Transcript Chapter 5.3 Notes: Use Angle Bisectors of Triangles

Geometry 11/17/14 Bellwork
Hint for #: 3
5.3 Use Angle Bisectors of Triangles
Objectives
 Use properties of angle bisectors
 Locate the incenter
Standards
 PS.1: Make sense of problems and persevere in solving them.
 G.PL.3: Prove and apply theorems about lines and angles,
including the following: vertical angles are congruent; when a
transversal crosses parallel lines, alternate interior angles are
congruent, alternate exterior angles are congruent, and
corresponding angles are congruent; when a transversal crosses
parallel lines, same side interior angles are supplementary; and
points on a perpendicular bisector of a line segment are exactly
those equidistant from the endpoints of the segment.
 G.PL.5: Explain and justify the process used to construct, with a
variety of tools and methods (compass and straightedge, string,
reflective devices, paper folding, dynamic geometric software,
etc.), congruent segments and angles, angle bisectors,
perpendicular bisectors, altitudes, medians, and parallel and
perpendicular lines.
Vocabulary
• Angle bisector: ray that divides
angle into 2 congruent angles
Vocabulary
• Point of concurrency: point of
intersection of segments, lines, or
rays
• Incenter: point of concurrency of
angle bisectors of a triangle
Angle Bisector Theorems
 Theorem 5.5 - Angle Bisector Theorem:
If a point is on the bisector of an angle, then it is equidistant from
the two sides of the angle.
Since AD is an  bisector,
then DE ≅ DF.
E
F
 Theorem 5.6 – Converse of the Angle Bisector Theorem:
If a point is in the interior of an angle and is equidistant from the
sides of the angle, then it lies on the bisector of the angle.
Example 1
EXAMPLE 1
Use the Angle Bisector Theorems
SOLUTION
Because DB  BA and AC  DC and
DB = DC = 18, DA bisects BAC by the Converse of the
Angle Bisector Theorem. So, mBAD = mCAD = 54°.
With a partner, do #1-3
Example
2: of Triangles
5.3 – Use Angle
Bisectors
Three spotlights from two congruent angles. Is
the actor closer to the spotlighted area on the
right or on the left?
Example 3
EXAMPLE 3
Use algebra to solve a problem
For what value of x does P lie on the bisector of  A?
SOLUTION
From the Converse of the Angle Bisector Theorem,
you know that if P lies on the bisector of A then P is
equidistant from the sides of A, so BP = CP.
BP = CP
x + 8 = 3x –6
7 =x
Set segment lengths equal.
Substitute expressions for segment lengths.
Solve for x.
Point P lies on the bisector of  A when x = 7.
GUIDED PRACTICE
Example
4
Do you have enough information to conclude that
QS bisects  PQR? Explain.
ANSWER
No; you need to establish that SR  QR and SP  QP.
Vocabulary and Another Theorem
 Theorem 5.7 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.
 The point of concurrency of the three angle bisectors of a triangle
is called the incenter of the triangle.
 The incenter always lies inside the triangle.
DP=EP=FP
A circle can be drawn with P as
It’s center and will just touch the
Sides of the triangle. The circle
Is said to be inscribed in the triangle.
Example 5
EXAMPLE 4
Use the concurrency of angle bisectors
In the diagram, G is the incenter of RST.
Find GW.
SOLUTION
By the Concurrency of Angle Bisectors of a
Triangle Theorem, the incenter G is equidistant
from the sides of  RST. So, to find GW, you can
find GU in  GUI. Use the Pythagorean Theorem,
a2 + b2 = c2.
Example 5 (continued):
EXAMPLE 4
c 2 = a 2 + b2
2
13 2 = GU + 12
2
Pythagorean Theorem
2
169 = GU + 144
25 =
5=
Substitute known values.
Multiply.
GU 2
Subtract 144 from each side.
GU
Take the positive square root of each side.
Because GU = GW, GW = 5.
Geometry 11/17/14 Homework
Section 5.3:
Pages 313-314:
Exercises: 4-24 Even