The document discusses various aspects of triangles, including bisectors, medians, altitudes, and inequalities related to these concepts. It covers indirect proof methods, triangle inequality theorem, and inequalities involving two triangles, providing examples and exercises for understanding these principles. Additionally, it presents questions and answers on properties and relationships within triangles, highlighting fundamental geometric theorems.

2. Bisectors, Indirect Proof Indirect Proof The Triangle Inequalities
Medians, & Inequality Involving Two
Altitudes Triangles
2 points 2 points 2 points 2 points 2 points
3 points 3 points 3 points 3 points 3 points
4 points 4 points 4 points 4 points 4 points
5 points 5 points 5 points 5 points 5 points
6 points 6 points 6 points 6 points 6 points
7 points 7 points 7 points 7 points 7 points
8 points 8 points 8 points 8 points 8 points
9 points 9 points 9 points 9 points 9 points
Team 1 : Team 2 :

3. 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

4. Bisectors, Medians, & Altitudes: 3 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!
The Question Is : M
Time’s bisector.
Find m<GJN if JN is an angle
Up! J
The Answer Is :
150

5. Bisectors, Medians, & Altitudes: 4 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
The Question Is :
Time’s find the value of
If HM is an altitude of ∆GHJ,
Up! J
z.
The Answer Is :
19

6. Bisectors, Medians, & Altitudes: 5 Points
The Question Is :
All of the angle bisectors of a triangle meet at
the _________. Stop!
Time’s Up!
The Answer Is :
incenter

7. Bisectors, Medians, & Altitudes: 6 Points
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

8. Bisectors, Medians, & Altitudes: 7 Points
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

9. Bisectors, Medians, & Altitudes: 8 Points
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

10. Bisectors, Medians, & Altitudes: 9 Points
The Question Is :
What are the special segments of triangles?
Stop!
Time’s Up!
The Answer Is :
Perpendicular bisectors, medians, angle
bisectors, and altitudes

11. 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

12. Indirect Proof: 3 Points
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

13. Indirect Proof: 4 Points
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

14. Indirect Proof: 5 Points
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.

15. Indirect Proof: 6 Points
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

16. Indirect Proof: 7 Points
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.

17. Indirect Proof: 7 Points
The Question Is :
State the assumption you would make to start an
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.

18. Indirect Proof: 9 Points
The Question Is :
State the assumption you would make to start an
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

19. 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.

20. Inequalities & Triangles: 3 Points
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

21. Inequalities & Triangles: 4 Points
The Question Is :
What does the “Comparison Property” say?
Stop!
The Answer Is :
Time’s Up!
a < b, a = b, or a > b

22. Inequalities & Triangles: 5 Points
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

23. Inequalities & Triangles: 6 Points
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

24. Inequalities & Triangles: 7 Points
The Question Is :
Determine the angle with the
8
Greatest measure: <8, <5, <7. 7
Stop! 5
The Answer Is : Time’s Up!
<5

25. Inequalities & Triangles: 8 Points
The Question Is :
6
Determine the angle with the
8
Greatest measure: <6, <7, <8. 7
Stop!
The Answer Is :Time’s Up!
<8

26. Inequalities & Triangles: 9 Points
The Question Is :
Determine the angle with the 6
Greatest measure: <1, <6, <9. 1
Stop! 9
The Answer Is : Time’s Up!
<1

27. 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.

28. The Triangle Inequality : 3 Points
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.

29. The Triangle Inequality : 4 Points
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

30. The Triangle Inequality : 5 Points
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

31. The Triangle Inequality : 6 Points
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

32. The Triangle Inequality : 7 Points
The Question Is :
Find the range for the measure of the third side: 5 and
11.
Stop!
The Answer Is :Time’s Up!
6 < n <16

33. The Triangle Inequality : 8 Points
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

34. The Triangle Inequality : 9 Points
The Question Is :
Find the range for the measure of the third side of the
triangle: 21 and 47.
Stop!
The Answer Is :Time’s Up!
26 < n < 68

35. 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.

36. Inequalities Involving 2 Triangles: 3 Points
The Question Is :
What is another name for the “SAS Inequality
Theorem?”
Stop!
The Answer Is :
Time’s Up!
The “Hinge Theorem.”

37. Inequalities Involving 2 Triangles: 4 Points
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.

38. Inequalities Involving 2 Triangles: 5 Points
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

39. Inequalities Involving 2 Triangles: 6 Points
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

40. Inequalities Involving 2 Triangles: 7 Points
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

41. Inequalities Involving 2 Triangles: 8 Points
The Question Is : 4 C
B
Write an inequality relating 6
the given pair of angles
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

42. Inequalities Involving 2 Triangles: 9 Points
The Question Is : 4 C
B
Write an inequality relating 6
the given pair of angles
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