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In the following problems, you will draw a variety of triangles and take note of their differences. Note 2
a. Draw any triangle on your paper.
b. Draw a second triangle that is different in some way from your first one. Describe in just a word or two how it is different.
Draw a third triangle that is different from both of your other two. Describe how it is different.
Draw two more triangles, different from all the ones that came before.
To make “different” triangles, you have to change some feature of the triangle. Make a list of the features of triangles that you changed.
The following shows how triangles can be classified according to some of their features:
Angles:
Right Triangles |
Acute Triangles |
Obtuse Triangles |
One right (90°) angle | All angles less than 90° | One angle more than 90° |
Sides:
Equilateral Triangles |
Isosceles Triangles |
Scalene Triangles |
All three sides are the same length. | Two sides are the same length. | All three sides are different lengths. |
Size:
Similar Triangles |
Congruent Triangles |
Same shape, possibly different size | Same size and shape |
More than one feature can be combined into a triangle. Decide which of the following combinations are possible. If the combination is possible, draw a sketch on a piece of paper. If not, explain why not.
Note 2
If you’re working in a group, first work on Problems A1-A5 individually. Afterwards, break into small groups to compare the lists of the features of triangles that everyone created.
a. Answers will vary. The following is one example:
b. Answers will vary. One example is a triangle whose side lengths are longer.
Answers will vary. One example is a triangle that, unlike the previous two, has one right angle.
Answers will vary. Some examples are the following:
Answers will vary, but generally, either the measures of angles or lengths of sides need to be changed to make “different” triangles.
a. This is possible.
b. This is possible.
c. This is impossible. Inside a triangle, equal angles correspond to equal sides, so for a right triangle to be equilateral, it would have to have three right angles. Two right angles next to each other, however, form parallel lines, which would mean it would not be possible to complete such a triangle.
d. This is impossible. The picture below shows the right angle and obtuse angle next to each other, with the side in between laid out horizontally. The side that extends from the right angle is vertical, while the side that extends from the obtuse angle is pointed away from the side that extends from the right angle. Because these sides must be connected to form a triangle, this kind of triangle is impossible to make.