Download 5.3 Use Angle Bisectors of a Triangle

5.3 Use Angle Bisectors of a Triangle
angle bisector : a ray that divides an angle into two congruent
adjacent angles.
distance from a point to a line : the length of the perpendicular
segment from the point to the line
P
Thm 5.5 Angle Bisector Theorem
If a point is on the bisector of an angle,
then it is__________________ from the sides of the angle.
B
-If AD bisects ∠BAC and DB ⊥ AB
and DC ⊥ AC, then
D
A
C
Thm 5.6 Converse of the Angle Bisector Theorem
If a point is in the
interior of an angle
is equidistant from the sides
and
of the angle,
then it lies ______________________________
B
D
A
C
1
D
Ex 1: Find the measure of ∠BAD.
18
18
B
54º
A
Ex 2: Find the value of x. State the theorem used in each problem.
a.
(3x + 5)º
(4x ­ 6)º
b.
5x
6x - 5
C
D
C
B
A
Thm 5.7 Concurrency of Angle Bisectors of a Triangle
-The
∠ bisectors of a triangle
equidistant from the side
intersect at a point that is
of the triangle.
B
F
P
A
E
Incenter: the point of concurrency of the angle bisectors of
a triangle
Ex 3: The ∠ bisectors of ∆XYZ meet at M. Find MK.
X
L
Z
M
8
5
J
K
Y
D
C