chapter 5.1 common core - g.co.10 prove theorems about triangles…the segment joining the midpoint...

Chapter 5.1

Common Core - G.CO.10 Prove theorems about triangles…the segment joining the midpoint of two sides of a triangle is parallel to the third side and half the length.

Objectives – To use properties of midsegments to solve problems,

Chapter 5.1

Midsegment Thm – the segment connecting the midpoints of 2 sides of a triangle is parallel to the third side and is half as long.

A B

X Y

Chapter 5.2

Common Core – G.CO.9 & G.SRT.5 Prove theorems about lines and angles…points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. Use congruence…criteria for triangles to solve problems and prove relationships in geometric figures.

Objectives – To use properties of perpendicular bisectors and angle bisectors.

Chapter 5.2 NotesPerpendicular Bisector Thm – If a pt is on the ⊥ bisector of a segment, then it is equidistant from the endpoints of the segment.If C then C A P B A P B * If line CP is the ⊥ bisector of segment AB, then CA = CB

Converse of the Perpendicular Bisector Thm – If a pt is equidistant from the endpoints of a segment, then it is on the perpendicular bisector of the segment.

If then A B A B

D D

* If DA = DB, then D lies on the ⊥ bisector of Segment AB

Angle Bisector Thm – If a pt is on the bisector of an angle, then it is equidistant from the 2 sides of the angle.

If B then B D D

A C A C

* If m∠BAD = m∠CAD, then DB = DC,

Converse of the Angle Bisector Thm – If a pt is in the interior of an ∠ and is equidistant from the sides of the ∠, then it lies on the bisector of the angle.If B then B D D A C A C

*If DB = DC, then m∠BAD = m∠CAD

Chapter 5.3

Common Core - G.C.3 Construct the inscribed and circumscribed circles of a triangle.

Objectives – To identify properties of perpendicular bisectors and angle bisectors.

Chapter 5.3 NotesConcurrency of Perpendicular Bisectors of a Triangle - The perpendicular bisectors of a triangle intersect at

a point that is equidistant from the vertices of the triangle. (Circumcenter)

B

A C

*PA = PB = PC

Circumcenters

Acute Right Obtuse

Concurrency of Angle Bisectors of a Triangle – The angle bisectors of a triangle intersect at a

point that is equidistant from the sides of the triangle. (Incenter)

* PD = PE = PF

Incenters

Acute Right Obtuse

Chapter 5.4

Common Core - G.CO.10 & G.SRT.5 Prove theorems about triangles…the medians of a triangle meet at a point.

Objectives – To identify properties of medians and altitudes of a triangle.

Chapter 5.4 Notes

Concurrency of Medians of a Triangle – The medians of a triangle intersect at a point that is two thirds of the distance from each vertex to the midpoint of the opposite side.

(Centroid)

Centroids

Acute Right Obtuse

Concurrency of Altitudes of a Triangle – The lines containing the altitudes of a triangle are concurrent.

(Orthocenter)

Orthocenters

Acute Right Obtuse

Chapter 5.5

Common Core Common Core – G.CO.10 Prove theorems about triangles.

Objectives – To use indirect reasoning to write proofs.

Chapter 5.5 NotesIndirect Proof – is a proof in which you prove that a

statement is true by first assuming that its opposite is true. If this assumption leads to an impossibility then you have proved that the original statement is true.

1) Identify the statement that you want to prove true.2) Assume the statement is false, assume its opposite is

true.3) Obtain statements that logically follow your assumption.4) If you obtain a contradiction, then the original statement

must be true.

Chapter 5.6

Common Core Common Core – G.CO.10 Prove theorems about triangles.

Objectives – To use inequalities involving angles and sides of triangles.

Chapter 5.6 NotesThm – If one side of a triangle is longer than another

side, then the angle opposite the longer side is larger than the angle opposite the shorter side.

Thm – If one angle of a triangle is larger than another angle, then the side opposite the larger angle is longer than the side opposite the smaller angle.

Exterior Angle Inequality Thm – The measure of an exterior angle of a triangle is greater than the measure of either of the 2 nonadjacent interior angles.

A

1

C B

*m∠1 > m∠A and m∠1 > m∠B

Triangle Inequality – The sum of the lengths of any 2 sides of a triangle is greater than the length of the third side. A

AB + BC > ACAC + BC > AB

AB + AC > BC C B

Chapter 5.7

Common Core Common Core – G.CO.10 Prove theorems about triangles.

Objectives – To apply inequalities in two triangles.

Chapter 5.7 Notes

Hinge Thm –

X Y A BConverse of the Hinge Thm –

A B

11 12