# Grade 7 Three- and Two- Dimensional Geometry

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University of Waterloo                                                                                                                      Centre for Education in
Waterloo, ON Canada                                                                                                                     Mathematics and Computing
N2L 3G1                                                                                                                                     Faculty of Mathematics

THREE-           AND         TWO-DIMENSIONAL GEOMETRY
This resource may be copied in its entirety, but is not to be used for commercial purposes without permission from the Centre for Education in
Mathematics and Computing, University of Waterloo.
Play 2D to 3D Morphing first! (You’ll need a printer, paper, scissors and glue.)
Click on http://www.senteacher.org/Worksheet/12/Nets.xhtml or
You may go to www.wiredmath.ca for the link.

1.         Use Geometer’s Sketchpad or pencil and paper to sketch the front, side, back, top and bottom views.

a.                                                 b.                                                 c.

A polygon is a figure formed by three or more line segments.
Examples of polygons are triangles, quadrilaterals, pentagons, and octagons.

A polyhedron is a three-dimensional object that has polygons as faces.
Examples of polyhedra are cubes, tetrahedrons, octahedra, and dodecahedrons.

2.         Which polyhedron has each set of views?
Slice of History
a.         Bottom                                   Front                                  Side
Euclid (c.325 BC-c270 BC)
was a Greek geometer
who wrote The Elements,
the world’s most famous
geometry textbook.

b.           Top                                    Front                                  Back                   This book remained the
standard        geometry
textbook until the end of
the nineteenth century.
That’s over two thousand
years!

Expectations: i) construct related lines, using angle properties and Geometer’s Sketchpad; ii) develop an understanding of similarity, and distinguish similarity   1
and congruence. For more activities and resources from the University of Waterloo’s Faculty of Mathematics, please visit www.cemc.uwaterloo.ca.
3.         Identify and sketch as many 3-dimensional shapes with the top view as shown.

a.                                                                             b.

To Measure an Angle:
1. Select the three vertices in order that form the angle. For
example, for ABC, select vertex A, then vertex B and finally
vertex C. (The labels will show up later.)
2. Click “Measure” > “Angle”
3. The measurement will appear in a highlighted (pink) box.

To   Measure a Segment:
1.   Use the select tool and click on the segment to choose it.
2.   Click “Measure” > “Length”
3.   The measurement will appear in a highlighted (pink) box.

Note: You can drag the measurements to any part of the worksheet.
Before you measure something else, click on an empty part of the
screen to clear the highlighting.

4.         Use Geometer’s Sketchpad to draw each triangle.

a. An isosceles triangle with two 30 angles.
b. An equilateral triangle with 6 cm sides.
c. A right triangle with an acute angle of 45 angle.

a. A parallelogram with two 5 cm sides.
b. A quadrilateral with no parallel sides.
c. A trapezoid with a 60 angle.

Expectations: i) construct related lines, using angle properties and Geometer’s Sketchpad; ii) develop an understanding of similarity, and distinguish similarity   2
and congruence. For more activities and resources from the University of Waterloo’s Faculty of Mathematics, please visit www.cemc.uwaterloo.ca.
Regular polygon has all sides and angles equal.

6.         Use Geometer’s Sketchpad to draw each shape.

a. A hexagon with two 45 angles.
b. A regular pentagon.

A net is a pattern that can be folded to make a three-dimensional figure.

7.         Which of the following are not nets?

8.         Use Geometer’s Sketchpad to draw the net of each three-dimensional figure.

a. A square pyramid.                               b. An octahedron.                                  c. A tetrahedron.

Did You Know?

This figure was introduced by
the Italian psychologist Gaetano
Kanizsa.

While you see the white triangle
on top, it doesn’t actually exist!
The contours of the triangle are

Expectations: i) construct related lines, using angle properties and Geometer’s Sketchpad; ii) develop an understanding of similarity, and distinguish similarity   3
and congruence. For more activities and resources from the University of Waterloo’s Faculty of Mathematics, please visit www.cemc.uwaterloo.ca.
Congruent figures have the same size and shape.

9.         Which pair of triangles are congruent? Explain.

10.        Which pair of four-sided figures are congruent? Explain.

CHALLENGE YOURSELF!
11.        A box has four flaps labelled P, Q, R, and S. Without moving the
box, the flaps are folded over in the order R, S, Q, P. For a person                                                                  Q
standing at point X and looking down on the box from the top, the
appearance of the box is                                                                                           P
R
a.                      b.                     c.
S

X
d.                                e.

EXTENSION
12.        Which figure below can be obtained by rotating the figure on the right?

a.                       b.                        c.                        d.                          e.

Expectations: i) construct related lines, using angle properties and Geometer’s Sketchpad; ii) develop an understanding of similarity, and distinguish similarity   4
and congruence. For more activities and resources from the University of Waterloo’s Faculty of Mathematics, please visit www.cemc.uwaterloo.ca.

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