Download Congruence Shortcuts?

Warm-Up
If m<J + m<E + m<R = 180°, then construct
<R.
4.4 Prove Triangles Congruent by SAS and HL
4.5 Prove Triangles Congruent by ASA and AAS
Objectives:
1. To discover and use shortcuts for
showing that two triangles are 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
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
Well, we know that SSS is a
valid shortcut, and I’ll give you
the hint that 2 others in the list
do not work.
We will test the remaining 5 in
class. 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 an Angle
5. Put point of compass on B and pencil on
C. Make a small arc.
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.
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 1
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.
Therefore, rt. triangles have theorems
such as HL (hypotenuse-leg) and LL
(leg-leg)
5 cm
12
13 cm
13 cm
5 cm
12
Example 2
Determine whether the triangles are
congruent in each pair.
Yes, SAS
No
Example 3
Determine whether the triangles are
congruent in each pair. Answer and explain
which theorem in your
notebook
Example 4
Explain the difference between the ASA and
AAS congruence shortcuts.
Answer in your notebook.
Example 5
TRY IT in your notebook!
I will pick someone at random to
work it on the board  Ain’t life
GRAND!