqAlgCHgEOASwCLABaAYACgACgANQAgAAAAAAAAAArAFUAgACrANUBAAErAAcAAAACAFUAAAMAA6sAAwAHAAAzESERJSERIVUCq%2F2rAgD%2BAAOr%2FFVVAwAAAQCAAFUC1QKrAAsASQEYsgwBARQTELEAA%2FaxAQT1sAo8sQMF9bAIPLEFBPWwBjyxDQPmALEAABMQsQEG5LEBARMQsAU8sQME5bELBfWwBzyxCQTlMTATIREzESEVIREjESGAAQBVAQD%2FAFX%2FAAGrAQD%2FAFb%2FAAEAAAIAgADrAtUCFQADAAcAZRgBsAgQsAbUsAYQsAXUsAgQsAHUsAEQsADUsAYQsAc8sAUQsAQ8sAEQsAI8sAAQsAM8ALAIELAG1LAGELAH1LAHELAB1LABELAC1LAGELAFPLAHELAEPLABELAAPLACELADPDEwEyE1IR0BITWAAlX9qwJVAcBV1VVVAAEAgAFVAtUBqwADADAYAbAEELEAA%2FawAzyxAgf1sAE8sQUD5gCxAAATELEABuWxAAETELABPLEDBfWwAjwTIRUhgAJV%2FasBq1YAAAAAAQAAAAEAANV4zkFfDzz1AAMEAP%2F%2F%2F%2F%2FWOhNz%2F%2F%2F%2F%2F9Y6E3MAAP8gBIADqwAAAAoAAgABAAAAAAABAAAD6P9qAAAXcAAA%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%2F)format(‘truetype’)%3Bfont-weight%3Anormal%3Bfont-style%3Anormal%3B%7D%3C%2Fstyle%3E%3C%2Fdefs%3E%3Ctext%20font-family%3D%22Times%20New%20Roman%22%20font-size%3D%2218%22%20text-anchor%3D%22middle%22%20x%3D%2213.5%22%20y%3D%2216%22%3E180%3C%2Ftext%3E%3Ctext%20font-family%3D%22m

Maths Gcse Wjec-Eduqas Higher

Circle-Theorem-Proofs Wjec-Eduqas Higher

Circle theorem proofs

How do I prove circle theorems using radii to form isosceles triangles?

How do I prove that the angle in a semicircle is 90°?

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