Chapter 5: Relationships in Triangles

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Chapter 5 Relationships in Triangles
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Lesson 5.1

• Bisectors, Medians, and Altitudes

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Perpendicular Bisector
Definition Facts to Know Point of Concurrency Example
A line, segment or ray that passes through the midpoint of the opposite side and is perpendicular to that side Any point on a perpendicular bisector is equidistant from the endpoints Circumcenter The point where 3 perpendicular bisectors intersect - the circumcenter is equidistant from all vertices of the triangle
A
E
B
D
C
BD CD AD BC E is the circumcenter- AE
BE CE
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Median
Definition Facts to Know Point of Concurrency Example
A segment that goes from a vertex of the triangle to the midpoint of the opposite side The median splits the opposite side into two congruent segments Small 1/3 median Big 2/3 median 2 x small big Centroid The point where 3 medians intersect
A
E
B
D
C
BD CD E is the centroid- ED 1/3 AD AE 2/3
AD 2 ED AE
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Angle Bisector
Definition Facts to Know Point of Concurrency Example
A line, segment, or ray that passes through the middle of an angle and extends to the opposite side Any point on an angle bisector is equidistant from the sides of the triangle Incenter The point where 3 angle bisectors intersect -the incenter is equidistant from all sides of the triangle
A
F
E
G
B
D
C
BAD CAD G is the incenter- EG
FG
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Altitude
Definition Facts to Know Point of Concurrency Example
A segment that goes from a vertex of the triangle to the opposite side and is perpendicular to that side Orthocenter The point where 3 altitudes intersect
A
B
D
C
AD BC
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• C. Find the measure of EH.

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• A. Find QS.

B. Find ?WYZ.
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In the figure, A is the circumcenter of ?LMN.
Find x if m?APM 7x 13.
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In ?XYZ, P is the centroid and YV 12. Find YP
and PV.
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In ?LNP, R is the centroid and LO 30. Find LR
and RO.
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Lesson 5.2

• Inequalities and Triangles

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Foldable

• Fold the paper into three sections (burrito fold)
  Then fold the top edge down about ½ an inch
• Unfold the paper and in the top small rectangles
  label each column

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Exterior Angle Inequality Inequality with Sides Inequality with Angles
Exterior Angle Remote Int. Remote Int. The exterior angle is greater than either of the remote interior angles by themselves rem. Int. lt ext. Ex The biggest side is across from the biggest angle The smallest side is across from the smallest angle Ex -The biggest angle is across from the biggest side/ the smallest angle is across from the smallest side Ex
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List the angles of ?ABC in order from smallest
to largest.
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List the sides of ?ABC in order from shortest to
longest.
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What is the relationship between the measures of
?A and ?B?
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Lesson 5.4

• The Triangle Inequality

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Triangle Inequality Theorem

• The sum of the lengths of any two sides of a
  triangle is greater than the length of the third
  side

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Triangle Inequality Theorem Problems

• Determine if the measures given could be the
  sides of a triangle.
• 16, 17, 19
• 16 17 33 yes, the sum of the two smallest
  sides is larger than the third side
• 6, 9, 15
• 6 9 15 no, the sum of the two smallest sides
  is equal to the other side so it cannot be a
  triangle

• Find the range for the measure of the third side
  given the measures of two sides.
• 7.5 and 12.1
• 12.1- 7.5 lt x lt 12.1 7.5
• 4.6 lt x lt 19.6
• 9 and 41
• 41-9 lt x lt 41 9
• 32 lt x lt 50

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• Determine whether it is possible to form a
  triangle with side lengths 5, 7, and 8.

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Is it possible to form a triangle with the given
side lengths of 6.8, 7.2, 5.1? If not, explain
why not.
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• Find the range for the measure of the third side
  of a triangle if two sides measure 4 and 13.

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In ?PQR, PQ 7.2 and QR 5.2. Which measure
cannot be PR? A 7 B 9 C 11 D 13
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Lesson 5.3

• Indirect Proof

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Steps to Completing an Indirect Proof

• Assume that ______________ (the conclusion is
  false)
• Then _______________ (show that the assumption
  leads to a contradiction) This contradicts the
  given information that ________________.
• Therefore, __________________ (rewrite the
  conclusion) must be true.

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B. State the assumption you would make to start
an indirect proof for the statement 3x 4y 1.
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Example Indirect Proof

• Given 5x lt 25
• Prove x lt 5

x 5.
1. Assume that
2. Then
x 9
And 5(9) 45
45gt 25
This contradicts the given info that 5x lt 25
3. Therefore,
x lt 5 must be true.
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Example Indirect Proof

• Given m is not parallel to n
• Prove m 3 m 2

m
3
2
n
1. Assume that
m 3 m 2
2. Then,
angles 2 and 3 are alternate interior angles

• When alternate interior angles are congruent then
  the lines that make them are parallel.

This contradicts the given info that m is not
parallel to n
3. Therefore, m 3 m 2 must be true.
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• Write an indirect proof to show that if 2x 11
  lt 7, then x gt 2.
• Given 2x 11 lt 7
• Prove x gt 2

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Lesson 5.5

• Inequalities Involving Two Triangles

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On the other side of the foldable from Lesson 2
(3 column chart)
SAS Inequality Theorem SSS Inequality Theorem Examples
(Hinge Theorem) When 2 sides of a triangle are congruent to 2 sides of another triangle, and the included angle of one triangle is greater than the included angle of the other triangle Then, the side opposite the larger angle is larger than the side opposite the smaller angle When 2 sides of a triangle are congruent to 2 sides of another triangle, and the 3rd side of a triangle is greater than the 3rd side of the other triangle Then, the angle opposite the larger side is larger than the angle opposite the smaller side Ex Ex Ex
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A. Compare the measures AD and BD.
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B. Compare the measures ?ABD and ?BDC.
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ALGEBRA Find the range of possible values for a.
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Find the range of possible values of n.