Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Euler angles wikipedia , lookup
Rational trigonometry wikipedia , lookup
History of geometry wikipedia , lookup
Noether's theorem wikipedia , lookup
Trigonometric functions wikipedia , lookup
Riemann–Roch theorem wikipedia , lookup
Four color theorem wikipedia , lookup
Brouwer fixed-point theorem wikipedia , lookup
Integer triangle wikipedia , lookup
History of trigonometry wikipedia , lookup
Properties of parallelogram Sides ? Diagonals ? Angles ? Theorem A diagonal of a parallelogram divides it into two congruent triangles C D Diagonal ? A B Triangles ? Proof Given :AC is a diagonal of the parallelogram ABCD To prove:ΔABC ≡ ΔCDA Proof :In ΔABC and ΔCDA D ∟BCA =∟DAC (Why?) ∟BAC =∟DCA (Why?) AC = CA (Why?) A B ΔABC ≡ ΔCDA (ASA)-proved The diagonal AC divides parallelogram ABCD into two congruent triangles ABC and CDA C Property -1 (Theorem 8.2) In a parallelogram opposite sides are equal C D In parallelogram ABCD AB =DC A B BC=AD Property -2 (Theorem 8.4) In a parallelogram opposite angles are equal C D In parallelogram ABCD ∟A = ∟C ∟B = ∟D A B Property -3 (Theorem 8.6) The diagonals of a parallelogram bisect each other The diagonals AC and BD C D O A B bisect each other at O, then OA= OC OB = OD Question What are the conditions for a quadrilateral to become a parallelogram? A quadrilateral is a parallelogram 1) If each pair of opposite sides are equal 2) If each pair of opposite angles are equal 3) If the diagonals of the quadrilateral bisect each other Another Condition (Theorem 8.8) A Quadrilateral is a parallelogram if a pair of its opposite sides is equal and parallel C D A B ABCD is a parallelogram if AB = DC and AB II DC Or ( if AD=BC and ABIIBC ) Conditions ? A quadrilateral is a parallelogram 1) If each pair of opposite sides are equal 2) If each pair of opposite angles are equal 3) If the diagonals of the quadrilateral bisect each other 4) If a pair of opposite side is equal and parallel Midpoint theorem (Theorem 8.9) The line segment joining the midpoints of any two sides of a triangle is parallel to the third side A If E and F are midpoints of E B sides AB and AC of triangle ABC ,then EF II BC F C Converse of midpoint theorem ( 8.10) The line drawn through the midpoint of one side of a triangle, parallel to another side bisects the third side A E B If E is the midpoint of AB and EF II BC then F is the mid point of AC (i.e.AF =FC ) F C Summary A diagonal of a parallelogram divides it into two congruent triangles In a parallelogram opposite sides are equal. In a parallelogram opposite angles are equal. The diagonals of a parallelogram bisect each other. A Quadrilateral is a parallelogram if a pair of its opposite sides is equal and parallel The line segment joining the midpoints of any two sides of a triangle is parallel to the third side The line drawn through the midpoint of one side of a triangle, parallel to another side bisects the third side