Chapter 1: Congruency

Question 1

What do we assume in this course?

Answer 1

1. A full rotation about a point is 360°, the angle along a line is 180°, a right angle is 90°.
2. Given any 2 distinct points, there is a unique line passing through them.
3. Given a pt and a distance, there is a unique circle whose centre is the point and whose radius is the distance.
4. Triangle Inequality: The sum of the length of 2 sides of a triangle is always greater than the length of the other side.
5. 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.

Question 2

What is the notation for this course?

Answer 2

• 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

Question 3

Thm 1.2.1:

Answer 3

Vertically opposite angles are equal.

Pf provided.

Question 4

Define parallel lines.

Answer 4

Two lines that either do not intersect or are the same line.

Question 5

Thm 1.3.1/2:

Answer 5

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)

Question 6

Thm 1.3.3:

Answer 6

The sum of angles in a triangle is 180°.

Pf provided.

Question 7

Thm 1.3.4 (Exterior Angle Thm):

Answer 7

An exterior angle of a triangle is equal to the sum of the opposite interior angles.
Pf provided.

Question 8

Corollary 1.3.5 9Exterior Angle Inequality):

Answer 8

An exterior angle of a triangle is greater that either of it’s opposite interior angles.

Question 9

Define congruency.

Answer 9

They are basically the same, but sitting in different positions or orientations (≅),

Question 10

Axiom 1.2.2 (SAS Congruency):

Answer 10

Two triangles are congruent if two sides and the angle between them…

Question 11

Thm 1.2.3 (SSS Congruency):

Answer 11

Two triangles are congruent if the three sides…

Question 12

Thm 1.2.4 (ASA Congruency):

Answer 12

Two triangles are congruent if two angles and the included side of one…

Question 13

Thm 1.4.1 (SAA Congruency):

Answer 13

Two triangles are congruent if two angles and a side of one…

Question 14

Define Isosceles triangle.

Answer 14

Two sides of equal length.

Question 15

Thm 1.2.6/7 (Isosceles Triangle Thm):

Answer 15

In △ABC, AB=AC iff ∠ABC=∠ACB.

Pf provided.

Question 16

Thm 1.2.9/10 (Angle Side Inequality):

Answer 16

In △ABC, ∠ABC > ∠ACB iff AC is longer than AB.

Pf provided.

Question 17

Thm 1.5.7 (Open Jaw Inequality):

Answer 17

Given △ABC and △DEF, with AB≅DE and BC≅EF, then ∠ABC < ∠DEF iff AC

Question 18

Thn 1.4.2 (SSA+ Congruency):

Answer 18

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.

Question 19

Exercise 1.5.3:

Answer 19

Prove that the hypotenuse of a right triangle is its longest side.
Soln provided.

Question 20

Corollary 1.4.3:

Answer 20

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+).

Question 21

Define a Quadrilateral and the 3 classification terms.

Answer 21

ABCD denotes the quad. with edges AB, BC, CD, DA. (Not same as ABDC).
Classified by convex, simple, and non-simple.

Question 22

Thm 1.3.10:

Answer 22

The sum of interior angles of a simple quad. is 360°.
Pf provided.
(False if non-simple, where sum <360°)

Question 23

Refer to Logic video for defns terminology and proof types.

Answer 23

Refer to Logic video for defns terminology and proof types.