Name
Date
Notes Formal Geometry
Chapter 5 Relationships in Triangles
Period
5.1 Bisectors of Triangles
A. Definitions
1. Perpendicular Bisector:
5.1 Perpendicular Bisector Theorem: If a point is on the _________________ bisector of a
segment, then it is equidistant from the endpoints.
Ex.
5.2 Converse of the Perpendicular Bisector Theorem: If a point is equidistant from the endpoints
of a segment, then it is on the _________________ bisector of the segment.
Ex.
2. Concurrent Lines:
3. Point of Concurrency:
4. Circumcenter:
5.3 Circumcenter Theorem: The perpendicular bisectors of a triangle intersect at a point called the
________________________ that is ____________________ from the vertices of the
_________________.
Ex.
5.4 Angles Bisector Theorem: If a __________ is on the bisector of an angle, then it is equidistant
from the _________ of the angle.
Ex.
1
5.5 Converse of the Angle Bisector Theorem: If a ______ in the interior of an angle is equidistant
from the sides of the angle, then it is on the _______________ of the angle.
Ex.
5. Incenter:
5.6 Incenter Theorem: The ___________ _______________ of a trigangle intersect at a point called
the _________________ that is equidistant from each side of the triangle.
Ex.
5.2 Medians and Altitudes of Triangles
A. Definitions:
1. Median:
2. Centriod:
5.7 Centroid Theorem:
3. Altitude:
4. Orthocenter:
Date:
Assignment: Fill in the notes for sections 5.1 and 5.2 complete with pictures =)
5.3 Inequalities in One Triangle
A. Opener:
1. Which angle is the largest? (Draw arrow)
2. Which side is the longest? (Draw arrow)
3. What is the relationship between the longest side and the largest angle?
2
B. Definitions:
1. 5.8 Exterior Angles Inequality Theorem:
2. 5.9 Angle-Side Relationships in Triangles Theorem:
3. 5.10 Angle-Side Converse Theorem:
C. Examples:
1. dfd
a. Df
b. Df
2. df
3. ad
3
4.
Date:
Assignment: p 346 #8-12(x2) (no more than 2 angles for each answer), 14-28 (x2), 50-51
5.4 Indirect Proof or Proof by Contradiction
A. Opener:
Is the base of an isosceles triangle always, sometimes or never the longest side of the triangle? Explain.
B. Definitions:
1. Indirect Reasoning:
2. Indirect Proof/Proof by Contradiction:
How to write an indirect proof:
4
C. Examples: State the assumption for starting an indirect proof.
1. Sd
2. Df
3. Df
4.
5.
Step 1: Given:
Prove:
Indirect Proof:
Step 2: Make a table
Step 3: Look at both cases
6.
5
7.
Date:
Assignment: p 355 #11-17, 22, 27, 49
5.5 The Triangle Inequality Theorem
A. Opener:
Open the baggie with straws. Give each member of your group all the straws of one
color. Create triangles with your straws. What conclusions can you draw?
B. Definitions:
1. 5.11 Triangle Inequality Theorem:
C. Examples:
1. 8 in., 15 in., 17 in.
2. 6 m, 8 m, 14 m
6
2.
3.
4.
Date:
Assignment: p 363 #6-20 (x2), 56-63
7
5.6 Inequalities in Two Triangles
A. Opener:
P 350 #1-3.
B. Definitions:
1. 5.13 Hinge Theorem: If two sides of a triangle are congruent to two sides of another triangle, and the
included angle of the first is larger than the included angle of the second triangle, then:
2. 5.14 Converse of the Hinge Theorem: If two sides of a triangle are congruent to two sides of another
triangle, and the third side in the first is longer than the third side in the second triangle, then the
included angle measure of the first triangle is greater than the included angle measure in the second
triangle.
C. Examples:
1. Compare the given measures:
8
2. Find the value of each variable.
3.
9
4.
5.
Date:
Assignment: p 372 #10-22 (x2), 24, 26
10