Download Section 9.3

Chapter 9
Congruence, Symmetry and Similarity
Section 9.3
Constructions and Congruence
In the last section we ended by making the comment that if all the corresponding
parts of two shapes were congruent then so are the two shapes. In this section we
show that in a triangle all of the parts (angles and sides) being congruent is not
required, we can get by with only some of them if they are arranged in certain
patterns.
Compass and Straightedge
The “tools” we use to copy parts of a triangle
are a compass and a straightedge.
A compass is a device used to draw circles or
parts of circles called arcs.
A straightedge is like a ruler but with no
markings on it. A ruler or yard stick is often
used but you must ignore the markings.
Copying a segment
The compass and straightedge can be used together
to transfer a segment of a given length onto a line.
This is done in two steps:
A
1. Put point on A and open till mark is on B
2. Lift off and put point on C and mark point D
B
C
D
Side-Side-Side (SSS) Triangle Congruence
If three sides of a triangle are congruent to the three corresponding sides of
another triangle, then the two triangles are congruent. We show this by showing
how segments from one triangle can be translated (copied by a compass and
straightedge) to form the other triangle.
A
B
D
C
E
F
Steps to copy a triangle by coping the sides:
1. Copy segment
BC to locate point F
2. Make arc of length AC with point at F
3. Copy segment
AB
to locate point D on the arc from step (2)
4. Use straightedge to fill in segment DF
Now we have, ABC  DEF
Copying an Angle
Copying an angle can be accomplished by copying a triangle that is included in
that angle.
A
B
C
Steps to copy an angle:
1. Swing arc on the original angle (ABC) and without changing it make same
arc on the other ray you want to copy it onto.
2. Make arc from where the arc in step (1) passed through the original angle and
transfer it to the ray you want to copy it onto.
3. With your straightedge draw the line that connect the endpoint and where the
arcs cross.
Side-Angle-Side (SAS) Triangle Congruence
If two sides and the included angle of one triangle are congruent to two corresponding sides
and the included angle of another triangle with the corresponding sides being congruent,
then the triangles are congruent.
The included angle of two sides of a triangle is the angle that is formed by the two
sides of the triangle. It can not just be any two congruent sides and an angle, but
the angle that is between the two sides.
Below we show how to use a compass and straightedge to copy the side-angleside of a triangle.
A
Steps to copy a triangle by copying a side-angle-side:
1. Copy ABC with vertex at point E.
B
C
2. Use straightedge to draw in
3. Copy AB onto
D
E
F
4. Copy BC onto
ED
EF
5. Draw segment
DF
6. ABC  DEF
ED
Angle Bisector
A compass and
straightedge can be used
to construct both angle
bisectors and
perpendicular bisectors
of segments.
Perpendicular Bisector
Base Angles of Isosceles Triangles are Congruent
A
In an isosceles triangle the angles made with the noncongruent side and one of the congruent sides are
called the base angles. In the triangle to the right
ABD and ACD are the base angles. The base
angles are congruent. The reason for this is as
follows:
1. Construct angle bisector for CAB and call the
point of intersection with BC point D.
2. BAD  CAD (Side-Angle-Side)
3. ABD  ACD (They are the corresponding parts
of the congruent triangles.)
B
D
C
The picture to the right is a circle with center
point A and a triangle inscribed in it. Inscribed
in a circle means all the vertices of the triangle
are on the circle. The angle that is on the
bottom right of the triangle is 70. Use what
you know about congruent angles and the
measure of the interior angles of triangles to
determine what the measures of all the other
angles are.
D
y
C
z
x
u
v
A
70
B
u = 70
The key here is to realize DAB is isosceles since two of the sides are radii of the
circle and u and 70 are base angles.
v = 40
The sum of interior angles of a triangle are 180. v+u+70=v+140=180
x = 140
The angles v and x are supplementary.
y = 20 and z = 20
This is the hardest. The angles y and z have the same measure since they are
both base angles of an isosceles triangle. The sum of interior angles of a triangle
are 180. y+z+x = y+y+160 = 2y+160=180