Preparing for the Regents Examination Geometry, AK

CHAPTER
3
3-1 Inductive Reasoning
(page 42)
1 For a polygon with n sides and vertices, from
each vertex, diagonals can be drawn to n 3
other vertices. This can happen n times. Each
diagonal is counted twice from a to b and
n(n 3)
from b to a. There are _
2 diagonals.
Test: Triangle 0
Quadrilateral
4(4 3)
_
2
2
5(5 3)
Pentagon _ 5
2
2 n 5
1 2 3 4 5
5 4 3 2 1
−−−−−−−−−−−
6 6 6 6 6 5(6) n(n 1) twice
the sum
The sum of the integers from 1 through
n(n 1)
n _
2 .
3 Answers will vary. 9, 15, 21 are some possible
answers.
4 Answers will vary. 3, 9, 15, 21 are some possible
answers.
3-2 Definitions and Logic
(page 43)
1 Line segments are congruent if and only if
they have the same measure.
2 A point of a line segment is the midpoint if
and only if it divides the line segment into
two congruent segments.
8 Chapter 3: Introduction to Geometric Proof
Introduction to
Geometric Proof
3 A line is an angle bisector if and only if
it divides an angle into two congruent
parts.
4 Lines are perpendicular if and only if the
lines form right angles.
5 A figure is a polygon if and only if it is a
closed figure in the plane formed by three
or more segments joined at their endpoints.
3-3 Deductive Reasoning
(page 44)
1 1. Given
2. Perpendicular lines form right angles.
3. Right triangles have one right angle.
2 EG divides FEH into two adjacent angles
with the same measure. FEG and HEG
are congruent because they have the same
measure. An angle bisector is the ray that
divides an angle into two adjacent congruent
angles.
3 1. Given
2. Line segments equal in measure are
congruent.
3. A midpoint divides a line segment into
two congruent segments.
4 Since −−
PR bisects SPQ, it divides the angle
into two congruent angles. SPR QPR.
If two angles are congruent, they have the
same measure.