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Warm Up

1) What is the difference between a perpendicular bisector and an angle bisector?

2) True or False: A segment bisects another segment and creates four right triangles.

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Paper Folding Bisectors

page 272 baby blue sectionshare a book with your partner

1) Make a conjecture about the bisectors of the angles of a triangle.

2) Make a conjecture about the perpendicular bisectors of the sides of a triangle.

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Concurrent- When three or more lines intersect in one point

Point of Concurrency- The point at which they intersect

3) Is there any relationship between the lengths of the point of concurrency and the vertex or sides of the triangle?

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Theorems:

The perpendicular bisectors of the sides of a triangle are concurrent at a point equidistant from the vertices.

The bisectors of the angles of a triangle are concurrent at a point equidistant from the sides.

(Gizmo Application )

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Find the center of the circle that circumscribes XYZ.

Concurrent Lines, Medians, and AltitudesLESSON 5-3

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(continued)

The lines y = 4 and x = 3 intersect at the point (3, 4).

This point is the center of the circle that circumscribes XYZ.Quick Check

Concurrent Lines, Medians, and AltitudesLESSON 5-3

Additional Examples

Vocab:

Circumcenter of a triangle

Circumscribed about a triangle

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Circumcenter of a Triangle- The point of concurrency of the perpendicular bisectors of a triangle

Circle is circumscribed about the triangle.

Incenter of a Triangle- The point of concurrency of the angle bisectors of a triangle

We can inscribe a circle in the triangle.

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Median of a Triangle

Is a segment whose endpoints are a vertex and the midpoint of the opposite side.

Are the medians of one triangle concurrent? If so, is there a relationship between the median and a point of concurrency?

Centroid- Point of Concurrency of the medians

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The centroid is the point of concurrency of the medians of a triangle.

The medians of a triangle are concurrent at a point that is two thirds the distance from each vertex to the midpoint of the opposite side. (Theorem 5-8)

WM = WX Theorem 5-823

16 = WX Substitute 16 for WM.23

24 = WX Multiply each side by .32

M is the centroid of WOR, and WM = 16. Find WX.

Because M is the centroid of WOR, WM = WX.23

Concurrent Lines, Medians, and AltitudesLESSON 5-3

Additional Examples

Quick Check

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Altitude of a Triangle

is the perpendicular segment from a vertex to the line containing the opposite side.

Theorem: The lines that contain the altitudes of a triangle are concurrent.

In your notes, draw the altitude of a triangle for an Acute Triangle, Right Triangle and Obtuse Triangle.

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Because LX = XM, point X is the midpoint of LM, and KX is a median of KLM.

So KX is both a median and an altitude.

Is KX a median, an altitude, neither, or both?

Concurrent Lines, Medians, and AltitudesLESSON 5-3

Additional Examples

Quick Check

Because KX is perpendicular to LM at point X, KX is an altitude.

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Practice 5-3#1, 2, 5-11

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Assignment

• P275-277

• #1-4, 9, 12-16, 19-22

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Find the relationship

Create a completed triangle using straws. Try different length straws

Find a relationship between the lengths of the sides that complete a triangle and that would not complete a triangle.

What conclusion can you draw about the smallest angles and largest angles with respect to the sides?

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What do we notice…..

Triangle inequality Theorem.

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

A

B C

AB + BC > AC

BC + AC > AB

AC + AB > BC

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Examples

Tell whether a triangle can have sides with the given lengths. Explain.

1.3, 5, 7

2. 4, 6, 11

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Workbook

• Page 347

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Comparison Property of InequalityIf a = b + c and c > 0 then a > b

Corollary to the Triangle Exterior Angle TheoremThe measure of an exterior angle of a triangle is greater than the measure of each of its remote interior anglesm <1 > m<2 and m<1 > m<3 If two sides of a triangle are not congruent, then the larger angle lies opposite the longer side If two angles of a triangle are not congruent, then the longer side lies opposite the larger angle.   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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Assignment Due Monday, 11/28

• Textbook Page 292 #1-27 odd

• Try to get this done in class, ideally your homework is to enjoy this break with family and friends!