Absolute geometry – quadrilaterals
Proof of SASAS
The proof of the SASAS congruence theorem is in your text, but I'll run it through a
breakdown with illustrations. See if you can fill in the reasons for each step, then watch
the live version when you're done.
Theorem statement: For convex quadrilaterals,
Proof:
Statement: Suppose we have two convex quadrilaterals under the correspondence
and that
Picture:
Reason:
Statement: (by definition) The quadrilaterals have diagonals AC and XZ shown below.
For convenience, number the angles as shown.
Picture:
Statement: + ABC ≅+ XYZ
Picture:
Reason:
Statement: (1 ≅ (3, (5 ≅ (7, AC ≅ XZ
Picture:
Reason:
Statement: A is in the interior of (BCD , and X is in the interior of (YZW . So
m(1 + m( 2 = m(BCD and m(3 + m( 4 = m(YZW .
Picture:
Reason:
Statement: Since m(BCD = m(YZW (equivalent with given congruence (C ≅ (Z )
and m(1 = m(3 (equivalent with established congruence (1 ≅ (3 ), we have
m( 2 = m( 4 and therefore ( 2 ≅ ( 4 .
Picture:
Reason:
Statement: + ACD ≅+ XZW
Picture:
Reason:
Statement: (D ≅ (W , ( 6 ≅ (8, AD ≅ XW .
Picture:
Reason:
Statement: C is in the interior of (BAD , and Z is in the interior of (YXW . So
m(5 + m( 6 = m(BAD and m(7 + m(8 = m(YXW .
Picture:
Reason:
Statement: Since m(5 = m(7 (equivalent with established congruence (5 ≅ ( 7 ) and
m(6 = m(8 (equivalent with established congruence ( 6 ≅ (8 ), we have
m(5 + m( 6 = m( 7 + m(8 so m(BAD = m(YZW and therefore (BAD ≅ (YZW .
Picture:
Reason:
Therefore
◊ABCD ≅ ◊XYZW