Overview of Section 1.3 The Parallel Postulate and Its Consequences
Euclid's Fifth Postulate -- the Parallel Postulate
If a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which the angles are less than the two right angles.
Playfair's Axiom is one of many equivalent statements for the Parallel Postulate:
Axiom 1.2 The Parallel Postulate (Playfair's Axiom): Given a line and a point not on the line, there exists a unique line through the point parallel to the given line.
Theorem 1.18: (Converse of Theorem 1.17) If two parallel lines are cut by a transversal, then a pair of corresponding angles is congruent.
Theorem 1.19: Two lines in a plane are parallel if and only if a pair of corresponding angles formed by a transversal is congruent.
Theorem 1.20: Two lines in a plane are parallel if and only if a pair of alternate interior angles formed by a transversal is congruent.
Theorem 1.21: Two lines are parallel if and only if a pair of interior angles on the same side of the transversal is supplementary.
Theorem 1.22: The sum of the measures of the interior angles of a triangle is 180 degrees.
Example 1.5. Two lines cut by a transversal are given. The segment AC is on the transversal and the segments CB and AB lie along the angle bisectors of the interior angles. NOTE THE TYPO. The book states AC is along the angle bisector. The problem is to determine the measure of angle B.
Now Solve This 1.9
1. Find an expression for the sum of the measures of the interior angles of a convex n-gon in terms of n.
Hint: Construct the diagonals of the convex n-gon from one of the vertices to create non-overlapping triangles.
2. What is the sum of the measures of the Exterior angles of a convex n-gon. Prove your answer.
Theorem 1.23: A Saccheri quadrilateral is a rectangle.
Theorem 1.24: The measure of an exterior angle in a triangle is equal to the sum of the measures of its two remote angles.
Theorem 1.25: If a transversal is perpendicular to one of two parallel lines, it is also perpendicular to the other line.
Theorem 1.26: In a parallelogram:
1. Each diagonal divides the parallelogram into two congruent triangles.
2. Each pair of opposite sides is congruent.
3. The diagonals bisect each other.
Theorem 1.27:
1. A quadrilateral in which each pair of opposite sides is congruent is a parallelogram.
2. A quadrilateral in which the diagonals bisect each other is a parallelogram.
3. A quadrilateral in which each pair of opposite angles is congruent is a parallelogram.
4. A quadrilateral in which each pair of opposite sides is parallel and congruent is a parallelogram.
Now Solve This 1.10
1. Prove Theorems 1.24 through 1.27.
2. Theorem 1.27 (4 parts) rephrased in "necessary and sufficient condition" language
Construction 1.5: Construction of a line through P (not on l) Parallel to l.
Essentially, with P as one vertex and one side on the line, construct a parallelogram (or rhombus).
Parallel Projections
Vertical projection -- along a perpendicular
Projection of P in any direction
Projection of P on parallel
Projection from a line k to a line l in the direction of m
Theorem 1.28: Parallel projection preserves betweenness
Theorem 1.29: A parallel projection preserves congruence of segments belonging to the same line
Corollary 1.6: If three or more parallel lines intercept congruent segments on one transversal, then they intercept congruent segments on any other transversal.
Construction 1.6: Division of a segment into any number n of congruent parts.
Theorem 1.30: A line through the midpoint of one side of a triangle and parallel to the second side bisects the third side.
Theorem 1.31: The Midsegment Theorem
The segment connecting the midpoints of two sides of a triangle is parallel to the third side and half as long as that side.
Now Solve This 1.11
We have seen that the diagonals of a parallelogram bisect each other. Suppose they bisect each other at O.
1. If we draw a line through O and it intersects two parallel sides of the parallelogram at points P and Q, respectively, prove that OP = OQ
2. The statement in Problem 1 is a generalization of a fact stated in the proof of Theorem 1.29. Which fact? Why is Problem 1 a generalization of the fact?
Medians of a Triangle
Theorem 1.32: The medians of a triangle are concurrent in a point whose distance from a vertex is two-thirds of the length of the median from that vertex.
Now Solve This 1.12
Alternative proof that any two medians intersect at a point that divides each median into segments whose lengths are in a ratio 2:1.
Example 1.6
What kind of quadrilateral is obtained when the midpoints of consecutive sides of the quadrilateral are connected? See GSP file.
Prove that the resulting figure is a parallelogram
Does the quadrilateral have to be convex?
What if the quadrilateral is not a simple closed figure, but 'crosses itself?'
Do the vertices of the quadrilateral for any of the above need to be in the same plane?
Example 1.7
Exploration of when the parallelogram in Example 1.6 is a rhombus or a rectangle.
Example 1.8
1. State and prove the Midsegment Theorem for trapezoids.
2. Prove that the segment connecting the two midpoints of the diagonals of a trapezoid is parallel to the bases of the trapezoid and find the length of this segment if the bases are of lengths a and b. Open GSP File to explore.
Now Solve This 1.13
Treasure Island Problem
Problem Set 1.3 (37 Problems)
