10-1 & 10-2: Congruence Criteria
Objectives:
Assignment:
1. To discover and use • P. 146: 8-11
shortcuts for showing • P. 153: 11-15
that two triangles are • P. 165: 1-11
congruent
Congruent Triangles (CPCTC)
Two triangles are congruent triangles if and
only if the corresponding parts of those
congruent triangles are congruent.
Corresponding
sides are
congruent
Corresponding
angles are
congruent
Warm-Up
Now back to the subject
of roof trusses. Would
it be necessary for the
manufacturer of a set
of trusses to check
that all the
corresponding angles
were congruent as
well as the sides?
Warm-Up
In other words, is it
sufficient that the
pieces of wood (the
sides of each triangle)
are all the same
length?
Congruent Triangles
Checking to see if 3 pairs of corresponding
sides are congruent and then to see if 3
pairs of corresponding angles are
congruent makes a total of SIX pairs of
things, which is a lot! Surely there’s a
shorter way!
Congruence Shortcuts?
Will one pair of congruent sides be
sufficient? One pair of angles?
Congruence Shortcuts?
Will two congruent parts be sufficient?
Congruent Shortcuts?
Will three congruent parts be sufficient?
Congruent Shortcuts?
Will three congruent parts be sufficient?
Included Angle
Included Side
Congruent Shortcuts?
Will three congruent parts be sufficient?
Investigation: Shortcuts
Shortcuts?:
SSS
SSA
SAS
ASA
AAS
AAA
To test the these 6 possible
shortcuts, we will use
compass and straightedge
constructions. For each of
these, you will be given
three pieces to form a
triangle. If the shortcut
works, one and only one
triangle can be made with
those parts.
Copying a Segment
We’re going to try making two congruent
triangles by simply copying the three sides
using only a compass and a straightedge.
First, let’s learn how to copy a segment.
Copying a Segment
1. Draw segment AB.
Copying a Segment
2. Draw a line with point A’ on one end.
Copying a Segment
3. Put point of compass on A and the pencil
on B. Make a small arc.
Copying a Segment
4.
Put point of compass on A’ and use the compass
setting from Step 3 to make an arc that intersects the
line. This is B’.
Investigation 1
Now apply the construction for copying a
segment to copy the three sides of a
triangle.
Investigation 1
1. Use your straight edge to construct a
triangle.
2. Now draw a line with A’ on one end.
Investigation 1
3. Copy segment AB onto your line to make
A’B’.
Investigation 1
4. Put point of compass on A and pencil on
C. Copy this distance from A’.
Investigation 1
5. Put point of compass on B and pencil on
C. Copy this distance from B’. This is C’
Investigation 1
6. Finish your new triangle by drawing
segments A’C’ and B’C’.
Investigation 1
7. Copy the original triangle ABC onto a
piece of patty paper. Compare this
triangle to triangle A’B’C’. Are they
congruent?
Side-Side-Side Congruence Postulate
SSS Congruence Postulate:
If the three sides of one triangle are congruent to
the three sides of another triangle, then the two
triangles are congruent.
SSS Congruence Postulate
Example 1
Decide whether the triangles are congruent.
Explain your reasoning.
Example 2
Example 3
Explain why the bench with the diagonal
support is stable, while the one without the
support can collapse.
Copying an Angle
Draw angle A on your paper. How could you
copy that angle to another part of your
paper using only a
compass and a
straightedge?
Copying an Angle
1. Draw angle A.
Copying an Angle
2. Draw a ray with endpoint A’.
Copying an Angle
3. Put point of compass on A and draw an
arc that intersects both sides of the angle.
Label these points
B and C.
Copying an Angle
4. Put point of compass on A’ and use the
compass setting from Step 3 to draw a
similar arc on the ray.
Label point B’ where
the arc intersects
the ray.
Copying an Angle
5. Put point of compass on B and pencil on
C. Make a small arc.
Copying an Angle
6. Put point of compass on B’ and use the
compass setting from Step 5 to draw an
arc that intersects the
arc from Step 4.
Label the
new point
C’.
Copying an Angle
7. Draw ray A’C’.
Example 4
If m<J + m<E + m<R = 180°, then construct
<R.
Investigation: SAS
Given the two
sides and the
included
angle shown
can you
construct one
and only one
triangle
ROW?
Investigation: ASA
Given the two
angles and
the included
side shown
can you
construct one
and only one
triangle
SAM?
Investigation: AAS
Given the two
angles and
the nonincluded side
shown can
you construct
one and only
one triangle
JER?
On this one, you’ll have to find
the size of the third angle. How
are you going to do that?
Investigation: SSA
Given the two
sides and the
non-included
angle shown
can you
construct one
and only one
triangle
RAN?
On this one, construct 𝐴𝑁 and
∠𝑁, then find a place for 𝑅𝐴.
Investigation: AAA
Given three
angles shown
can you
construct one
and only one
triangle
NAH?
Pick a size for 𝑁𝐴, and then copy
the rest of the angles. I made it
easy; the angles are all the same.
Congruence Shortcuts
Side-Side-Side (SSS) Congruence Postulate:
If the three sides of one triangle are congruent to
the three sides of another triangle, then the two
triangles are congruent.
Congruence Shortcuts
Side-Angle-Side (SAS) Congruence Postulate:
If two sides and the included angle of one triangle
are congruent to two sides and the included
angle of another triangle, then the two triangles
are congruent.
Congruence Shortcuts
Angle-Side-Angle (ASA) Congruence Postulate:
If two angles and the included side of one triangle
are congruent to two angles and the included
side of another triangle, then the two triangles
are congruent.
Congruence Shortcuts
Angle-Angle-Side (AAS) Congruence Theorem:
If two angles and a non-included side of one
triangle are congruent to the corresponding two
angles and the non-included side of another
triangle, then the two triangles are congruent.
Example 5
What is the length of
the missing leg in the
each of the right
triangles shown?
Notice that the pieces given
here correspond to SSA,
which doesn’t work.
Because of the Pythagorean
Theorem, right triangles are
an exception.
5 cm
13 cm
13 cm
5 cm
And One More!
Hypotenuse-Leg (HL) Congruence Theorem:
If the hypotenuse and a leg of a right triangle are
congruent to the hypotenuse and leg of another
right triangle, then the two triangles are
congruent.
Example 6
Determine whether the triangles are
congruent.
Example 7
Determine whether the triangles are
congruent.
Example 8
Explain the difference between the ASA and
AAS congruence shortcuts.
Example 9
10-1 & 10-2: Congruence Criteria
Objectives:
Assignment:
1. To discover and use • P. 146: 8-11
shortcuts for showing • P. 153: 11-15
that two triangles are • P. 165: 1-11
congruent