11X1 T13 02 angle theorems 1 (2011)

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<ul><li><p>Angle Theorems </p></li><li><p>Angle Theorems (4) The angle subtended by an arc (or chord) at the centre is double the </p><p>angle subtended by the arc (or chord) at the circumference. </p></li><li><p>Angle Theorems (4) The angle subtended by an arc (or chord) at the centre is double the </p><p>angle subtended by the arc (or chord) at the circumference. </p><p>C A </p><p>B </p><p>O </p></li><li><p>Angle Theorems (4) The angle subtended by an arc (or chord) at the centre is double the </p><p>angle subtended by the arc (or chord) at the circumference. </p><p> arc sameon ncecircumfereat twicecentre,at 2 ABCAOC</p><p>C A </p><p>B </p><p>O </p></li><li><p>Angle Theorems (4) The angle subtended by an arc (or chord) at the centre is double the </p><p>angle subtended by the arc (or chord) at the circumference. </p><p> arc sameon ncecircumfereat twicecentre,at 2 ABCAOC</p><p>ABCAOC 2 :Prove</p><p>C A </p><p>B </p><p>O </p></li><li><p>Angle Theorems (4) The angle subtended by an arc (or chord) at the centre is double the </p><p>angle subtended by the arc (or chord) at the circumference. </p><p> arc sameon ncecircumfereat twicecentre,at 2 ABCAOC</p><p>ABCAOC 2 :Prove</p><p>Proof: Join BO and produce to X </p><p>C A </p><p>B </p><p>O </p><p>X </p></li><li><p>Angle Theorems (4) The angle subtended by an arc (or chord) at the centre is double the </p><p>angle subtended by the arc (or chord) at the circumference. </p><p> arc sameon ncecircumfereat twicecentre,at 2 ABCAOC</p><p>ABCAOC 2 :Prove</p><p>Proof: Join BO and produce to X </p><p> radii , isosceles is OBOAAOB</p><p>C A </p><p>B </p><p>O </p><p>X </p></li><li><p>Angle Theorems (4) The angle subtended by an arc (or chord) at the centre is double the </p><p>angle subtended by the arc (or chord) at the circumference. </p><p> arc sameon ncecircumfereat twicecentre,at 2 ABCAOC</p><p>ABCAOC 2 :Prove</p><p>Proof: Join BO and produce to X </p><p> radii , isosceles is OBOAAOB isosceles s'base OABOBA</p><p>C A </p><p>B </p><p>O </p><p>X </p></li><li><p>Angle Theorems (4) The angle subtended by an arc (or chord) at the centre is double the </p><p>angle subtended by the arc (or chord) at the circumference. </p><p> arc sameon ncecircumfereat twicecentre,at 2 ABCAOC</p><p>ABCAOC 2 :Prove</p><p>Proof: Join BO and produce to X </p><p> radii , isosceles is OBOAAOB isosceles s'base OABOBA OABOABOBAAOX exterior </p><p>C A </p><p>B </p><p>O </p><p>X </p></li><li><p>Angle Theorems (4) The angle subtended by an arc (or chord) at the centre is double the </p><p>angle subtended by the arc (or chord) at the circumference. </p><p> arc sameon ncecircumfereat twicecentre,at 2 ABCAOC</p><p>ABCAOC 2 :Prove</p><p>Proof: Join BO and produce to X </p><p> radii , isosceles is OBOAAOB isosceles s'base OABOBA OABOABOBAAOX exterior </p><p> 2 OBAAOX C </p><p>A </p><p>B </p><p>O </p><p>X </p></li><li><p>Angle Theorems (4) The angle subtended by an arc (or chord) at the centre is double the </p><p>angle subtended by the arc (or chord) at the circumference. </p><p> arc sameon ncecircumfereat twicecentre,at 2 ABCAOC</p><p>ABCAOC 2 :Prove</p><p>Proof: Join BO and produce to X </p><p> radii , isosceles is OBOAAOB isosceles s'base OABOBA OABOABOBAAOX exterior </p><p> 2 OBAAOX C </p><p>A </p><p>B </p><p>O </p><p>X </p><p> methodsimilar by 2 OBCCOX </p></li><li><p>Angle Theorems (4) The angle subtended by an arc (or chord) at the centre is double the </p><p>angle subtended by the arc (or chord) at the circumference. </p><p> arc sameon ncecircumfereat twicecentre,at 2 ABCAOC</p><p>ABCAOC 2 :Prove</p><p>Proof: Join BO and produce to X </p><p> radii , isosceles is OBOAAOB isosceles s'base OABOBA OABOABOBAAOX exterior </p><p> 2 OBAAOX </p><p> 2 ABCAOC </p><p>C A </p><p>B </p><p>O </p><p>X </p><p> methodsimilar by 2 OBCCOX </p></li><li><p>(5a) The angle in a semicircle is a right angle. </p></li><li><p>(5a) The angle in a semicircle is a right angle. </p><p>A </p><p>B </p><p>O </p><p>C </p></li><li><p>(5a) The angle in a semicircle is a right angle. </p><p> semicircle ain 90 ACBA </p><p>B </p><p>O </p><p>C </p></li><li><p>(5a) The angle in a semicircle is a right angle. </p><p> semicircle ain 90 ACB</p><p>diameter is :Data AOBA </p><p>B </p><p>O </p><p>C </p></li><li><p>(5a) The angle in a semicircle is a right angle. </p><p> semicircle ain 90 ACB</p><p>diameter is :Data AOB90 :Prove ACB</p><p>A </p><p>B </p><p>O </p><p>C </p></li><li><p>(5a) The angle in a semicircle is a right angle. </p><p> semicircle ain 90 ACB</p><p>diameter is :Data AOB90 :Prove ACB</p><p>Proof: straight 180AOB</p><p>A </p><p>B </p><p>O </p><p>C </p></li><li><p>(5a) The angle in a semicircle is a right angle. </p><p> semicircle ain 90 ACB</p><p>diameter is :Data AOB90 :Prove ACB</p><p>Proof: straight 180AOB</p><p>arc sameon ncecircumfere</p><p>at twicecentreat 2 ACBAOB</p><p>A </p><p>B </p><p>O </p><p>C </p></li><li><p>(5a) The angle in a semicircle is a right angle. </p><p> semicircle ain 90 ACB</p><p>diameter is :Data AOB90 :Prove ACB</p><p>Proof: straight 180AOB</p><p>arc sameon ncecircumfere</p><p>at twicecentreat 2 ACBAOB</p><p> 90ACB</p><p>A </p><p>B </p><p>O </p><p>C </p></li><li><p>(5a) The angle in a semicircle is a right angle. </p><p> semicircle ain 90 ACB</p><p>diameter is :Data AOB90 :Prove ACB</p><p>Proof: straight 180AOB</p><p>arc sameon ncecircumfere</p><p>at twicecentreat 2 ACBAOB</p><p> 90ACB</p><p>A </p><p>B </p><p>O </p><p>C </p><p>Exercise 9B; 1 ace etc, 2, 6, 8ac, 9ab, 10ac, 11ac, 12, 13 </p></li></ul>