11x1 t13 02 angle theorems 1 (2011)

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  • Angle Theorems

  • Angle Theorems (4) The angle subtended by an arc (or chord) at the centre is double the

    angle subtended by the arc (or chord) at the circumference.

  • Angle Theorems (4) The angle subtended by an arc (or chord) at the centre is double the

    angle subtended by the arc (or chord) at the circumference.

    C A

    B

    O

  • Angle Theorems (4) The angle subtended by an arc (or chord) at the centre is double the

    angle subtended by the arc (or chord) at the circumference.

    arc sameon ncecircumfereat twicecentre,at 2 ABCAOC

    C A

    B

    O

  • Angle Theorems (4) The angle subtended by an arc (or chord) at the centre is double the

    angle subtended by the arc (or chord) at the circumference.

    arc sameon ncecircumfereat twicecentre,at 2 ABCAOC

    ABCAOC 2 :Prove

    C A

    B

    O

  • Angle Theorems (4) The angle subtended by an arc (or chord) at the centre is double the

    angle subtended by the arc (or chord) at the circumference.

    arc sameon ncecircumfereat twicecentre,at 2 ABCAOC

    ABCAOC 2 :Prove

    Proof: Join BO and produce to X

    C A

    B

    O

    X

  • Angle Theorems (4) The angle subtended by an arc (or chord) at the centre is double the

    angle subtended by the arc (or chord) at the circumference.

    arc sameon ncecircumfereat twicecentre,at 2 ABCAOC

    ABCAOC 2 :Prove

    Proof: Join BO and produce to X

    radii , isosceles is OBOAAOB

    C A

    B

    O

    X

  • Angle Theorems (4) The angle subtended by an arc (or chord) at the centre is double the

    angle subtended by the arc (or chord) at the circumference.

    arc sameon ncecircumfereat twicecentre,at 2 ABCAOC

    ABCAOC 2 :Prove

    Proof: Join BO and produce to X

    radii , isosceles is OBOAAOB isosceles s'base OABOBA

    C A

    B

    O

    X

  • Angle Theorems (4) The angle subtended by an arc (or chord) at the centre is double the

    angle subtended by the arc (or chord) at the circumference.

    arc sameon ncecircumfereat twicecentre,at 2 ABCAOC

    ABCAOC 2 :Prove

    Proof: Join BO and produce to X

    radii , isosceles is OBOAAOB isosceles s'base OABOBA OABOABOBAAOX exterior

    C A

    B

    O

    X

  • Angle Theorems (4) The angle subtended by an arc (or chord) at the centre is double the

    angle subtended by the arc (or chord) at the circumference.

    arc sameon ncecircumfereat twicecentre,at 2 ABCAOC

    ABCAOC 2 :Prove

    Proof: Join BO and produce to X

    radii , isosceles is OBOAAOB isosceles s'base OABOBA OABOABOBAAOX exterior

    2 OBAAOX C

    A

    B

    O

    X

  • Angle Theorems (4) The angle subtended by an arc (or chord) at the centre is double the

    angle subtended by the arc (or chord) at the circumference.

    arc sameon ncecircumfereat twicecentre,at 2 ABCAOC

    ABCAOC 2 :Prove

    Proof: Join BO and produce to X

    radii , isosceles is OBOAAOB isosceles s'base OABOBA OABOABOBAAOX exterior

    2 OBAAOX C

    A

    B

    O

    X

    methodsimilar by 2 OBCCOX

  • Angle Theorems (4) The angle subtended by an arc (or chord) at the centre is double the

    angle subtended by the arc (or chord) at the circumference.

    arc sameon ncecircumfereat twicecentre,at 2 ABCAOC

    ABCAOC 2 :Prove

    Proof: Join BO and produce to X

    radii , isosceles is OBOAAOB isosceles s'base OABOBA OABOABOBAAOX exterior

    2 OBAAOX

    2 ABCAOC

    C A

    B

    O

    X

    methodsimilar by 2 OBCCOX

  • (5a) The angle in a semicircle is a right angle.

  • (5a) The angle in a semicircle is a right angle.

    A

    B

    O

    C

  • (5a) The angle in a semicircle is a right angle.

    semicircle ain 90 ACBA

    B

    O

    C

  • (5a) The angle in a semicircle is a right angle.

    semicircle ain 90 ACB

    diameter is :Data AOBA

    B

    O

    C

  • (5a) The angle in a semicircle is a right angle.

    semicircle ain 90 ACB

    diameter is :Data AOB90 :Prove ACB

    A

    B

    O

    C

  • (5a) The angle in a semicircle is a right angle.

    semicircle ain 90 ACB

    diameter is :Data AOB90 :Prove ACB

    Proof: straight 180AOB

    A

    B

    O

    C

  • (5a) The angle in a semicircle is a right angle.

    semicircle ain 90 ACB

    diameter is :Data AOB90 :Prove ACB

    Proof: straight 180AOB

    arc sameon ncecircumfere

    at twicecentreat 2 ACBAOB

    A

    B

    O

    C

  • (5a) The angle in a semicircle is a right angle.

    semicircle ain 90 ACB

    diameter is :Data AOB90 :Prove ACB

    Proof: straight 180AOB

    arc sameon ncecircumfere

    at twicecentreat 2 ACBAOB

    90ACB

    A

    B

    O

    C

  • (5a) The angle in a semicircle is a right angle.

    semicircle ain 90 ACB

    diameter is :Data AOB90 :Prove ACB

    Proof: straight 180AOB

    arc sameon ncecircumfere

    at twicecentreat 2 ACBAOB

    90ACB

    A

    B

    O

    C

    Exercise 9B; 1 ace etc, 2, 6, 8ac, 9ab, 10ac, 11ac, 12, 13