Relationships in Triangles

Bisectors, Indirect Proof Indirect Proof The Triangle Inequalities
Medians, & Inequality Involving Two
Altitudes Triangles

2 points 2 points 2 points 2 points 2 points

Team 1 : Team 2 :

Bisectors, Medians, & Altitudes: 2 Points
In ∆GHJ, HP=5x-16, H
N
PJ=3x+8, m<GJN=6y-3,
m<NJH=4y+23, &
m<HMG=4z+14 G P
Stop!
M
Time’s Up!
The Question Is :
J
GP is a median of ∆GHJ. Find HJ.

The Answer Is :
88

N
m<NJH=4y+23, &
m<HMG=4z+14 G P
Stop!
The Question Is : M
Time’s bisector.
Find m<GJN if JN is an angle
Up! J

The Answer Is :
150

N
m<NJH=4y+23, &
m<HMG=4z+14 G P
Stop!
M
The Question Is :
Time’s find the value of
If HM is an altitude of ∆GHJ,
Up! J
z.
The Answer Is :
19


The Question Is :
All of the angle bisectors of a triangle meet at
the _________. Stop!
Time’s Up!
The Answer Is :
incenter

CP is an altitude, CQ is the C
angle bisector of <ACB, and R is
the midpoint of AB.

The Question Is : Stop!
and m<QCB=42+x.
Time’s Up! P
Find m<ACQ if m<ACB=123-x
A Q R B

The Answer Is :
m<ACQ = 55

CP is an altitude, CQ is the C
angle bisector of <ACB, and R is
the midpoint of AB.

The Question Is : Stop!
Find AB if AR=3x+6 and
RB=5x-14 Time’s Up! P
A Q R B

The Answer Is :
AB = 72


The Question Is :
In ∆RST, if the point P is the midpoint of RS,
Stop!
then PT is a(n) ________.

The AnswerTime’s Up!
Is :
median


The Question Is :
What are the special segments of triangles?
Stop!
Time’s Up!
The Answer Is :
Perpendicular bisectors, medians, angle
bisectors, and altitudes

Indirect Proof: 2 Points
The Question Is :
When using _________, you assume that the
conclusion false and then show that this assumption
Stop!
leads to a contradiction of the hypothesis, or some
other accepted fact, such a definition, postulate,
Time’s Up!
theorem, or corollary.

The Answer Is :
Indirect reasoning

The Question Is :
True or False: Another name for an indirect
proof is a proof by contradiction.
Stop!
Time’s Up!
The Answer Is :
True

The Question Is :
What are the three steps of writing an indirect proof.
The Answer Is :
1. Assume that the conclusion is false
Stop!
2. Show that this assumption leads to a
Time’s Up!
contradiction of the hypothesis, or some other
fact, such as a definition, postulate, theorem, or
corollary
3. Point out that because the false conclusion leads
to an incorrect statement, the
original conclusion must be true

The Question Is :
State the assumption you would make to
start an indirect proof of the following
Stop!
statement: EF is not a perpendicular bisector.
Time’s Up!
The Answer Is :
EF is a perpendicular bisector.

The Question Is :
State the assumption you would make to
start an indirect proof of the following
Stop!
statement: If 5x < 25, then x < 5.
Time’s Up!
The Answer Is :
x>5

The Question Is :
State the assumption you would make to start an
indirect proof of the following statement: If a
rational number is any number that can be
Stop!
expressed as a/b, where a and b are integers and b
is not equal to 0, 6 is a rational number.
Time’s Up!
The Answer Is :
6 cannot be expressed as a/b, where
a & b are integers and b is not equal
to 0.

The Question Is :
indirect proof of the following statement: The angle
bisector of the vertex angle of an isosceles triangle
Stop!
is also an altitude of the triangle.

Time’s Up!
The Answer Is :
The angle bisector of the vertex angle of an isosceles
triangle is not an altitude of the triangle.

The Question Is :
indirect proof of the following statement: Two lines
Stop!
that are cut by a transversal so that alternate
interior angles are congruent are parallel.

The Answer Is :
Time’s Up!
The lines are not parallel

Inequalities & Triangles: 2 Points
The Question Is :
What is the definition of an inequality?

The Answer Is :
Stop!
For any real numbers a and b,Up!if and only if
Time’s a > b
there is a positive number c such that a = b + c.

The Question Is :
What are the four properties of Inequalities for
Real Numbers?
Stop!
The Answer Is :
Time’s Up!
Comparison Property, Transitive Property,
Addition and Subtraction Properties, &
Multiplication and Division Properties

The Question Is :
What does the “Comparison Property” say?

Stop!
The Answer Is :
Time’s Up!
a < b, a = b, or a > b

The Question Is : 7
4
Use the Exterior Angle
Inequality Theorem to list
Stop!
all angles that satisfy the
stated condition: measures
less than m<1 Time’s Up! 5
6
8
3
2
1

The Answer Is :
<4, <5, <6

The Question Is : 7
4
Determine the angle with the
Greatest measure: <1, <2, <3.
Stop!
The Answer Is : Time’s Up! 5
8
2
1
6 3
<1

The Question Is :
8
Greatest measure: <8, <5, <7. 7
Stop! 5

The Answer Is : Time’s Up!
<5

The Question Is :
6
8
Stop!
The Answer Is :Time’s Up!
<8

The Question Is :
Determine the angle with the 6

Stop! 9

The Answer Is : Time’s Up!
<1

The Triangle Inequality : 2 Points

The Question Is :
What is the “Triangle Inequality Theorem?”

The Answer Is :
Stop!
The sum of the lengths of anyUp!
Time’s two sides of a triangle is
greater than the length of the third side.


The Question Is :
What is the “Triangle Inequality Theorem” used to
determine?
Stop!
The Answer Is :
Time’s Up!
Whether three segments can form a triangle.


The Question Is :
If a line is horizontal, the shortest distance from a point
to that line will be along a _______ line. Likewise, the
Stop!
shortest distance from a point to a vertical line lies
along a ________ line.
Time’s Up!
The Answer Is :
vertical, horizontal


The Question Is :
The ________ segment from a point to a line is the
shortest segment from the point to the line.
Stop!
Time’s Up!
The Answer Is :
perpendicular


The Question Is :
Can the following lengths be the lengths of the sides of
a triangle: 1, 2, 3? Explain.
Stop!
The Answer Is :
Time’s Up!
No; 1 + 2 is not greater than 3


The Question Is :
Find the range for the measure of the third side: 5 and
11.
Stop!
6 < n <16


The Question Is :
Determine whether or not the given measures can be
Stop!
the lengths of the sides of a triangle: 9, 21, 20. Explain.

Time’s Up!
The Answer Is :
Yes; 9 + 20 > 21


The Question Is :
Find the range for the measure of the third side of the
triangle: 21 and 47.
Stop!
26 < n < 68

Inequalities Involving 2 Triangles: 2 Points
The Question Is :
What does the “SAS Inequality” state?

The Answer Is : Stop!
Two sides of a triangle are congruent to two sides of
another triangle. Time’s Up! in the first
If the included angle
triangle has a greater measure than the included angle
in the second triangle, then the 3rd side
Of the first triangle is longer than the
3rd side of the second.

The Question Is :
What is another name for the “SAS Inequality
Theorem?”
Stop!
The Answer Is :
Time’s Up!
The “Hinge Theorem.”

The Question Is :
What does the “SSS Inequality Theorem” state?
The Answer Is :
Stop! to 2 sides of
If 2 sides of a triangle are congruent
Time’s other, then the angle
rd
Up!
another triangle, and the 3 side in one triangle is
rd
longer than the 3 side in the
between the pair of congruent sides in the 1st triangle
is greater than the corresponding angle in the 2nd
triangle.

15 A
The Question Is : D 20

Write an inequality
relating AB and CD. 50 B

Stop!
The Answer Is : 15

AB < CD
Time’s Up!
C

The Question Is : x+5

Write an inequality 45
relating AB and CD. 3x - 7
Stop!
The Answer Is :
Time’s Up!
7/3 < x <6

The Question Is : 4 C
B
Write an inequality relating 6
the given pair of angles
Stop!
or segments: AB, FD. 9 6
9

D

The Answer Is :
Time’s Up!
A
F
6
10
AB > FD

B
9
or segments: m<BDC,
m<FDB.
Stop!
9 6
D

Time’s Up!
A
10 F
6

The Answer Is :
M<BDC < m<FDB

B

m<DBF.
Stop!
or segments: m<FBA, 9 6
9

D

The Answer Is :
Time’s Up!
A
10 F
6

m<FBA > m<DBF

Relationships in Triangles

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Relationships in Triangles