Two tribes live on an island. Members of one tribe always tell the truth and members of the other tribe always lie. You arrive on the island and meet two islanders named Flora and Fred. Flora says, “Only one of us is from the tribe that always lies.” Which tribe does Fred come from? ~ X not possible because Flora would have lied ~ Possibly correct ~ X not possible because Flora would have been telling the truth when she was actually a liar ~ Possibly correct Therefore, either correct option makes Fred a liar. Fun with Reasoning Flora Fred T T T L L T L L
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Two tribes live on an island. Members of one tribe always tell the truth and members of the other tribe always lie. You arrive on the island and meet two.
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Two tribes live on an island. Members of one tribe always tell the truth and members of the other tribe always lie.
You arrive on the island and meet two islanders named Flora and Fred.
Flora says, “Only one of us is from the tribe that always lies.”
Which tribe does Fred come from?
~ X not possible because Flora would have lied
~ Possibly correct
~ X not possible because Flora would have been
telling the truth when she was actually a liar
~ Possibly correct
Therefore, either correct option makes Fred a liar.
Fun with Reasoning
Flora Fred
T T
T L
L T
L L
StarterFactorize x2 + x - 12
= (x + 4)(x – 3)
Expand (2x + 3)(x – 4)
= 2x2 – 5x – 12
Geometric Reasoning
ANGLE PROPERTIES OF LINES AND TRIANGLES
Angle Definitions• Acute• Obtuse
• Reflex
• Less than 90º• More than 90º, less
than 180º• More than 180º
Angle NotationUse Capital Letters at vertex of angle
Use lower case for case for opposite side
Angles can also be described as
BÂC or BAC
Angle Rules
• Straight line
• Vertically Opposite
• At a point
• Triangle
• Parallel lines
• angles of polygons
1
3
4
5
2 6
Angles on a Straight Line
• Angles on a straight line add to 180o
• x + 117o = 180o ( ‘s on line)• x = 63o
‘s on line
Vertically Opposite
• Vertically Opposite angles are equal
• xo = 40º (Vert opp ‘s)
• yo + 40o = 180º ( ‘s on line)
• yo = 140º
Vert opp ‘s
Angles at a Point
• Angles at a point add to 360o
u + 100º + 90º + 75º = 360º
u + 265º = 360º
u = 360º - 265º
u = 95º ( ‘s at pt)
‘s at pt
Angles of a triangle
• The sum of all angles in a triangle = 180º
50º + 70º +s = 180º
120º + s = 180º
s = 180º -120º
s = 60º ( Sum of )
Sum of
Exterior Angles of a Triangle
• The exterior angle of a triangle is the sum of the two interior opposite angles
Ext of
tº = 50º + 70ºtº = 120º (Ext of )
Special Triangles
• Isosceles – 2 sides are equal • 2 base angles are equal
22 + i + j = 180º but i = j (isosceles)22 + 2 i = 180º2i = 180º - 22º2i = 158º i = 79º , j = 79º
Base ‘s isos
Equilateral Triangles
• 3 equal sides → 3 equal angles 180º / 3 = 60º
n + p + o = 180º
But as equilateral, n = p = o
So 3n = 180º
n = 60º = p =o
equilat
Practice Problems
GAMMA Text - Exercise 31.01 – pg. 448-450
• Q #1 ~ basic (you can skip this if you want)
• Q #2-17 ~ good achievement questions
• Q #18-25 ~ gets increasingly more difficult
IWB Gamma Mathematics Ex 18.01 pg 447
Starter
Simplify
Simplify
62
35
9
3
yx
yx
3
3
3y
x
46
23
x
x
2
1
Note 2: – Properties of Parallel Lines
58°
58°x y
103°
103°
sr
Parallel line angles
• Corresponding angles on parallel lines are equalw = 55o
• Alternate angles on parallel lines are equalg = 38o
• Co-interior angles on parallel lines add to 180o
y + 149º =180º y = 180º -149º y = 31º
Example – Parallel Lines
A walkway and its hand rail both slope upwards at an angle of 6º. Calculate the size of the
co-interior angles of the bars, base and handrails
x = 90º – 6º = 84º
6º Risex
y
y = 90º + 6º = 96º
Practice Problems - GAMMA
• Ex 31.02 pg. 451 # 6a, c, 7a, c, 8• Ex 31.03 pg 453 # 3, 5
Another interesting feature of tangents and circles……..
When you form a quadrilateral from 2 tangents and 2 radii, the result is always a cyclic quadrilateral !
Starter
B C
DA
O 53
yz
v
u
xw
w + x = 90º (ے in a semi circle = 90)v = 180 – 90 – 53 = 37º (sum ے in ∆ )t = 90 – 37 = 53ºx = t = 53º (from the same arc =)u = 53º (ے from the same arc =)z = 180 – 53 – 53 = 74º (sum ے in ∆ )y = 180 – 74 = 106º (ےon a line)t
Notice that these are all isoceles triangles – however we did not need to know this to solve for these angles – EXAMPLE of a PROOF!