Download Example 1: Using the SAS and SSS Congruence Shortcuts

Mathematician: _________________________
Date: _______________
Core-Geometry: 4.4 SAS and HL
Warm-up:
1. Rewrite the following linear equations using
slope-intercept form: y = mx + b
2x – 3y = 12
3x + 2y = 12
Are these line perpendicular, parallel, skew or other?
2. Provide two numbers that multiply to -24 and sum to 10.
Review
1. Construct a copy of the triangle below:
2. List and draw all 7 triangles in this unit.
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Revisit: Core Geometry: Investigation of Triangle Congruence Shortcuts
If you prove that two triangles are congruent using the definition of congruence, then
you need to show that all six parts (three angles and three sides) of both triangles are
congruent.
Fortunately there are “shortcuts”, and we can use less than six corresponding parts to
prove two triangles are congruent. We will start by comparing just three parts of each
triangle.
1.
List the six possible combinations of three parts (angles and/or sides) in the first
column of the table on the results sheet. Flipping the order creates identical possible
combinations. For example, Angle – Angle – Side is the same as Side – Angle – Angle
because they are the same three parts in reverse order.
2.
Investigate each possible congruence shortcut by using the activity found on the
following website: http://illuminations.nctm.org/Activity.aspx?id=3504
Instructions:
 Select three triangle parts from the top, right menu to start. If you choose side
AB, angle A, and angle B, you will be working on Angle – Side – Angle. If instead
you choose side AB, angle A, and angle C, you will be working on
Angle – Angle – Side. This creates those parts in the work area. (Note: The tool
does not allow you to select more than three parts. If you select the wrong part,
simply unselect it before choosing another part.)
 Click and drag a dot to move the part to a new location. Click and drag a side's
endpoint or angle's arrow to rotate the part. The center of rotation is the side's
midpoint or the angle's vertex, respectively. Move the parts of the triangle so
that points labeled with the same letter touch. To help place parts, points
marked with the same letter snap together. When angles snap, the rays are
extended to the edge of the work area.
 When you create a closed triangle, the points merge and center is filled in.
 Once a triangle is formed with the original three parts, the triangle moves to the
bottom, right corner of the work area, and congruent parts appear.
 Form a second triangle with these congruent parts.
 After a second triangle is formed, you will be asked if they are congruent. You
can test congruence by manipulating either triangle. Click and drag within the
triangle to move it to a new location. Click and drag a vertex to rotate the
triangle. Use the Flip button to reflect the triangle horizontally. First click on the
triangle you would like to reflect, and then click the Flip button.
i. If you can create two different triangles with the same parts, then those
parts do not prove congruence. Careful - two
triangles might be mirror images but still congruent,
therefore you may have to flip your triangles to see
how they are congruent. For example, all the
triangles to the right are congruent.
ii. If, however, the second triangle can only be formed
congruent to the first, then that arrangement of
three parts is a congruence shortcut.
 If the two triangles are congruent, you will be asked if it's
possible to make a triangle that is not congruent to the original. If you create a
third congruent triangle, you will be given the option to try again.
 The Reset button clears the work area and creates new sides and angles for the
selected parts.
 The New button clears your selection and work area
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Three Parts
AAA
Name
AngleAngleAngle
Does it prove
congruence?
Are you
kidding?
NO
WAY!!
AAS
AngleAngleSide
ASA
AngleSideAngle
SSA
SideSideAngle
SAS
SideAngleSide-
SSS
SideSideSide-
YES
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Sketch
4.4 Proving Triangles Congruent by SAS and HL
An included angle is the angle between two _____________.
In the investigation last class and this class you saw that there is only _______ way
to form a triangle given _______ side lengths and an _________________ ______________.
Any _________ triangles with these three parts ______________________ must be
______________________.
In the diagram above, D ________ @ D ________
Example 1: Using the SAS and SSS Congruence Shortcuts
Decide whether enough information is given to prove that the triangles are
congruent. If there is enough information, explain which shortcut you would use
and write a triangle congruency statement.
There is one shortcut we did not investigate. This shortcut only works with
______________ triangles. In a right triangle, the sides adjacent to the right angle are
called _____________. The side opposite the right angle is called the
_______________________.
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If the ______________________ and ____________ of a right triangle are congruent to the
______________________ and ____________ of a second right triangle then the two
triangles are _____________________.
In the diagram below, D ________ @ D ________
Example 2: Using the HL Congruence Shortcut
Decide whether enough information is given to prove that the triangles are
congruent using the HL shortcut. If there is enough information, write a triangle
congruency statement.
Hmwk 3.3
p.243 Ex 4.4 # 1-7 odd, 9-14, 16-18, 20-22
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