Download Core Geometry: Investigation of Triangle Congruence Shortcuts

Mathematician: _________________________
Date: _______________
Core-Geometry: 4.3 Prove Triangles
Congruent by SSS
Warm-up:
1. Provide two linear equations that are parallel.
2. Provide an equation of a line that has an undefined slope.
Review
1. Determine the distance between (10, -3) and (15, 5).
2. Determine the midpoint of AB where A is (5, 1) and B is (-1, -5).
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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
Name
Does it prove
congruence?
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Sketch
4.3 Proving Triangles Congruent by SSS
In the investigation you saw that there is only _______ way to form a triangle given
_______________ side lengths. Any _________triangles with the same three
_______________ lengths must be _______________________________.
Example 1 Using the SSS Congruence Shortcut
Decide whether the congruence statement is true. Explain your reasoning.
a)
b)
Example 2 Triangles and Coordinate Geometry
Using the coordinates listed, decide if the triangles below are congruent. If they
are, write a congruence statement for the congruent triangles.
Hmwk 3.2
p.236 Ex 4.3 # 1-9 odd, 17, 19
Quiz for lessons 4.1-4.3 # 1-5
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