Group Laws/Defintions
Simplifying Compositions
Euclid's Postulates
Angle Pairs/Transversals
Constructions
100
What is the operation in a group that is associative? (This means these don't matter).
Parentheses/grouping
100
Fill in the blank
In Dn: rn= j500 = ____
e
100
Name Postulate #1
A straight line segment can be drawn joining any two points.
100
If 𝛉 and 𝛟 are vertical angles and we know that 𝛉 = 26.5°, what is 𝛟?
𝛟 = 26.5°
100
Identify This Construction:
Bisecting a line segment
200
Which subgroup of D6 is not cyclic?
A. {e, r2, r4}
B. {e, r, r2, r3, r4, r5}
C. {e, r3, j, r3j}
D. {e, r5j}
C. {e, r3, j, r3j}
200
Fill in the blank
In Dn: r37j = jr___
n-37
200
Name Postulate #2
Any straight line segment can be extended infinitely in a straight line.
200
If 𝝰 and 𝝱 are consecutive interior angles, and we know that 𝝰 = 178°, what is 𝝱?
𝝱 = 2°
200
Sketch the construction for bisecting an angle.
300
What is the operation in a dihedral group?
Composition
300
Find the base element in Dn that is equivalent to
In D4: r121j3176r5j9r47
r3j
300
Name Postulates #3 and #4
Any straight line segment can be drawn as a circle having the segment as the radius and one endpoint as the center.
All right angles are congruent.
300
∠BAD and ∠CAD are complementary (add up to 90°). Solve for x and find angle 𝛉. Explain with angle pairs.
(3x - 10) + (x + 20) = 90 (Given that angles were complementary)
4x + 10 = 90
4x = 80
x = 20
∠DAB = (20 + 20) = 40
Since angle 𝛉 is a vertical angle of ∠DAB, then 𝛉 = 40°.
300
Sketch the construction for copying an angle.
400
Which is not a subgroup of D8?
A. {e, r4, rj, r5j}
B. {e, r4j, j, r4}
C. {e, r5, j, r5j}
D. {e, r2, r4, r6}
C. {e, r5, j, r5j}
400
Find the base element in Dn that is equivalent to
In D4: r77j999r2j9r6
r3
400
Name Postulate #5
If two lines are drawn which intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines must intersect each other on that side if extended far enough (Parallel Postulate).
400
The lines highlighted with the triangles are parallel. Line AC is a transversal. Solve for x and find angle 𝜸. Explain with angle pairs.
(4x + 23) is a corresponding angle with C.
C + (x + 42) = 180 (supplementary angles)
(4x + 23) + (x + 42) = 180
5x + 65 = 180
5x = 115
x = 23
𝜸 = (23 + 42) = 65° (alternate exterior angles)
400
Prove the construction for creating a perpendicular line through a point.
Answer may vary: one example is to use SSS for the triangles CDE and CDF, to show that the angles at C are congruent and supplementary, thus equalling 90 degrees.
500
In Dn: Which of these numbers are the ones that will generate the subgroup of all rotations?
the numbers coprime to n
500
Find the base element in Dn that is equivalent to
In D6: r999j999r999j999r999
r3
500
Explain why doesn't each of Euclid’s Postulates work in spherical geometry (Need to mention all 5)!
#1 - gives curve
#2 - gives circle
#3 - gives bowl or dome (any description of a cross section with a curved radius and a circle for a base)
#4 - creates a situation where a triangle is great than 180°
#5 - “parallel lines” intersect at the north and south poles (thinking of longitude lines on Earth).
500
The following sets of lines are parallel: AB and CD, AC and BD, and FG and CO. Solve for x and find the angles 𝝷, 𝝱, and 𝜡 of Triangle AFE. Explain using ALL necessary angle pairs.
(4x + 12) is a corresponding angle with ∠ABG.
(3x - 28) + (4x + 12) = 180 (Supplementary Angles)
7x - 16 = 180
7x = 196
x = 28
∠BAF = 3(28) - 28 = 56° (Alternate Interior Angles)
64 + 𝝷 + ∠BAF = 180 (Supplementary Angles)
64 + 𝝷 + 56 = 180
𝝷 = 60°
∠COG = 66° (Vertical Angles)
∠CFG = 66° (Alternate Interior Angles)
𝝱 = 66° (Vertical Angles)
𝝷 + 𝝱 + 𝜡 = 180° (Triangle Angle Sum)
60 + 66 + 𝜡 = 180°
𝜡 = 54°
500
Prove the construction for creating a parallel line.
Identifies that DE = CG, DF = CH, and EF = GH so the triangles formed by those sides are congruent. Thus angle EDF = angle GCH. So by our transversal rules, these lines are parallel.
