Transcript Chapter 5.1 Notes

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
X
B
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 Notes
Perpendicular 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
D
A
B
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
then
B
B
D
A
C
D
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
then
B
D
A
C
B
D
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 Notes
Concurrency 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 Notes
Indirect 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 Notes
Thm – 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 > AC
AC + 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
B
Converse of the Hinge Thm –
A
11
B
12