Download Math 310

Math 310
Section 10.1
Congruence and Constructions
Congruence vs. Similarity
Def
Similar means that two objects have the same
shape, but not necessarily the same size.
Congruence means that two objects have both
the same size and the same shape.
Ex
Similar shapes:
Ex
Congruent shapes:
Congruent Segments & Angles
Def
Segments AB is congruent to segment CD iff
mAB = mCD (their measures)
<ABC is congruent to <DEF iff
m(<ABC) = m(<DEF)
Circles
Circles are the backbone of constructions,
therefore, we begin constructions with some
info about circles.
A Circle
Def
A circle is the loci (think “set of ”) all points
equidistant from a given center.
Parts of a Circle
radius
minor
arc
major
arc
diameter
Parts of a Circle (cont)
semicircle
center
semicircle
O
Why Circles?
So why are circles so important to constructions?
By the very definition of circles, they allow us to
copy distances, ie create congruent segments.
And, as we will see, the use of two circles, one
can copy an angle. Therefore, it is the use of
circles that allow us to create congruent
geometric objects. (restricted primarily of course
to segments and angles)
Construction: Congruent Segments
Before Angles
Before we can copy angles however, we need one
more definition and one more postulate, both
regarding triangles.
Congruent Triangles
Def
Two triangles are congruent if all of their parts are
congruent. That is to say, triangle ABC is
congruent to triangle EFG iff m<A = m<E,
m<B = m<F, m<C = m<G, and sides
mAB
= mEF, mBC = mFG, and mCA = mGE.
Note: Order here is very important.
Ex
A
C
Let us suppose the following two
figures are congruent and that
they are drawn to scale.
B
It would then be inappropriate to say that
side AB was congruent to side HI. Clearly
side AB is congruent to side IG and side
CA is congruent to side HI. Thus is we
call the upper triangle ABC, the lower
triangle must then be called IGH.
I
H
G
SSS
It would be highly complicated and tiring to prove
for every set of triangles (or other figures) that
all their corresponding sides and angles were
congruent every time. Therefore,
mathematicians have discovered certain
conditions which guarantee that all parts are
congruent. The first of these is called the SideSide-Side Congruence Postulate, or SSS for
short.
SSS Congruence Postulate
Thrm
If the three sides of one triangle are congruent,
respectively, to the three sides of a second
triangle, then the triangles are congruent.
Ex
Use the SSS congruence postulate and
your compass to demonstrate that these
two triangles are congruent. Then name
the triangles so that corresponding parts
match up. Assume the triangles are
drawn to scale.
Copying Triangles
Now, since we have SSS, to copy a triangle we
simply need to copy all three lengths of the
triangle and we are guaranteed that the angles
will be copied also.
Construction: Congruent Triangles
I will demonstrate two possibilities.
 You are given a triangle to copy
 You are given the measurements of a triangle to
Construct
How Bout Angles?
If you are given an angle, simply attaching another
side yields a triangle, and thus by copying the
triangle we can also copy the angle!
Construction: Congruent Angles
Triangle Inequality
The sum of the measures of any two sides of a
triangle must be greater than the measure of the
third side.
SAS Congruence Postulate
Thrm
If two sides and the included angle of one triangle
are congruent to the two sides and the included
angle of another triangle, respectively, then the
two triangles are congruent.
Ex
W
O
60
Given the following diagrams, state
why the two triangles are congruent,
and then, taking the name of the
triangle at left to be WOC, what is
the name of the triangle below?
C
B
D
60
R
What do you do with these?
Constructions allow us to see properties of the
geometric objects we are constructing. By
constructing a geometric figure to given
specification, we are, in essence, proving that
what we have constructed satisfies those
conditions. Alternately, to construct an object,
we can follow a proof of its properties.
Perpendicular Bisector
Def
The perpendicular bisector of a segment is a line
passing through the midpoint of the segment,
perpendicular to the segment.
Construction: Perpendicular Bisector
Perpendicular Bisector Theorems
Thrm
Any point equidistant from the endpoints of a
segment is on the perpendicular bisector of the
segment.
Any point on the perpendicular bisector of a
segment is equidistant from the endpoints of
the segment.
Altitude of a Triangle
Def
The altitude of a triangle is a segment drawn
perpendicularly from one side of a triangle
through the vertex opposite it.
altitude
altitude
Question
How many altitudes does a triangle have?
3
Isosceles Triangle Theorems
Thrm
The angles opposite the congruent sides are
congruent. (Base angles of an isosceles triangle
are congruent.)
The angle bisector of an angle formed by two
congruent sides contains the altitude of the
triangle and is the perpendicular bisector of the
third side of the triangle.
Ex
If these sides are
congruent…
…then these angles
are congruent.
And if this angle is
bisected by this
segment…
…then this segment
is the altitude of the
triangle and the
perpendicular
bisector of this side.
Circumscribe & Circumcenter
Def
To circumscribe a circle about some polygon is to
construct the circle so that each vertex of the
polygon lies on the circle, and the polygon is
contained by the circle.
The circumcenter of a triangle is the point that is
equidistant from all three vertices of a triangle.
(i.e. a circle can be circumscribed about the
triangle)
Construction: Circumcenter