lesson title: properties of parallelogramcontent.njctl.org/courses/math/geometry-2015-16/... ·...

Geometry – Quadrilaterals -1- NJCTL.org

Name _____________________________ Date _____________________

Lab: Properties of Parallelograms .

Materials:

1. Handouts

2. Geoboard with rubber bands

3. Protractor

Part 1

1. On a coordinate grid, plot and label the points A(6, 2), B(17, 5), C(13, 12), and D(2, 9).

2. What general shape will be formed if you connect A to B, B to C, C to D, and D to A? Connect each pair of points to verify your answer.


3. Use the distance formula, 2 2

1 2 1 2( ) ( )d x x y y , to determine the length of each of the

segments.

Length of AB =

Length of BC =

Length of CD =

Length of DA =

4. What can you conclude about the lengths of the segments in this figure?

Part 2

1. On a coordinate grid, plot and label the points A(-7, 5), B(-2, -4), C(8, -8), and D(3, 1).



3. Use the Slope formula, 1 2

1 2

( )

( )

y ym

x x

, to determine the slopes of each of the segments.

Slope of AB =

Slope of BC =

Slope of CD =

Slope of DA =

4. What can you conclude about the slopes of the segments in this figure?

Part 3

1. On a coordinate grid, plot and label the points A(-3, 0), B(-1, 5), C(5, 4), and D(3, -1).



3. Diagonals are segments that connect two non-adjacent vertices of a polygon. How many

diagonals will the figure above have? Use a straightedge to draw the diagonals and verify your answer.

4. Using the Midpoint formula, 1 2 1 2

midpoint ,2 2

x x y y

, find the midpoint of diagonal AC

. Plot and label this midpoint on the above coordinate plane as point F.


1 2 1 2( ) ( )d x x y y , to determine the length of each of these

segments.

Length of AF =

Length of CF =

From these lengths, what can you verify?



segments.

Length of BF =

Length of DF =

From these lengths, what can you conclude?

7. What can you conclude about the diagonals in this figure?


Part 4

• Consecutive angles of a quadrilateral are two angles that share a common ray. • Opposite angles of a quadrilateral are two angles that are located on either end of a diagonal;

they are directly across from one another. Think of your Geoboard as the first quadrant of the coordinate plane, like the picture below

1. Assume that the bottom edge of the Geoboard is the x-axis and the left edge is the y-axis. Stretch the rubber band around these 4 coordinates to create a parallelogram.

A(1, 1) B(3, 1) C(5, 5) D(3, 5)

2. Use a protractor to measure the angles at each of the vertices and record them below.

m∠A =

m∠ B =

m∠ C =

m∠ D =

3. Do you notice any similarities or differences among the angle measurements? Explain. 4. Now stretch the rubber band around these 4 coordinates to create a different parallelogram.

E(2, 1) F(2, 4) G(4, 5) H(4, 2)

x-axis

y-axis


5. Use a protractor to measure the angles at each of the vertices and record them below

m∠ E =

m∠ F =

m∠ G =

m∠ H = 6. Do you notice any similarities or differences among the angle measurements in the second

parallelogram? Explain. 7. Based on your responses to the #1-6, what general statement(s) can you make about a

property or properties found in all parallelograms regarding angle measurements?

8. Using your protractor draw a parallelogram below using only the angle measurements to guide you. Label the angle measurements in your parallelogram.


Name ________________________________ Exit Card

On a coordinate grid, plot and label the points A(-3, 3), B(8, 5), C(2, -1).

1. Using the information that you discovered in the previous activities, where could you place point D in order to make a parallelogram? 2. Please verify your solution by showing the slopes of the sides, the lengths of the sides, the midpoint of the diagonals as well as the angle measures. 3. After you have found one point that makes a parallelogram, is it possible to find another point that will make a different parallelogram with the three given points? How many different points exist that will form a parallelogram with the three given points? List as many of these points as you can.


Name __ANSWERS____________________ Date _____________________

Lab: Properties of Parallelograms .

Materials:

1. Handouts

2. Geoboard with rubber bands

3. Protractor

Part 1

1. On a coordinate grid, plot and label the points A(6, 2), B(17, 5), C(13, 12), and D(2, 9).


Parallelogram; see shape above.



1 2 1 2( ) ( )d x x y y , to determine the length of each of the

segments.

Length of AB = 11.40

Length of BC = 8.06

Length of CD = 11.40

Length of DA = 8.06

4. What can you conclude about the lengths of the segments in this figure? Opposite sides are equal/congruent.

Part 2

1. On a coordinate grid, plot and label the points A(-7, 5), B(-2, -4), C(8, -8), and D(3, 1).


Parallelogram; see shape above.

3. Use the Slope formula, 1 2

1 2

( )

( )

y ym

x x

, to determine the slopes of each of the segments.


Slope of AB = -1.8 = -9/5

Slope of BC = -0.4 = -2/5

Slope of CD = -1.8 = -9/5

Slope of DA = -0.4 = -2/5

4. What can you conclude about the slopes of the segments in this figure? Opposite sides have equal slopes, making them parallel.

Part 3

1. On a coordinate grid, plot and label the points A(-3, 0), B(-1, 5), C(5, 4), and D(3, -1).

2. What general shape will be formed if you connect A to B, B to C, C to D, and D to A? Connect each pair of points to verify your answer. Parallelogram; see shape above.

3. Diagonals are segments that connect two non-adjacent vertices of a polygon. How many diagonals will the figure above have? Use a straightedge to draw the diagonals and verify your answer. 2 diagonals; see figure in grid on previous page.


4. Using the Midpoint formula, 1 2 1 2

midpoint ,2 2

x x y y

, find the midpoint of diagonal AC

. Plot and label this midpoint on the above coordinate plane as point F. see figure in grid on previous page for picture. Coordinates of F(1, 2).



segments.

Length of AF = 4.47

Length of CF = 4.47

From these lengths, what can you verify? Segment AC (one diagonal) is bisected.



segments.

Length of BF = 3.61

Length of DF = 3.61

From these lengths, what can you conclude? Segment BD (other diagonal) is bisected.

7. What can you conclude about the diagonals in this figure? The diagonals bisect each other; intersect at the midpoint.


Part 4

• Consecutive angles of a quadrilateral are two angles that share a common ray. • Opposite angles of a quadrilateral are two angles that are located on either end of a diagonal;

they are directly across from one another. Think of your Geoboard as the first quadrant of the coordinate plane, like the picture below

1. Assume that the bottom edge of the Geoboard is the x-axis and the left edge is the y-axis. Stretch the rubber band around these 4 coordinates to create a parallelogram. See figure above to the right.

A(1, 1) B(3, 1) C(5, 5) D(3, 5)

2. Use a protractor to measure the angles at each of the vertices and record them below.

m∠A = 63.43o

m∠ B = 116.57 o

m∠ C = 63.43 o

m∠ D = 116.57 o

3. Do you notice any similarities or differences among the angle measurements? Explain.

Opposite angles are congruent/equal, and adjacent angles are supplementary. 4. Now stretch the rubber band around these 4 coordinates to create a

different parallelogram. See figure to the right. E(2, 1) F(2, 4) G(4, 5) H(4, 2)

x-axis

y-axis

Note: If the students’ measurements

are close, give them full credit.


5. Use a protractor to measure the angles at each of the vertices and record them below

m∠ E = 63.43o

m∠ F = 116.57 o

m∠ G = 63.43o

m∠ H = 116.57 o

6. Do you notice any similarities or differences among the angle measurements in the second

parallelogram? Explain. They have the same measurements as the angles of parallelogram ABCD. The opposite angles are congruent/equal, and the adjacent angles are supplementary.

7. Based on your responses to the #1-6, what general statement(s) can you make about a

property or properties found in all parallelograms regarding angle measurements? The opposite angles are congruent/equal, and the adjacent angles are supplementary.

8. Using your protractor draw a parallelogram below using only the angle measurements to guide you. Label the angle measurements in your parallelogram. See student work; answers will vary.

Note: If the students’ measurements

are close, give them full credit.


Name _ANSWERS________________________ Exit Card

On a coordinate grid, plot and label the points A(-3, 3), B(8, 5), C(2, -1). SEE ALL GRAPHS BELOW…ANY ONE OF THEM WOULD WORK.

1. Using the information that you discovered in the previous activities, where could you place point D in order to make a parallelogram? Any of the answers below would work.

Figure 1: D(-9, -3) Figure 2: D(3, 9) Figure 3: D(13, 1)

2. Please verify your solution by showing the slopes of the sides, the lengths of the sides, the midpoint of the diagonals as well as the angle measures.

Lengths of the

sides Slopes of the

sides Midpoints of the diagonals

Angle Measures

Figure 1

G (-0.5, 1)

Slope AD1 = 1.00

Slope D1C = 0.18

Slope BC = 1.00

Slope AB = 0.18

mÐCD1A = 34.70°

mÐBCD1 = 145.30°

mÐABC = 34.70°

mÐD1AB = 145.30°

D1A = 8.49 cm

CD1 = 11.18 cm

BC = 8.49 cm

AB = 11.18 cm


Figure 2

H (2.5, 4)

Figure 3

I (5, 2)

3. After you have found one point that makes a parallelogram, is it possible to find another point that will make a different parallelogram with the three given points? How many different points exist that will form a parallelogram with the three given points? List as many of these points as you can.

Yes, it is possible to find other points to make a parallelogram. There are 3 total points that you can make to form a parallelogram with the original 3 points.

D(-9, -3), D(3, 9), or D(13, 1)

AD2 = 8.49 cm

BD2 = 6.40 cm

BC = 8.49 cm

AC = 6.40 cm

Slope AD2 = 1.00

Slope D2B = –0.80

Slope BC = 1.00

Slope AC = –0.80

mÐBD2A = 96.34°

mÐCBD2 = 83.66°

mÐACB = 96.34°

mÐD2AC = 83.66°

Slope AC = –0.80

Slope CD3 = 0.18

Slope D3B = –0.80

Slope AB = 0.18

mÐD3CA = 131.04°

mÐBD3C = 48.96°

mÐABD3 = 131.04°

mÐBAC = 48.96°

AC = 6.40 cm

CD3 = 11.18 cm

BD3 = 6.40 cm

AB = 11.18 cm

lesson title: properties of parallelogramcontent.njctl.org/courses/math/geometry-2015-16/... ·...

Documents