In this activity, students are given various quadrilaterals and asked to decompose each into two identical triangles by drawing a line segment. They observe the kinds of quadrilaterals for which this is possible. To check whether the two triangles in a quadrilateral are identical, students trace one triangle on tracing paper and then rotate it to match the other triangle. The process prepares students to see any triangle as occupying half of a parallelogram, and consequently, as having one half of its area.

To generalize about quadrilaterals that can be decomposed into identical triangles, students need to analyze the features of the given shapes and look for structure (MP7).

There are a number of geometric observations in this unit that must be taken for granted at this point in students' study of mathematics. This is one of those instances. Students have seen only examples of a parallelogram being decomposable into two copies of the same triangle, or have verified this conjecture only through physical experimentation, but for the time being it can be considered a fact. Starting in grade 8, they will begin to prove some of the observations they have previously taken to be true.

The discussion should serve two goals: to highlight how quadrilaterals can be decomposed into triangles and to help students make generalizations about the types of quadrilaterals that can be decomposed into two identical triangles. Consider these questions:

• “How did you decompose the quadrilaterals into two triangles?” (Connect opposite vertices by drawing a diagonal.)
• “Did the strategy of drawing a diagonal work for decomposing all quadrilaterals into two triangles?” (Yes.) "Are all of the resulting triangles identical?" (No.)
• “What do A, B, and D have that C and E do not?” (A, B, and D have two pairs of parallel sides that are of equal lengths. They are parallelograms.)
• "What type of quadrilateral can always be decomposed into two congruent triangles?" (a parallelogram)
• “What features must a quadrilateral have for it to be decomposable into two identical triangles?” (two pairs of congruent sides and one pair of congruent angles)

If students point out that a kite also can be decomposed into two congruent triangles, ask "What features do kites and parallelograms have in common?"

If time permits, discuss how students verified the congruence of the two triangles.

• “How did you check if the triangles are identical? Did you simply stack the traced triangle or did you do something more specific?” (They may notice that it is necessary to rotate one triangle—or to reflect one triangle twice—before the triangles could be matched up.)
• “Did anyone use another way to check for congruence?” (Students may also think in terms of the parts or composition of each triangle. For example, they might say that both triangles have all the same side lengths and have a right angle.)

In this activity, students compose quadrilaterals using pairs of identical triangles. Previously, students saw that a triangle can be seen as half of a familiar quadrilateral. This activity prompts them to think the other way—to see that two identical triangles of any kind can always be joined to produce a parallelogram. Both explorations prepare students to make connections between the area of a triangle and that of a parallelogram in a subsequent lesson.

A key understanding to uncover here is that two identical copies of a triangle can be joined along any corresponding side to compose a parallelogram. This means more than one parallelogram can be formed by the same pair of triangles.

As students work, look for different compositions of the same pair of triangles. Select students who use different approaches to share.

When manipulating the cutouts, students are likely to notice that right triangles can be composed into rectangles (and sometimes squares) and that non-right triangles produce parallelograms that are not rectangles.

As before, students make generalizations here that they don't yet have the tools to justify. This is appropriate at this stage. Later in their study of mathematics, they will learn to verify what they now take as facts.

In the digital version of the activity, students use an applet to compose pairs of triangles into quadrilaterals. The applet allows students to move and rotate the triangles.

The focus of this discussion would be to clarify whether or not two copies of each triangle can be composed into a rectangle or a parallelogram, and to highlight the different ways in which two triangles could be composed into a parallelogram. 

Ask a few students who composed different parallelograms from the same pair of triangles to share. Invite the class to notice how these students ended up with different parallelograms. To help them see that a triangle can be joined along any side of its copy to produce a parallelogram, ask questions such as:

• “Here is one way of composing Triangles S into a parallelogram. Did anyone else do it this way? Did anyone obtain a parallelogram in a different way?”
• “How many different parallelograms can be created with any two copies of a triangle? Why?” (3 ways, because there are 3 sides along which the triangles could be joined.)
• “What kinds of triangles can be used to compose a rectangle? How?” (Right triangles, by joining two copies along the side opposite the right angle.)
• “What kinds of triangles can be used to compose a parallelogram? How?” (Any triangle, by joining two copies along any sides with the same length.)