relationships in triangles

Chapter 5: Relationships in Triangles

Bisectors, Medians, & Altitudes

Indirect Proof Indirect Proof The Triangle Inequality

Inequalities Involving Two Triangles

2 points

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Team 2 :Team 1 :

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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!

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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

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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

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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!