VocabVocab
• Concurrent Lines• When three or more lines intersect
they are called concurrent lines
• Point of Concurrency• The point of intersection for
concurrent lines
• Concurrent Lines• When three or more lines intersect
they are called concurrent lines
• Point of Concurrency• The point of intersection for
concurrent lines
Perpendicular BisectorPerpendicular Bisector
• A line, segment, or ray that passes through the midpoint of the side and is perpendicular to that side. • Any point on a perpendicular bisector is
equidistant from the endpoints of the segment.(theorem 5.1)
• Any point that is equidistant from the endpoints of a segment lies on the perpendicular bisector of that segment. (theorem 5.2)
• A triangle has three perpendicular bisectors.
• A line, segment, or ray that passes through the midpoint of the side and is perpendicular to that side. • Any point on a perpendicular bisector is
equidistant from the endpoints of the segment.(theorem 5.1)
• Any point that is equidistant from the endpoints of a segment lies on the perpendicular bisector of that segment. (theorem 5.2)
• A triangle has three perpendicular bisectors.
Circumcenter Circumcenter
• The intersection of the three perpendicular bisectors, their point of concurrency.
• The circumcenter of a triangle is equidistant from the vertices of a triangle. (Theorem 5.3)
• The intersection of the three perpendicular bisectors, their point of concurrency.
• The circumcenter of a triangle is equidistant from the vertices of a triangle. (Theorem 5.3)
j
k lLines j, k, and l are ⊥ .bisctors
= = This means AG GC GB
:Also = & = & = BE CE AF CF AD BD
: Also all⊥ bisctors form right angles with the segments they. bisect
G
F
E
D
B
A
C
Angle Bisector TheoremsAngle Bisector Theorems
• Any point on the angle bisector is equidistant from the sides of the angle. (theorem 5.4)
• Any Point equidistant from the sides of an angle lies on the angle bisector. (theorem 5.5)
• Any point on the angle bisector is equidistant from the sides of the angle. (theorem 5.4)
• Any Point equidistant from the sides of an angle lies on the angle bisector. (theorem 5.5)
IncenterIncenter
• The intersection of the three angle bisectors.
• The incenter of a triangle is equidistant from each side of the triangle.
• The intersection of the three angle bisectors.
• The incenter of a triangle is equidistant from each side of the triangle.
jk
l
65
43
21
Segments j, k, and l are allangle bisectors.
This means BF = DF = EF
Also: <1 = <2, <3 = <4, <5 = <6
Also: GH = GII
H
DB
E
F
C
A
G
MedianMedian
• A segment whose endpoints are a vertex of a triangle and the midpoint of the side opposite the vertex.
• A segment whose endpoints are a vertex of a triangle and the midpoint of the side opposite the vertex.
BD is a median of ABCAD ≅ DC
D
B
A C
CentroidCentroid
• The point of intersection of the three medians of a triangle
• The centroid is the point of balance for any triangle
• The centroid is located two thirds of the distance from a vertex to the midpoint of the side opposite the vertex on a median. (Theorem 5.7)
• The point of intersection of the three medians of a triangle
• The centroid is the point of balance for any triangle
• The centroid is located two thirds of the distance from a vertex to the midpoint of the side opposite the vertex on a median. (Theorem 5.7)
Centroid ExampleCentroid ExampleBD , AF, CE is a median of ABCtherefore:
AG = 2
3AF and GH =
1
3AF
BG = 2
3BD and GD =
1
3BG
CG = 2
3CE and GE =
1
3CE
G
D
FE
B
A C
////
2
3BP =BE
23(10) =BE
203
=BE
2(EN )=AE2(x) =x+17x=17
AN =AE + ENAN =x+17 + xAN =(17) +17 + (17)AN =51
2
3
1
3
AltitudeAltitude
• A segment formed from the vertex to the line containing the opposite side and is perpendicular to the opposite side.
• Every triangle has three altitudes, their intersection is know as the orthocenter.
• A segment formed from the vertex to the line containing the opposite side and is perpendicular to the opposite side.
• Every triangle has three altitudes, their intersection is know as the orthocenter. Segment BD is an
altitude, therefore∠ BDC and∠ BDA are
both right angles
D
B
A
C
Exterior Angle Inequality Theorem
Exterior Angle Inequality Theorem
• If an angle is an exterior angle of a triangle then its measure is greater than the measure of either of its corresponding remote interior angles.
(theorem 5.8)
• If an angle is an exterior angle of a triangle then its measure is greater than the measure of either of its corresponding remote interior angles.
(theorem 5.8)
m∠4 > m∠1
m∠4 > m∠243
2
1
Triangle Sides and Angles Triangle Sides and Angles
• If one side of a triangle is longer than another side, then the angle opposite the longer side has a greater measure than the angle opposite the shorter side.
• If one angle of a triangle has a greater measure than another angle, then the side opposite the greater angle is longer than the side opposite the lesser angle.
• If one side of a triangle is longer than another side, then the angle opposite the longer side has a greater measure than the angle opposite the shorter side.
• If one angle of a triangle has a greater measure than another angle, then the side opposite the greater angle is longer than the side opposite the lesser angle.
ExampleExample
9
7
5
B
A
C
∠C is the largest angle
∠B is the middle angle
∠A is the smallest angle
38°65°
77°
E
DF
DF is the largest side
DE is the middle side
EF is the smallest side
Triangle Inequality Theorem
Triangle Inequality Theorem
• The sum of the lengths of any two sides of a triangle is greater than the length of the third side. • Always check using the two smallest
sides, they must be larger than the third. If this is true the numbers will represent a triangle.
• The sum of the lengths of any two sides of a triangle is greater than the length of the third side. • Always check using the two smallest
sides, they must be larger than the third. If this is true the numbers will represent a triangle.
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