Intro to Geometry

Question 1

State the first 4 of Euclids Axioms

Answer 1

1) Through any two points there is a unique line 2) Its possible to draw a unique circle of any given radius around any given point 3)Its possible to extend any line segment continuously to a larger line segment 4) All right angles are equal

Question 2

When are points collinear

Answer 2

When there is a straight line between all of them

Question 3

Given 3 points ABC in the plane, how do we measure the angle

Answer 3

Question 4

Lemma; Let a and b be a pair of intersecting lines. Then the two angles where we go counter clockwise from a to b are equal

Answer 4

alpha + beta = pi

alpha^prine + beta = pi

therfore alpha = alpha^prine

Question 5

Define the angle ab of two intersecting lines a and b

Answer 5

L(ab) is the value of either of the two equal angles from a to b anti-clockwise

Question 6

When are two lines a and b perpendicular

Answer 6

When L(ab) = L(ba) = pi/2

Question 7

What is the perpendicular bisector of AB

Answer 7

The unique line which passes through the midpoint of AB and is perpendicular to AB

Question 8

Give three examples of isometric maps and one which is not

Answer 8

Reflection, Rotation and Translation are

Dilation isnt

Question 9

Define the notation for a rotation, translation and a dilation

Answer 9

Rotation - R_{(O, theta)} where O is the central point

T_v: R to R ie (x,y) to (x + v1, y + v2)

D_{(O, alpha)} dilates a map by alpha from centre O

Question 10

Define an isometry

Answer 10

A map f:Plane to the Plane if for any two points A and B in the plane d(A,B) = d(f(A),f(B))

Question 11

When is an isometry orientation preserving

Answer 11

L(ABC) = L(f(A)f(B)f(C))

Question 12

Define Congrunecy

Answer 12

Question 13

When are two line segments congruent

Answer 13

When they are of equal length, | AB | = | CD | and theres an isometry taking A to C and B to D

Question 14

What is Axiom 7

Answer 14

Given a line l and a point P, there exists an isometry which leaves l fixed and moves P to the other side of l

Question 15

When is a triangle ABC clockwise orientated

Answer 15

If moving from A to B to C takes you clockwise

Question 16

Lemma; Isometries preserve internal angles of a triangle. More precisely, let ABC and A’B’C’ be such that ABC ~ A’B’C’ then alpha = alpha^prine beta = beta^prine gamma = gamma^prine

Answer 16

Question 17

Prove the SAS Congruence criterion, where | BA | = | B’A’ |, | BC | = | B’C’ | and beta=beta^prine

Answer 17

Question 18

Prove the ASA Congruence criterion, where | BC | = | B’C’ |, beta=beta^prine and gamma=gamma^prine

Answer 18

Question 19

When is a triangle an isosceles

Answer 19

When it has two sides that are the same length

Question 20

Lemma; Let ABC be a triangle in the plane. Then | AC | = | BC | if and only if alpha = beta

Answer 20

If alpha = beta. Then ABC is congruent to BAC by ASA so | AC | = | BC |

If | AC | = | BC | then ABC is congruent to BAC by SAS so alpha = beta

Question 21

Prove the SSS congruence criterion

Answer 21

Question 22

Lemma; For any two points A and B in the plane the perpendicular bisector of AB is the locus of points equidistant from A and B

Answer 22

Question 23

Theorem; In any triangle ABC the perpendicular bisectors meet at a point O. The point O is the centre of the unique circle passing through A, B and C

Answer 23

Question 24

Define a tangent

Answer 24

A line l is tangent to a circle C at point P if it meets C only at P

Question 25

Lemma; Let l be a line and P a point not on l. Then there exists a unique line through P which is perpendicular to l, and the shortest distance from P to l is along this line

Answer 25

Question 26

Lemma; Through any point P on a circle C there is a unique line tangent to C. It is the line perpendicular to OP where O is the centre of the circle

Answer 26

Question 27

State the Parallel Postulate

Answer 27

If a straight line intersecting two straight lines makes the interior angles on one of the sides which are less than two right angles in total, then the two straight lines, if produced indefinitely, meet on that side

Question 28

State Playfairs Axiom

Answer 28

Given a line l and a point P not on l, there is a unique line through P parallel to l

Question 29

Theorem; Euclids 5th Axiom implies Playfairs Axiom

Answer 29

Question 30

Let l₁ and l₂ be a pair of parallel lines. Let l₃ be a line falling on l₁ and l₂. Then alpha = beta^prine and beta = alpha^prine

Answer 30

Question 31

Lemma; The interior angles of a triangle sum to pi

Answer 31

Question 32

Let ABCD be a parallelogram with AB||CD and AC||BD, then |AC| = |BD| and |AB| = |CD|

Answer 32

Question 33

Give the three statement definitions of Area

Answer 33

1) The area of a rectangle is the product of the lengths of two adjacent sides
2) If two figures are disjoint or meet along their edges, then the area of their union is the sum of their areas
3) Congruent figures have equal areas

Question 34

Lemma; The area of a parallelogram is base times height

Answer 34

Question 35

Lemma; The area of a triangle is a half times the base times the height

Answer 35

Question 36

When are two triangles similar

Answer 36

When are the angles are equal, denoted by ~

Question 37

State and Prove Pythagorus’ Theorem

Answer 37

Question 38

Define a great circle

Answer 38

A great circle, or a geodesic on a sphere is the intersection of the sphere with a plane passing through its origin

Question 39

When are two points antipodal

Answer 39

When the straight line connecting them passes through the origin

Question 40

Lemma; Through any two non-antipodal points on a sphere there passes a unique great circle

Answer 40

Question 41

What is Ambient and Intrinsic notion?

Answer 41

Ambient - straight line connecting A and B

Intrinsic - Smallest arc of any great circle connecting A and B

Question 42

Give 2 examples of spherical isometries

Answer 42

Rotation about an axis through the centre of the sphere

Any reflection in a plane passing through the centre of the sphere

Question 43

Define the spherical angle BAC

Answer 43

Question 44

State the sum of angles of a triangle ABC on a sphere of radius r

Answer 44

L(A) + L(B) + L(C) = pi + area(ABC)/r²