Grade 9: Geometry of 2D Shapes
April 2013
Grade 9
Geometry of 2D Shapes
Goals:
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Classify triangles by angles and sides
Classify quadrilaterals by angles, sides, and diagonals
Classify other polygons by sides
Use and recall clear definitions for terms related to polygons
Find and articulate minimum conditions for triangle congruence
Special properties of triangles and quadrilaterals
Solve problems involving missing segment and angle lengths
Terminology:
Triangles:
 Acute triangle
 Right triangle
 Obtuse triangle
 Scalene triangle
 Isosceles triangle
 Equilateral triangle
Important Theorems:
 An exterior angle of triangle equals the
sum of opposite interior angles
 Isosceles and equilateral triangle
properties
 Sum of interior angles of triangles is 180°
 Pythagorean Theorem (covered later)
 Special quadrilateral properties
 Sum of interior angles of any polygon
Quadrilaterals
 Diagonal
 Square
 Rectangle
 Kite
 Rhombus
 Parallelogram
 Trapezium
Congruent Triangles:
 Congruent Triangles
 Included Angle
 Included Side
 SSS
 SAS
 AAS
 ASA
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Grade 9: Geometry of 2D Shapes
Classifying 2D Shapes
Classifying Triangles By Sides:
Name
Scalene Triangle
Definition
A triangle with all 3 sides of
a different length
Isosceles Triangle
A triangle with at least 2
congruent sides
Equilateral Triangle
A triangle with 3 congruent
sides
Drawing
Classifying Triangles By Angles:
Name
Acute Triangle
Definition
A triangle with all three
angles less than 90°
Right Triangle
A triangle with a right angle
in it
Obtuse Triangle
A triangle with an angle of
greater than 90°
Equiangular Triangle
A triangle with all three
angles congruent (i.e. 60°)
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Drawing (add markings)
Grade 9: Geometry of 2D Shapes
Classifying Quadrilaterals by Sides
Name
Trapezium
Definition
A quadrilateral in which one
pair of opposite sides is
parallel.
Kite
A quadrilateral in which two
pairs of adjacent sides are
the same length.
Parallelogram
A quadrilateral in which both
pairs of opposite sides are
parallel.
Rhombus
A quadrilateral with four
congruent sides.
Rectangle
A quadrilateral with four
right angles.
Square
A quadrilateral with four
congruent sides and a right
angle.
Drawing
Properties and Diagonals of Quads
Name
Trapezium
Drawing (add markings)
Special Properties
 One pair of parallel sides
 If the nonparallel sides are congruent, then the
diagonals are congruent.
Kite
 One diagonals bisects the other
 Diagonals are perpendicular
 2 sets of adjacent sides are congruent
Parallelogram
 Diagonals bisect each other
 Opposite sides are both parallel and congruent
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Grade 9: Geometry of 2D Shapes
Rhombus
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Diagonals bisect each other
Diagonals
All sides are congruent
Opposite sides are parallel
If one angle is 90°, then the rhombus is also a
square
Rectangle
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Diagonals bisect each other
Diagonals are congruent
Opposite sides are parallel and congruent
If adjacent sides are congruent, then the rectangle
is also a square
Square
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Diagonals bisect each other
Diagonals are perpendicular
Diagonals are congruent
All sides are congruent
Opposite sides are parallel
Other Polygons
Property
Definition
Example
Convex
All interior angles are less than or equal to 90°
Concave
There is an interior angle of more than 90°
Regular
All sides and angles are congruent
Not Regular
Not all sides and angles are congruent
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Grade 9: Geometry of 2D Shapes
Exercise 1
1. Draw a simple, small sketch of each of the following. If the figure is impossible to draw,
then write “not possible.”
1.1. A scalene right triangle
1.11. A rhombus which is also rectangle
1.2. An isosceles right triangle
1.12. A rhombus which is NOT a
1.3. An obtuse, isosceles triangle
parallelogram
1.4. An equiangular, obtuse triangle
1.13. A square which is also a kite
1.5. An equilateral triangle which is
1.14. A regular hexagon
also isosceles
1.15. A dodecagon which is NOT regular
1.6. A triangle with an angle of more
1.16. A pentagon which is concave
than 180°
1.17. A concave, regular dodecagon
1.7. A quadrilateral with 7 sides
1.18. A concave quadrilateral
1.8. A trapezoid with a pair of
1.19. A convex pentagon
congruent sides
1.20. A rectangle with an angle of more
1.9. A rectangle which is also a square
than 180°
1.10. A rectangle which is NOT a square
2. Decide if each statement is TRUE or FALSE. You may wish to recall definitions above or
study the chart to the right.
2.1.
2.2.
2.3.
2.4.
2.5.
2.6.
2.7.
2.8.
2.9.
2.10.
2.11.
2.12.
2.13.
2.14.
2.15.
2.16.
All squares are also trapeziums.
All rectangles are also parallelograms.
All rhombuses are also parallelograms.
All rectangles are also squares
All parallelograms are also trapeziums.
All squares are also kites.
All quadrilaterals have exactly four sides
It is possible for a trapezium to also be a
parallelogram
If a quadrilateral has one pair of
congruent sides, then it must be a kite.
If a quadrilateral has one pair of parallel
sides, then it must be a trapezium
If a quadrilateral has one pair of parallel
sides, then it must be a square.
If a quadrilateral has four congruent sides, then it must be a rhombus.
If a quadrilateral has a right angle, then it must be a rectangle.
If a quadrilateral has two congruent sides, then it must be a kite.
If a quadrilateral has opposite angles congruent, then it must be a rhombus
If a quadrilateral has opposite angles congruent, then it must be a parallelogram.
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Grade 9: Geometry of 2D Shapes
3. Who am I? Give the name which most correctly suits the description:
3.1. I have four sides. I have four right angles.
3.2. All of my sides are congruent. I have 3 sides.
3.3. I have three angles. Here are their measures: 177°, 1°, 2°.
3.4. I have four angles. Here are their measures: 90°, 90°, 90°, 90°.
3.5. I have three sides. Here are their lengths: 5 m, 12 m, 5 m.
3.6. I have three sides. Here are their lengths: 2 km, 3 km, 2,5 km.
3.7. I have four sides. Here are their lengths: 2 mm, 2 mm, 3 mm, 3 mm.
3.8. I have four sides. Here are their lengths: 4 cm, 4 cm, 4 cm, 4 cm.
3.9. My diagonals are the same length.
3.10. All of my sides and angles are congruent. I have 7 sides.
3.11. One of my diagonals bisects the other.
3.12. I have two pairs of opposite, congruent sides.
3.13. My diagonals are congruent. They also bisect each other, and they’re
perpendicular to each other.
3.14. I have three angles. Here are their measures: 90°, 45°, 45°.
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Grade 9: Geometry of 2D Shapes
Special Properties of Triangles
The following triangle theorems are ones which you should write in your geometry toolbox,
know, and be able to apply in exercises:
Sum of Interior Angles – The sum of the interior angles of any triangle is 180°
Equilateral Triangles – In an equilateral (or
equiangular) triangle, all sides are equal and all
angles are 60°
Isosceles Triangle – In an isosceles triangle, the
base angles are congruent.
Pythagorean Theorem – In a right triangle, 𝑎2 + 𝑏 2 = 𝑐 2 , where 𝑎 and 𝑏 are legs
and 𝑐 is the hypotenuse. (Note: We’ll cover Pythagoras’ Theorem in more depth later
on.)
Legs: sides of a right triangle
which are adjacent to the
right angle.
Hypotenuse: The side of a
right triangle which is
opposite the right angle.
NB: The Pythagorean
Theorem does NOT apply if
there’s no right angle!
Conclusion: By the Pythagorean Theorem,
𝑎2 + 𝑏 2 = 𝑐 2
Exterior Angle Theorem – The exterior angle of a triangle equals the sum of the
opposite two interior angles.
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Grade 9: Geometry of 2D Shapes
Exercise 2
1. Classify each triangle by its angles and sides.
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Grade 9: Geometry of 2D Shapes
2. Find the missing angle length:
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Grade 9: Geometry of 2D Shapes
3. Find the value of 𝑥:
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Grade 9: Geometry of 2D Shapes
4. Solve for 𝑥:
5. Find the measure of angle A:
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Grade 9: Geometry of 2D Shapes
Special Properties of Quadrilaterals
To assist in learning the properties below, learners should be very familiar with the
quadrilateral definitions and properties identified in the first section.
Sum of Interior Angles
The sum of the interior angles of any quadrilateral is 360°
Parallelogram
The opposite sides of parallelograms are parallel and equal
Parallelogram
The opposite angles of parallelograms are equal
Rhombus
The opposite sides of a rhombus are parallel and equal
Rhombus
The opposite angles of a rhombus are equal
Bisecting Diagonals
The diagonals of a square, rectangle, and parallelogram and
rhombus bisect each other
Perpendicular Diagonals
The diagonals of a square, rhombus, and kite are perpendicular
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Grade 9: Geometry of 2D Shapes
Exercise 3
1. Classify each quadrilateral according to its sides.
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Grade 9: Geometry of 2D Shapes
2. Find the measure of each missing angle:
3. Solve for 𝑥:
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Grade 9: Geometry of 2D Shapes
4. In each Parallelogram, find the missing measure marked with “?”.
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Grade 9: Geometry of 2D Shapes
5. For each polygon
a) Classify the polygon according to its number of sides.
b) Then state if it is regular or not.
c) Then state if it is convex or concave.
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Grade 9: Geometry of 2D Shapes
Congruent Triangles
Two triangles are called congruent triangles if they have the same angles and side lengths.
For example, the two triangles below are congruent, and so we write ∆ABC ≅ ∆DEF. In this
section, we’ll look at the minimum conditions for two triangles to be congruent.
The next page details the 4 ways triangles can be shown congruent. They are: SSS, ASA, SAS,
and AAS.
NB There are two important conditions which are NOT sufficient to prove triangles
congruent. The first one is SSA (or ASS), and the other is AAA.
These triangles satisfy the AAA condition, but are clearly NOT congruent!
These triangles satisfy the SSA condition, but are NOT congruent!
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Grade 9: Geometry of 2D Shapes
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Grade 9: Geometry of 2D Shapes
Exercise 4
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Grade 9: Geometry of 2D Shapes
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Grade 9: Geometry of 2D Shapes
19. For each triangle, identify the included side between the two marked angles
19.1.
19.3.
19.2.
19.4.
20. For each triangle, identify the included angle between the two marked sides.
20.1.
20.3.
20.2.
20.4.
21. For each set of congruent triangles, carefully write a congruence statement (NB: Order
matters! ∆ABC ≅ ∆DEF means that segments AB and DE are congruent, BC and EF are
congruent, etc. We must make sure that the triangle order corresponds.)
21.1.
21.2.
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Grade 9: Geometry of 2D Shapes
21.3.
21.4.
22. Determine whether each of the following triangles are congruent. If they are, give the
congruence condition (SSS, SAS, ASA, AAS). If there isn’t a congruence condition, then
write “not congruent.”
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Grade 9: Geometry of 2D Shapes
23. Determine whether each of the following triangles are congruent. If they are, give the
congruence condition (SSS, SAS, ASA, AAS). If there isn’t a congruence condition, then
write “not congruent.”
24. Determine whether each of the following triangles are congruent. If they are, give the
congruence condition (SSS, SAS, ASA, AAS). If there isn’t a congruence condition, then
write “not congruent.”
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