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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. If not, explain why not.

- a scalene acute triangle
- an isosceles acute triangle
- an equilateral acute triangle
- a scalene obtuse triangle
- an isosceles obtuse triangle
- an equilateral obtuse triangle
- an equilateral triangle that is also isosceles

A certain quadrilateral has one diagonal that is 2 inches long and another diagonal that is 3 inches long. A diagonal is a line segment connecting any two non-adjacent vertices.

- Draw two such quadrilaterals. What, if anything, do they have in common? How are they different?
- Draw such a quadrilateral where the diagonals bisect each other but are not perpendicular. What does it look like?
- Draw such a quadrilateral where the diagonals are perpendicular but do not bisect each other. What does it look like?
- Draw such a quadrilateral where the diagonals bisect each other and are perpendicular. What does it look like?

Create a quadrilateral with diagonals that are the same length and bisect each other. What kind of quadrilateral is it? Can you explain why?

Create a quadrilateral with diagonals that are the same length, bisect each other, and are perpendicular. What kind of quadrilateral is it? Can you explain why?

Are the triangles different or congruent? If you think they are congruent, try to draw a triangle that fits the description but is not congruent.

For each part below, draw two different triangles that fit the information given. What do you notice?

- One side is 2 inches long; another side is 3 inches long. The angle between them is 45°.
- One side is 2 inches long; another side is 3 inches long. The angle between them is 75°.
- One side is 2 inches long; another side is 3 inches long. The angle between them is 90°.

Stop! Do the above problem before you proceed. Use the tip text to help you solve the problem if you get stuck.

You have already seen the SSS (side-side-side) congruence test for triangles: If the three sides of one triangle have the same lengths as the three sides of another triangle, then the two triangles are congruent. That is, they have exactly the same size and shape. Describe and name a new congruence test based on your work in Problem H5.

For each part below, draw two different triangles that fit the information given. What do you notice?

- The three angle measures are 45°, 45°, and 90°.
- The three angle measures are 60°, 60°, and 60°.
- The three angle measures are 100°, 30°, and 50°.

Stop! Do the above problem before you proceed. Use the tip text to help you solve the problem if you get stuck.

Use your work in Problem H7 to answer this question.

Is there an angle-angle-angle (AAA) congruence test for triangles? That is, if the three angles of one triangle have the same measures as the three angles of another triangle, are the two triangles necessarily congruent? Explain your answer.

Stop! Do the above problem before you proceed. Use the tip text to help you solve the problem if you get stuck.

**Steen, Lynn Arthur (1990). Pattern. In On the Shoulders of Giants: New Approaches to Numeracy. Edited by Lynn Arthur Steen (pp. 1-10). Washington, D.C.: National Academy Press.**

Reproduced with permission from the publisher. Copyright © 1990 by National Academy Press. All rights reserved.

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Pattern

Continued…

- It is possible. For example:

- It is possible. For example:

- It is possible. For example:

- It is possible. For example:

- It is possible. For example:

- This is impossible, because equal sides correspond to equal angles. This would mean that a triangle would have two consecutive obtuse angles. The sides extending from these two angles could not be connected for the same reason as in Problem A5, part (d).

- It is possible. In fact, all equilateral triangles are also isosceles triangles.

- Constructions and answers will vary. But, if the diagonals have different lengths (such as 2 and 3 units), then the figure can never be a square or a rectangle.
- Constructions will vary, but the figure will always be a parallelogram. One possible example is the following:

- Constructions will vary, but the figure will either be a kite or a random quadrilateral. One possible example follows:

- In this case, there is only one possibility — a rhombus:

You should find that the result must be a rectangle. Here’s one explanation for that fact: The sum of the interior angles in any quadrilateral is 360°. Because the line segments marked “a” are all equal, the angles opposite to them inside the respective triangles are equal. Therefore, the sum of angles 4A + 4B = 360°; i.e., 4(A + B) = 360°; i.e., A + B = 90°. Hence, all of the interior angles are right angles, and the quadrilateral is indeed a rectangle.

The quadrilateral in question is a rectangle as described in the solution to Problem H3. In addition, two adjacent isosceles right triangles with hypotenuses a and b respectively are congruent since they have congruent legs, and the congruent (right) angles between them. So we must have a = b, and therefore the quadrilateral is a square.

In parts a-c, it is impossible to draw two different triangles. In other words, if we fix two sides and the angle between them, we uniquely determine a triangle.

This type of congruence can be called SAS (side-angle-side) congruence: If two triangles have two sides equal in length, and the angles between those sides are equal in their degree measure, then the two triangles are congruent.

In parts a-c, we can create two or more distinct triangles by keeping the angles fixed and changing the lengths of the sides. For example, we can build a second triangle where each side is twice as long as the original and the angles will remain the same size.

No. Problem H7 shows that two triangles can have the same size angles without being congruent. The triangles appear to have the same overall shape, but they might be larger or smaller than the original. They are not congruent, but they do have a relationship. They are called similar.