What do we assume in this course?
- A full rotation about a point is 360°, the angle along a line is 180°, a right angle is 90°.
- Given any 2 distinct points, there is a unique line passing through them.
- Given a pt and a distance, there is a unique circle whose centre is the point and whose radius is the distance.
- Triangle Inequality: The sum of the length of 2 sides of a triangle is always greater than the length of the other side.
- Parallel Postulate: Given two lines and a third cutting through both, if interior angles on the same side are less than 180°, then the two lines meet at a point on that same side.
- > Equivalent to Playfair’s Axiom: Given a line and a pt in the plane, there is exactly one line through P parallel to the line.
What is the notation for this course?
- AB is line segment
- AB (arrow top) is ray starting at A
- AB (two-sided arrow top) is line
- ∠BAC (sometimes ∠A) is interior angle. Outside angle is the reflex angle.
- △ABC is the triangle
Thm 1.2.1:
Vertically opposite angles are equal.
Pf provided.
Define parallel lines.
Two lines that either do not intersect or are the same line.
Thm 1.3.1/2:
If a transversal cuts 2 lines, the following are equivalent:
1. The adjacent angles total 180°
2. The alternate angles are equal
3. The corresponding angles are equal
4. The alternating exterior angles are equal
5. The two lines are parallel
(if one is true, all are true)
Thm 1.3.3:
The sum of angles in a triangle is 180°.
Pf provided.
Thm 1.3.4 (Exterior Angle Thm):
An exterior angle of a triangle is equal to the sum of the opposite interior angles.
Pf provided.
Corollary 1.3.5 9Exterior Angle Inequality):
An exterior angle of a triangle is greater that either of it’s opposite interior angles.
Define congruency.
They are basically the same, but sitting in different positions or orientations (≅),
Axiom 1.2.2 (SAS Congruency):
Two triangles are congruent if two sides and the angle between them…
Thm 1.2.3 (SSS Congruency):
Two triangles are congruent if the three sides…
Thm 1.2.4 (ASA Congruency):
Two triangles are congruent if two angles and the included side of one…
Thm 1.4.1 (SAA Congruency):
Two triangles are congruent if two angles and a side of one…
Define Isosceles triangle.
Two sides of equal length.
Thm 1.2.6/7 (Isosceles Triangle Thm):
In △ABC, AB=AC iff ∠ABC=∠ACB.
Pf provided.
Thm 1.2.9/10 (Angle Side Inequality):
In △ABC, ∠ABC > ∠ACB iff AC is longer than AB.
Pf provided.
Thm 1.5.7 (Open Jaw Inequality):
Given △ABC and △DEF, with AB≅DE and BC≅EF, then ∠ABC < ∠DEF iff AC
Thn 1.4.2 (SSA+ Congruency):
The SSA rule for congruency holds if the length of the side opposite the given angle is larger than or equal to the other side.
Pf provided.
Exercise 1.5.3:
Prove that the hypotenuse of a right triangle is its longest side.
Soln provided.
Corollary 1.4.3:
If the hypotenuse and one side of a right angle triangle are congruent to the hypotenuse and one side of another, then the triangles are congruent (SSA+).
Define a Quadrilateral and the 3 classification terms.
ABCD denotes the quad. with edges AB, BC, CD, DA. (Not same as ABDC).
Classified by convex, simple, and non-simple.
Thm 1.3.10:
The sum of interior angles of a simple quad. is 360°.
Pf provided.
(False if non-simple, where sum <360°)
Refer to Logic video for defns terminology and proof types.
Refer to Logic video for defns terminology and proof types.
