Geometric Proofs Midterm
Which of Euclid's five postulates apply to spheres?
1-4
Which of Euclids postulates make up absolute geometry?
1-4
If you are standing at a point (4,9), find all the points exactly 5 units away from you in Euclidean geometry.
12
Consider the Earth is a sphere with radius 6371 km. What is the shortest distance from Quito, Ecuador (0N, 78W) to Kampala, Uganda (0N, 32E).
12,231.44km
Consider the Earth is a sphere with radius 6371 km. What is the distance if traveling due East from Saint Petersburg, Russia (60N, 30E) to Anchorage, Alaska (60N, 150W).
150+30=180 180×111.2cos(60N)= 10,008km
Consider the Earth is a sphere with radius 6371 km. What is the formula you would use to find the shortest distance from Saint Petersburg, Russia (60N, 30E) to Anchorage, Alaska (60N, 150W)
150-30=120, 120×111.2cos(60N)
What is a point defined as?
1D object in space
If you are standing at a point (4,9), find all the points exactly 5 units away from you in taxicab geometry.
20
Consider the Earth is a sphere with radius 6371 km. What is the formula for the shortest distance between Paris (48N, 2E) to Seattle (48N, 122W)?
2π(6371)(124/360)
Consider the Earth is a sphere with radius 6371 km. What is the shortest distance from Saint Petersburg, Russia (60N, 30E) to Anchorage, Alaska (60N, 150W)
6672km
Consider the Earth is a sphere with radius 6371 km. Saint Petersburg, Russia (60N, 30E) to Anchorage, Alaska (60N, 150W), if you continue traveling due east from Alaska until you reach Russia from the west, what path will you have traversed?
Arctic circle
For each statement, give the reasoning Proof: suppose we know angle ABC is congruent to angle DEF and pairs: angle ABC and angle CBG, and angle DEF and angle FEH are linear 10. it follows that angle CBG is congruent to FEH as desired
CPCTC
Which type of geometry is defined as "the most direct way to get to where you want"?
Euclidean
Which type of geometry will be a circle that will look like a circle?
Euclidean
Which axioms are considered neutral geometry?
Hilbert
For each statement, give the reasoning Proof: for every line l and every point P there exists a line through P perpendicular to l 6. if Q≠A then triangle PAQ congruent to triangle P'AQ
SAS
For each statement, give the reasoning Proof: suppose we know angle ABC is congruent to angle DEF and pairs: angle ABC and angle CBG, and angle DEF and angle FEH are linear 4. then triangle ABC is congruent to triangle A'EC'
SAS
For each statement, give the reasoning Proof: suppose we know angle ABC is congruent to angle DEF and pairs: angle ABC and angle CBG, and angle DEF and angle FEH are linear 7. triangle AGC is congruent to A'G'C'
SAS
For each statement, give the reasoning Proof: suppose we know angle ABC is congruent to angle DEF and pairs: angle ABC and angle CBG, and angle DEF and angle FEH are linear 9. hence triangle CBG congruent to C'EG'
SAS
Which axioms are based off Euclidean geometry?
SMSG
What type of geometry states that you cannot cut through but go around? a. intrinsic b. extrinsic
a
Which of these definitions describes spherical geometry? a. the sum of the angles in a triangle are always greater than 180. Lines are defined as great circles. Satisfies 2 of 5 of Euclid's postulates b. the distance between two points is the sum of the absolute differences of their coordinates. Intrinsic distance. The distance goes unit by unit. c. based on four axioms of Euclidean and Hilbert's geometry without the parallel postulate. d. geometry with constant negative curvature. Similar to saddle surfaces. e. study of plane and solid figures on the basis of axioms and theorems. uses the distance formula. extrinsic distance. shortest distance from any pair of points.
a
Working with shortest distance between cities, what would you do for two different direction of cities? a. add them b. subtract them
a
What is a plane defined as?
a 2D flat object in space
What is a Great Circle defined as?
a circle whose center is the center of a sphere
What is a lemma?
a helping theorem
What is the definition of transversal?
a line that cuts across two or more lines
What is a definition?
a way to illustrate the meaning of a word
Provide an example for: things which are equal to the same thing are also equal to one another
a=90, b=90, a=b
What is the definition of scalene triangles?
all sides are different lengths
What is the definition of equilateral triangles?
all sides are the same
In hyperbolic geometry, the sum of the angles in a triangle is ________
always less than 180
In spherical geometry, the sum of the angles in a triangle is ________
always more than 180
Redefine: to produce a finite straight line continuously in a straight line
any line segment can be extended indefinitely
Redefine: that all right angles are equal to one another
any two right angles are congruent
Consider the Earth is a sphere with radius 6371 km. What is the formula you would use to find the shortest distance from Quito, Ecuador (0N, 78W) to Kampala, Uganda (0N, 32E)
arc length= 2πr(θ/360)
What is a postulate?
assumptive answer to a question
What do equilateral and isosceles triangles have in common?
at least two sides that are the same
For each statement, give the reasoning Proof: suppose we know angle ABC is congruent to angle DEF and pairs: angle ABC and angle CBG, and angle DEF and angle FEH are linear 1. find a point A' on ED with AB congruent to A'E
axiom C1
What type of geometry states that you just go straight to it? a. intrinsic b. extrinsic
b
Which of these definitions describes taxicab geometry? a. the sum of the angles in a triangle are always greater than 180. Lines are defined as great circles. Satisfies 2 of 5 of Euclid's postulates b. the distance between two points is the sum of the absolute differences of their coordinates. Intrinsic distance. The distance goes unit by unit. c. based on four axioms of Euclidean and Hilbert's geometry without the parallel postulate. d. geometry with constant negative curvature. Similar to saddle surfaces. e. study of plane and solid figures on the basis of axioms and theorems. uses the distance formula. extrinsic distance. shortest distance from any pair of points.
b
Working with shortest distance between cities, what would you do for two of the same direction for cities? a. add them b. subtract them
b
Which of these definitions describes absolute geometry? a. the sum of the angles in a triangle are always greater than 180. Lines are defined as great circles. Satisfies 2 of 5 of Euclid's postulates b. the distance between two points is the sum of the absolute differences of their coordinates. Intrinsic distance. The distance goes unit by unit. c. based on four axioms of Euclidean and Hilbert's geometry without the parallel postulate. d. geometry with constant negative curvature. Similar to saddle surfaces. e. study of plane and solid figures on the basis of axioms and theorems. uses the distance formula. extrinsic distance. shortest distance from any pair of points.
c
Which of these is NOT a difference between SMSG and Hilbert's axioms? a. SMSG is common in K12 while Hilbert is not b. SMSG does not have continuity, betweenness or area but Hilbert does c. SMSG does not include space while Hilbert does d. SMSG covers volume while Hilbert does not
c
Which of these is NOT one of Euclid's postulates? a. to draw a straight line from any point to any point b. that all right angles are equal to one another c. things which are equal to the same thing are also equal to one another d. to describe a circle with any center and distance
c
What do isosceles and scalene triangles have in common?
can have a right angle or obtuse
What do equilateral, isosceles and scalene triangles have in common?
can have all acute angles
What do equilateral and scalene triangles have in common?
can have an angle that is 60
When the scale factor of a dilation is less than one, ______ occurs
compression
For each statement, give the reasoning Proof: for every line l and every point P there exists a line through P perpendicular to l 3. there is a point P' on AX such that AP' congruent to AP
congruence 1
For each statement, give the reasoning Proof: for every line l and every point P there exists a line through P perpendicular to l 2. on the opposite side of l from P there is a ray AX such that angle XAB congruent to PAB
congruence 4
For each statement, give the reasoning Proof: for every line l and every point P there exists a line through P perpendicular to l 9. we can lay off an angle congruent to this right angle with vertex P and one side of l; the other side of this angle is part of a line through P perpendicular to l
congruence 4
What does a parallel projection preserve?
congruence, ratios of lengths of segments
What is hyperbolic geometry?
constant negative curvature
Which of these definitions describes hyperbolic geometry? a. the sum of the angles in a triangle are always greater than 180. Lines are defined as great circles. Satisfies 2 of 5 of Euclid's postulates b. the distance between two points is the sum of the absolute differences of their coordinates. Intrinsic distance. The distance goes unit by unit. c. based on four axioms of Euclidean and Hilbert's geometry without the parallel postulate. d. geometry with constant negative curvature. Similar to saddle surfaces. e. study of plane and solid figures on the basis of axioms and theorems. uses the distance formula. extrinsic distance. shortest distance from any pair of points.
d
Which of these is NOT one of Euclid's common notions? a. the whole is greater than the part b. things which coincide with one another are equal to one another c. if equals be added/subtracted from equals the wholes/remainders are equal d. to produce a finite straight line continuously in a straight line
d
For each statement, give the reasoning Proof: suppose we know angle ABC is congruent to angle DEF and pairs: angle ABC and angle CBG, and angle DEF and angle FEH are linear 5. Hence AC is congruent to A'C' and angle BAC is congruent to EA'C'
definition of congruent triangles
For each statement, give the reasoning Proof: suppose we know angle ABC is congruent to angle DEF and pairs: angle ABC and angle CBG, and angle DEF and angle FEH are linear 8. therefore, CG congruent to C'G' and angle BGC congruent to EG'C'
definition of congruent triangles
For each statement, give the reasoning Proof: for every line l and every point P there exists a line through P perpendicular to l 4. Let Q be the intersection PP' and l
definition of opposite sides of l
For each statement, give the reasoning Proof: for every line l and every point P there exists a line through P perpendicular to l 7. hence angle PQA congruent to P'QA therefore PP' perpendicular to l
definition of perpendicular
For each statement, give the reasoning Proof: for every line l and every point P there exists a line through P perpendicular to l 5. If Q=A then PP' perpendicular to l
definition of perpendicular lines
What is a corollary?
direct consequence of a proven theorem
Briefly describe Euclidean geometry
distance between two points on a plane where points, lines and planes can take any direct path
Briefly describe spherical geometry
distance between two points on a sphere, similar to latitude and longitude
Which of these definitions describes Euclidean geometry? a. the sum of the angles in a triangle are always greater than 180. Lines are defined as great circles. Satisfies 2 of 5 of Euclid's postulates b. the distance between two points is the sum of the absolute differences of their coordinates. Intrinsic distance. The distance goes unit by unit. c. based on four axioms of Euclidean and Hilbert's geometry without the parallel postulate. d. geometry with constant negative curvature. Similar to saddle surfaces. e. study of plane and solid figures on the basis of axioms and theorems. uses the distance formula. extrinsic distance. shortest distance from any pair of points.
e
When the scale factor of a dilation is greater than one, ______ occurs
enlargement
Which postulate when added makes up Euclidian geometry?
fifth
Redefine: to describe a circle with any center and distance
given two distinct points P and Q, a circle centered at P with radius PQ can be drawn
Redefine: to draw a straight line from any point to any point.
given two distinct points P and Q, there is exactly one line that passes through P and Q
Provide an example: things which coincide with one another are equal to one another
if one thing can be moved to coincide with another then they are equal
For each statement, give the reasoning Proof: for every line l and every point P there exists a line through P perpendicular to l 1. we assume first that P does not lie on l, and that A and B are any two points on l
incidence 2
For each statement, give the reasoning Proof: for every line l and every point P there exists a line through P perpendicular to l 8. assume now P is on l. Since there are points not lying on l, we can drop a perpendicular from one of them to l, thereby obtaining right angle
incidence 3
What is a circle?
infinitely many points that are equal distance from the origin
What are the undefined terms in SMSG that are NOT in Hilbert's axioms?
lie on, distance, angle measure, area, volume
Provide an example: if equals be added to equals, the wholes are equal
magnitudes of the same kind can be compared and added but magnitudes of different kinds cannot
Provide an example: if equals are subtracted from equals, the remainders are equal
magnitudes of the same kind can be compared and subtracted but magnitudes of different kinds cannot
What does a midpoint do?
perfectly bisects two points
What are the top 3 building blocks of geometry this class will be focused on?
planes, points, lines
What does Hilbert's incidence axioms describe?
points and lines
What are the undefined terms that exist in both SMSG and Hilbert's axoms?
points, planes, lines
What are the transformations that are isometries?
rotation, reflection, translation
What is a line defined as?
set of points infinitely close and extending in one dimension indefinitely
What is a path defined as?
shortest distance between two points on a sphere
What is an axiom?
something accepted without question
What is a theorem?
something that must be proven using definitions, postulates and other theorems
Which type of geometry is defined as "going unit by unit to get to where you want"?
taxi-cab
Which type of geometry will be a circle that actually looks like a diamond/square?
taxi-cab
The smaller the triangle, ________
the closer the sum is to 180
Briefly describe taxi-cab geometry
the path between two points is limited to a coordinate plane where the movements can only be horizontal or vertical
What is a transformation?
the process of manipulating a figure in a plane by a rotation, reflection, translation or dilation.
What does it mean for a transformation to be rigid?
the transformation preserves distance, size, and angles
Provide an example: the whole is greater than the part
this could be interpreted as a definition of "greater than"
What is an example of a shape that follows all three of Hilbert's incidence axioms?
triangle with three points
Redefine: that, if a striaght 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 are the angles less than the two right angles.
two lines are parallel if the transversal that cuts them has interior angles that equal to two 90 degree angles
What is the definition of parallel?
two lines perpendicular to the same line are parallel to each other
What is the definition of isosceles triangles?
two sides are the same
What is the order of the axiomatic system?
undefined terms, interpret system, model of system
Does Euclid's fifth postulate apply to spheres?
yes but there are no parallel lines on a sphere so it does not apply
What is the formula for Taxi Cab circles?
|x₁-x₂|+|y₁-y₂|
Write the distance formula.
√(x₂-x₁)²+(y₂-y₁)²
