Few of the theorems of angles and lines with proves.
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CONTENTS Theorem 1: Vertically opposite angles are equal in measure Theorem 2: The sum of all the angles of a triangle is 1800 Theorem 3: An exterior angle of a triangle equals the sum of the two interior opposite angles in measure. Theorem 4: If to sides of a triangle are equal in measure, then the angles opposite these sides are equal in measure Theorem 5: Opposite sides and opposite angles of a parallelogram are respectively equal in measure Theorem 6: A diagonal bisects the area of a parallelogram Theorem 7: The measure of the angle at the Centre of the circle is twice the measure of the angle at the circumference, standing on the same arc Theorem 8: A line through the Centre of a circle perpendicular to a chord bisects the chord Theorem 9: If two triangles are equiangular, the lengths of the corresponding sides are in proportion SINGH'S GLASSES
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Theorem: Vertically opposite angles are equal in measure. MAIN MENU 3 K Given: Intersecting lines L and K, with vertically opposite angles 1 and 2. To prove: Zl=Z2 Construction: Label angle 3 Proof: Zl+Z3=1800 Z2+Z3=1800 ZI+Ä=Ä Zl=Z2 Straight angle Straight angle .....Subtract Z 3 from both sides Q.E.D.
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Theorem: The measure of the three angles of a triangle sum to 1800 MAIN MENU a 3 Given: To Prove: The triangle abc with 1,2 and 3. Z 1 1800 4 1 b 5 2 Construction: Draw a line through a, Parallel to bc. Label angles 4 and 5. Proof: Zl=Z4 and Z2=Z5 But Z4+Z5+Z3=1800 1 1800 Alternate angles Straight angle Q.E.D.
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Theorem: An exterior angle of a triangle equals the sum of the two interior opposite angles in measure. a 1 2 b MAIN MENU 3 Given: A triangle with interior opposite angles 1 and 2 and the exterior angle 3. To prove: Construction: Proof: Label angle 4 Z 24- Z4=1800 Z 34- Z4=1800 Z 2+ Z 34- Z4 Three angles in a triangle Straight angle Q.E.D.
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Given: To prove: Construction: Proof: MAIN Theorem: If to sides of a triangle are equal in measure, then the angles MENU opposite these sides are equal in measure. a 2 d I adl = I ad The triangle abc, with ab = ac and base angles 1 and 2. Draw ad, the bisector of Zbac. Label angles 3 and 4. Consider abd and A acd: ab Aabd ac Aacd given construction common SAS Corresponding angles Q.E.D.
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Theorem: Opposite sides and opposite angles of a parallelgram are respectively equal in measure. b Given: To prove: a 1 Parallelogram abcd d MAIN MENU 3 2 Zabc = Zadc, Zbad = Zbcd Construction: Join a to c. Label angles 1,2,3 and 4. Consider A abc and adc : Proof: Z2 and Z4 ac ac A abc Aadc abl= dc and ad And Zabc = Zadc Similarly, Zbad = Zbcd bc Alternate angles common ASA Corresponding sides Corresponding angles Q.E.D.
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Theorem:A diagonal bisects the area of a parallelogram. ab = dc ad = bc ac = ac area Aabc = area adc Given: To prove: Proof: b Parallelogram abcd with diagonal [ac]. Area of abc = area of adc. ConsiderA abc and adc: abc adc MAIN MENU d Opposite sides Opposite sides Common sss Q.E.D.
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Theorem: The measure of the angle at the centre of the circle is twice the measure of the angle at the circumference, standing on the same arc. a 15 3 c MAIN MENU Given: To prove: Circle, centre o, containing points a, b and c. Zboc = 2 Zbac Construction: Join a to o and continue to d. Label angles 1,2,3,4 and 5. Proof: But Similarly, i.e. Consider aob: Z 1 + Z 5 = 2(Z2 + Z-4) Zboc = 2 Zbac Exterior angle Base angles in an isosceles Q.E.D.
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Theorem: A line through the centre of a circle perpendicular to a chord bisects the ca = cb cd = cd ad = bd chord. Given: To prove: c a b MAIN MENU Circle, centre c, a line L containing c, chord [ab], such that L Lab and Ln ab = d. ad = bd Construction: Label right angles 1 and 2. Proof: Consider Cda and cdb: = = 900 Cda A cdb Given Both radii common Corresponding sides Q.E.D.
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Theorem: If two triangles are equiangular, the lengths of the corresponding sides are in proportion. Two triangles with equal angles. MAIN MENU Given : To prove: Construction: a 2 5 labl lacl lbcl - Idfl - lefl Idel On ab mark off ax equal in length to de. On ac mark off ay equal to df and label the angles 4 and 5. Proof: z 1 = [xyl is parallel to [bc] x 3 2 1 c labl laxl labl Idel lacl layl lacl - Idfl As xy is parallel to bc. lbcl Similarly Q.E.D.
Discussion
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