Standards for TEACH Mathematical Content - Middle School · PDF fileMathematical Content ... Opposite sides of a parallelogram are congruent. Opposite angles of a parallelogram are
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G-SRT.2.5 Use congruence ... criteria for triangles to solve problems and to prove relationships in geometric figures.
Vocabularydiagonal
PrerequisitesTheorems about parallel lines cut by a transversal
Triangle congruence criteria
Math BackgroundIn this lesson, students extend their earlier work with triangle congruence criteria and triangle properties to prove facts about parallelograms. This lesson gives students a chance to use inductive and deductive reasoning to investigate properties of the sides, angles, and diagonals of parallelograms.
Students have encountered parallelograms in earlier grades. Ask a volunteer to define parallelogram. Students may have only an informal idea of what a parallelogram is (e.g., “a slanted rectangle”), so be sure they understand that the mathematical definition of a parallelogram is a quadrilateral with two pairs of parallel sides. You may want to show students how they can make a parallelogram by drawing lines on either side of a ruler, changing the position of the ruler, and drawing another pair of lines. Ask students to explain why this method creates a parallelogram.
Investigate parallelograms.
Materials: geometry software
Questioning Strategies• As you use the software to drag points
A, B, C, and/or D, does the quadrilateral remain a parallelogram? Why? Yes; the lines that form opposite sides remain parallel.
• What do you notice about consecutive angles in the parallelogram? Why does this make sense? Consecutive angles are supplementary. This makes sense because opposite sides are parallel, so consecutive angles are same-side interior angles. By the Same-Side Interior Angles Postulate, these angles are supplementary.
Teaching StrategySome students may have difficulty with terms like opposite sides or consecutive angles. Remind students that opposite sides of a quadrilateral do not share a vertex (that is, they do not intersect). Consecutive sides of a quadrilateral do share a vertex (that is, they intersect). Opposite angles of a quadrilateral do not share a side. Consecutive angles of a quadrilateral do share a side. You may want to help students draw and label a quadrilateral for reference.
Prove that opposite sides of a parallelogram are congruent.
Questioning Strategies• Why do you think the proof is based on drawing
the diagonal ___
DB ? Drawing the diagonal creates two triangles; then you can use triangle congruence criteria and CPCTC.
INTRODUCE
TEACH
1
2
Properties of ParallelogramsFocus on ReasoningEssential question: What can you conclude about the sides, angles, and diagonals of a parallelogram?
You may have discovered the following theorem about parallelograms.
Prove that opposite sides of a parallelogram are congruent.
Complete the proof.
Given: ABCD is a parallelogram.
Prove: ___
AB ≅ ____
CD and ___
AD ≅ ___
BC
REFLECT
2a. Explain how you can use the rotational symmetry of a parallelogram to give an argument that supports the above theorem.
2b. One side of a parallelogram is twice as long as another side. The perimeter of the parallelogram is 24 inches. Is it possible to find all the side lengths of the parallelogram? If so, find the lengths. If not, explain why not.
2
Statements Reasons
1. ABCD is a parallelogram. 1.
2. Draw ___
DB. 2. Through any two points there exists exactly one line.
3. ___
ABǁ ___
DC;___
ADǁ___
BC 3.
4. ∠ADB≅∠CBD;∠ABD≅∠CDB 4.
5. ___
DB≅___
DB 5.
6. 6. ASA Congruence Criterion
7. AB≅CD;AD≅BC 7.
Theorem
If a quadrilateral is a parallelogram, then opposite sides are congruent.
△ABD ≅ △CDB
Under a 180° rotation about the center of the parallelogram, each side is mapped
to its opposite side. Since rotations preserve distance, this shows that opposite
sides are congruent.
Yes; consecutive sides have lengths x, 2x, x, and 2x, so x + 2x + x + 2x = 24, or
6x = 24. Therefore x = 4 and the side lengths are 4 in., 8 in., 4 in., and 8 in.
Use the straightedge tool of your geometry software to draw a straight line. Then plot a point that is not on the line. Select the point and line, go to the Construct menu, and construct a line through the point that is parallel to the line. This will give you a pair of parallel lines, as shown.
Repeat Step A to construct a second pair of parallel lines that intersect those from Step A.
The intersections of the parallel lines create a parallelogram. Plot points at these intersections. Label the points A, B, C, and D.
Use the Measure menu to measure each angle of the parallelogram.
Use the Measure menu to measure the length of each side of the parallelogram. (You can do this by measuring the distance between consecutive vertices.)
Drag the points and lines in your construction to change the shape of the parallelogram. As you do so, look for relationships in the measurements.
REFLECT
1a. Make a conjecture about the sides and angles of a parallelogram.
1A
B
C
D
E
F
Properties of ParallelogramsFocus on ReasoningEssential question: What can you conclude about the sides, angles, and diagonalsof a parallelogram?
Recall that a parallelogram is a quadrilateral that has two pairs of parallel sides. You use the symbol � to name a parallelogram. For example, the figure shows �ABCD.
Teaching StrategyThe lesson concludes with the theorem that
states that opposite angles of a parallelogram are
congruent. The proof of this theorem is left as an
exercise (Exercise 1). Be sure students recognize
that the proof of this theorem is similar to the
proof that opposite sides of a parallelogram
are congruent. Noticing such similarities is an
important problem-solving skill.
Essential QuestionWhat can you conclude about the sides, angles, and diagonals of a parallelogram? Opposite sides of a parallelogram are congruent. Opposite angles of a parallelogram are congruent. The diagonals of a parallelogram bisect each other.
SummarizeHave students make a graphic organizer to
summarize what they know about the sides, angles,
and diagonals of a parallelogram. A sample is
shown below.
Exercise 1: Students practice what they learned
in part 2 of the lesson.
Exercise 2: Students use reasoning to extend what
they know about parallelograms.
Exercise 3: Students use reasoning and/or algebra
to find unknown angle measures.
Exercise 4: Students apply their learning to solve