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3.1 CAIA and the Results of a Parallel Postulate

We proved parallel lines exist in neutral geometry using only Euclid’s first four axioms.

Note

Right angles and perpendicularity work reasonably well in neutral geometry.

Since two lines perpendicular to the same line are parallel, we know that we construct some parallel lines. Now that we have a parallel postulate, we only need a few definitions to be ready to prove important results.

Median

A line passing through a vertex and the midpoint of the side opposite that vertex.

Parallel Segments

We call line segments parallel if they are contained in lines that are parallel.

Distance from a point to a line

Given a point \(P\) not on line \(l\), the distance between \(P\) and \(l\) is the distance \(PQ\) where \(Q\in l\) and \(\overleftrightarrow{PQ}\perp l\).

Trapezoid

A convex quadrilateral with exactly one pair of parallel sides.

Warning

Two correct but different definitions of the trapezoid exist, one where only one pair of sides can be parallel (as we are using), and another where at least one pair of sides are parallel. For the second, note that parallelograms are a subset of trapezoids. For the first, trapezoids and parallelograms are disjoint sets.

Isosceles Trapezoid

A trapezoid with non-parallel sides congruent.

Kite

A quadrilateral with two pairs of adjacent sides congruent.

Rhombus

A parallelogram with all sides congruent.

Theorems

  1. Converse of the Alternate Interior Angle Theorem. If two parallel lines are intersected by a transversal, then the alternate interior angles formed are congruent.

  2. The sum of the measures of the interior angles of triangle is \(180^\circ\).

    Tip

    For a neutral geometry, the angle sum for triangle was at most \(180^\circ\). What forces the equality in Euclidean space?

  3. Euclidean Exterior Angle Theorem. The measure of an exterior angle of a triangle is equal to the sum of the measures of the two remote interior angles.

  4. Parallelogram Decomposition Theorem. 1 In parallelogram \(ABCD\), \(\triangle{ABD} \cong \triangle{CDB}\) and \(\triangle{ABC}\cong\triangle{CDA}\).

    Note

    Two important corrolaries of the Parallelogram Decomposition theorem will be quite useful.

    Segments

    Opposite sides of a parallelogram are congruent.

    Angles

    Opposite angles of a parallelogram are congruent.

    Tip

    The next few important theorems identify key relationship about the diagonals of quadrilaterals.

    See also

    The Isosceles Triangle theorem and its corollary – both proved in neutral geometry – are useful in proving the theorems below.

  5. The diagonals of a quadrilateral bisect each other if and only the quadrilateral is a parallelogram.

  6. Exactly one diagonal of a quadrilateral is the perpendicular bisector of the other if and only if the quadrilateral is a kite.

    See also

    After Theorem 8, we can prove our Second Construction Theorem.

  7. The diagonals of a quadrilateral are perpendicular bisectors of one another if and only if the quadrilateral is a rhombus.

  8. The diagonals of a quadrilateral are congruent perpendicular bisectors of one another if and only if the quadrilateral is a square.

1

This theorem name is non-standard, invented and used in this document only.

3. Euclidean Geometry 3.2 Parallel Lines