6.2 Proving Quadrilaterals are Parallelograms

Theorems If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram. ABCD is a parallelogram.

Theorems If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram. ABCD is a parallelogram.

Theorems If an angle of a quadrilateral is supplementary to both of its consecutive angles, then the quadrilateral is a parallelogram. ABCD is a parallelogram. x°x° (180 – x) ° x°x°

Theorems If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram. ABCD is a parallelogram.

Given : AB ≅ CD, AD ≅ CB Prove: ABCD is a  Statements: 1. AB ≅ CD, AD ≅ CB. 2. AC ≅ AC 3. ∆ABC ≅ ∆CDA 4.  BAC ≅  DCA,  DAC ≅  BCA 5. AB ║CD, AD ║CB. 6. ABCD is a  Reasons: 1. Given 2. Reflexive Prop. of Congruence 3. SSS Congruence Postulate 4. CPCTC 5. Alternate Interior 6. Def. of a parallelogram.

X = 29 Z = = 102 Y = 180 – 78 – 29 =73 X = 31 Z = 105 Y = 44 X = 73 Z = = 107 Y = = 73

5y = 60 y = 12 3x = 180 3x = 180 3x = 60 X = 20

4x + y = 8 2x + y = 6 Solving System of Equations (Hint: subtract both equations) 2x = 2 X = 1 4 *1 + y = y= 8 Y = 4

x + y = 20 2x = 24 x = y = 20 y= 8

Using properties of parallelograms Show that A(2, -1), B(1, 3), C(6, 5) and D(7,1) are the vertices of a parallelogram.

Ex. 4: Using properties of parallelograms that opposite sides have the same length. AB=√(1 – 2) 2 + [3 – (- 1) 2 ] = √17 CD=√(7 – 6) 2 + (1 - 5) 2 = √17 BC=√(6 – 1) 2 + (5 - 3) 2 = √29 DA= √(2 – 7) 2 + (-1 - 1) 2 = √29 AB ≅ CD and BC ≅ DA. Because both pairs of opposites sides are congruent, ABCD is a parallelogram.