Relationships in Triangles
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Transcript of Relationships in Triangles
Chapter 5: Relationships in Triangles
Bisectors, Medians, & Altitudes
Indirect Proof Indirect Proof The Triangle Inequality
Inequalities Involving Two Triangles
2 points
3 points
4 points
5 points
6 points
7 points 7 points
6 points
5 points
4 points
3 points
2 points 2 points
3 points
4 points
5 points
6 points
7 points 7 points
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5 points
4 points
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2 points 2 points
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7 points
Team 2 :Team 1 :
8 points
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Bisectors, Medians, & Altitudes: 2 Points
The Answer Is :88
The Question Is :
GP is a median of ∆GHJ. Find HJ.
G
M
J
P
NHIn ∆GHJ, HP=5x-16,
PJ=3x+8, m<GJN=6y-3, m<NJH=4y+23, & m<HMG=4z+14 Stop!
Time’s Up!
The Answer Is :
150
The Question Is :
Find m<GJN if JN is an angle bisector.
Bisectors, Medians, & Altitudes: 3 Points
G
M
J
P
NHIn ∆GHJ, HP=5x-16,
PJ=3x+8, m<GJN=6y-3, m<NJH=4y+23, & m<HMG=4z+14
Stop! Time’s Up!
The Answer Is :
19
The Question Is :If HM is an altitude of ∆GHJ, find the value of z.
Bisectors, Medians, & Altitudes: 4 Points
G
M
J
P
NHIn ∆GHJ, HP=5x-16,
PJ=3x+8, m<GJN=6y-3, m<NJH=4y+23, & m<HMG=4z+14 Stop!
Time’s Up!
The Answer Is :incenter
The Question Is :All of the angle bisectors of a triangle meet at the _________. Stop!
Time’s Up!
Bisectors, Medians, & Altitudes: 5 Points
The Answer Is :
m<ACQ = 55
The Question Is :Find m<ACQ if m<ACB=123-x and m<QCB=42+x.
Bisectors, Medians, & Altitudes: 6 Points
CP is an altitude, CQ is the angle bisector of <ACB, and R is the midpoint of AB.
C
A P Q R B
Stop! Time’s Up!
The Question Is :Find AB if AR=3x+6 and RB=5x-14
The Answer Is :AB = 72
Bisectors, Medians, & Altitudes: 7 Points
CP is an altitude, CQ is the angle bisector of <ACB, and R is the midpoint of AB.
C
A P Q R B
Stop! Time’s Up!
The Question Is :In ∆RST, if the point P is the midpoint of RS, then PT is a(n) ________.
The Answer Is :median
Stop! Time’s Up!
Bisectors, Medians, & Altitudes: 8 Points
The Answer Is :Perpendicular bisectors, medians, angle bisectors, and altitudes
The Question Is :
What are the special segments of triangles?Stop!
Time’s Up!
Bisectors, Medians, & Altitudes: 9 Points
The Question Is :When using _________, you assume that the conclusion false and then show that this assumption leads to a contradiction of the hypothesis, or some other accepted fact, such a definition, postulate, theorem, or corollary.
The Answer Is :Indirect reasoning
Stop! Time’s Up!
Indirect Proof: 2 Points
The Question Is :True or False: Another name for an indirect proof is a proof by contradiction.
The Answer Is :True
Stop! Time’s Up!
Indirect Proof: 3 Points
The Question Is :What are the three steps of writing an indirect proof.
The Answer Is :1. Assume that the conclusion is false2. Show that this assumption leads to a
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
Stop! Time’s Up!
Indirect Proof: 4 Points
The Question Is :State the assumption you would make to start an indirect proof of the following statement: EF is not a perpendicular bisector.
The Answer Is :EF is a perpendicular bisector.
Stop! Time’s Up!
Indirect Proof: 5 Points
The Answer Is :x > 5
Indirect Proof: 6 Points
The Question Is :State the assumption you would make to start an indirect proof of the following statement: If 5x < 25, then x < 5.Stop!
Time’s Up!
The Answer Is :
6 cannot be expressed as a/b, wherea & b are integers and b is not equal to 0.
Indirect Proof: 7 PointsThe 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 expressed as a/b, where a and b are integers and b is not equal to 0, 6 is a rational number.
Stop! Time’s Up!
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 is also an altitude of the triangle.
The Answer Is :The angle bisector of the vertex angle of an isosceles triangle is not an altitude of the triangle.
Stop! Time’s Up!
Indirect Proof: 7 Points
The Question Is :
State the assumption you would make to start an indirect proof of the following statement: Two lines that are cut by a transversal so that alternate interior angles are congruent are parallel.
The Answer Is :
The lines are not parallel
Stop! Time’s Up!
Indirect Proof: 9 Points
The Answer Is :For any real numbers a and b, a > b if and only if there is a positive number c such that a = b + c.
The Question Is :What is the definition of an inequality?
Stop! Time’s Up!
Inequalities & Triangles: 2 Points
The Answer Is :Comparison Property, Transitive Property, Addition and Subtraction Properties, & Multiplication and Division Properties
The Question Is :What are the four properties of Inequalities for Real Numbers?
Stop! Time’s Up!
Inequalities & Triangles: 3 Points
The Answer Is :a < b, a = b, or a > b
The Question Is :What does the “Comparison Property” say?
Stop! Time’s Up!
Inequalities & Triangles: 4 Points
The Answer Is :
<4, <5, <6
The Question Is :
Use the Exterior Angle Inequality Theorem to list all angles that satisfy the stated condition: measures less than m<1
Inequalities & Triangles: 5 Points
123
86
5
47
Stop! Time’s Up!
The Answer Is :
<1
The Question Is :
Determine the angle with the Greatest measure: <1, <2, <3.
Inequalities & Triangles: 6 Points
123
86
5
47
Stop! Time’s Up!
The Answer Is :
<5
The Question Is :
Determine the angle with the Greatest measure: <8, <5, <7.
Inequalities & Triangles: 7 Points
8
57
Stop! Time’s Up!
The Answer Is :
<8
The Question Is :
Determine the angle with the Greatest measure: <6, <7, <8.
Inequalities & Triangles: 8 Points
8
6
7Stop! Time’s Up!
The Answer Is :
<1
The Question Is :
Determine the angle with the Greatest measure: <1, <6, <9.
Inequalities & Triangles: 9 Points
6
91
Stop! Time’s Up!
The Triangle Inequality : 2 Points
The Question Is :What is the “Triangle Inequality Theorem?”
The Answer Is :
The sum of the lengths of any two sides of a triangle is greater than the length of the third side.
Stop! Time’s Up!
The Answer Is :
Whether three segments can form a triangle.
The Question Is :What is the “Triangle Inequality Theorem” used to determine? Stop!
Time’s Up!
The Triangle Inequality : 3 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 shortest distance from a point to a vertical line lies along a ________ line.
The Answer Is :vertical, horizontal
Stop! Time’s Up!
The Triangle Inequality : 4 Points
The Question Is :The ________ segment from a point to a line is the shortest segment from the point to the line.
The Answer Is :perpendicular
Stop! Time’s Up!
The Triangle Inequality : 5 Points
The Answer Is :
No; 1 + 2 is not greater than 3
The Question Is :Can the following lengths be the lengths of the sides of a triangle: 1, 2, 3? Explain.Stop!
Time’s Up!
The Triangle Inequality : 6 Points
The Answer Is :
6 < n <16
The Question Is :Find the range for the measure of the third side: 5 and 11. Stop!
Time’s Up!
The Triangle Inequality : 7 Points
The Question Is :Determine whether or not the given measures can be the lengths of the sides of a triangle: 9, 21, 20. Explain.
The Answer Is :
Yes; 9 + 20 > 21
Stop! Time’s Up!
The Triangle Inequality : 8 Points
The Answer Is :
26 < n < 68
The Question Is :Find the range for the measure of the third side of the triangle: 21 and 47.
Stop! Time’s Up!
The Triangle Inequality : 9 Points
The Question Is :
Inequalities Involving 2 Triangles: 2 Points
What does the “SAS Inequality” state?
The Answer Is :Two sides of a triangle are congruent to two sides of another triangle. If the included angle in the first triangle has a greater measure than the included angle in the second triangle, then the 3rd sideOf the first triangle is longer than the 3rd side of the second.
Stop! Time’s Up!
The Question Is :
What is another name for the “SAS Inequality Theorem?”
The Answer Is :
The “Hinge Theorem.”
Stop! Time’s Up!
Inequalities Involving 2 Triangles: 3 Points
The Answer Is :
If 2 sides of a triangle are congruent to 2 sides of another triangle, and the 3rd side in one triangle is longer than the 3rd side in the other, then the angle between the pair of congruent sides in the 1st triangle is greater than the corresponding angle in the 2nd triangle.
The Question Is :
What does the “SSS Inequality Theorem” state?
Stop! Time’s Up!
Inequalities Involving 2 Triangles: 4 Points
The Answer Is :
AB < CD
The Question Is :
Write an inequality relating AB and CD.
Inequalities Involving 2 Triangles: 5 Points
D A
B
C
15
50
2015
Stop! Time’s Up!
The Answer Is :
7/3 < x <6
The Question Is :
Write an inequality relating AB and CD.
Inequalities Involving 2 Triangles: 6 Points
x + 5
3x - 745
Stop! Time’s Up!
The Answer Is :
AB > FD
The Question Is :
Write an inequality relatingthe given pair of anglesor segments: AB, FD.
Inequalities Involving 2 Triangles: 7 Points
D
A
BC
10
6
9
4
F
96
6
Stop! Time’s Up!
The Answer Is :
M<BDC < m<FDB
The Question Is :
Write an inequality relatingthe given pair of anglesor segments: m<BDC,m<FDB.
Inequalities Involving 2 Triangles: 8 Points
D
A
BC
10
6
9
4
F
96
6
Stop! Time’s Up!
The Answer Is :
m<FBA > m<DBF
The Question Is :
Write an inequality relatingthe given pair of anglesor segments: m<FBA,m<DBF.
Inequalities Involving 2 Triangles: 9 Points
D
A
BC
10
6
9
4
F
96
6
Stop! Time’s Up!