Geometry of 2D shapes (Grade 9) — Mathematics Grades 8–11

Properties of triangles and quadrilaterals, congruency, similarity, and unknown angles.

Gr 9 Term 2 — Geometry of 2D shapesTerm 2

Practise Geometry of 2D shapes in the app →

What gets asked

Calculate an unknown angle in a triangle or a quadrilateral.
Decide whether two triangles are congruent and name the condition.
Use similar triangles to calculate an unknown side.
Give the properties of a quadrilateral from its sides or diagonals.

You must be able to

Use a triangle's angle sum of 180 degrees or a quadrilateral's 360 degrees.
Use the rule that an exterior angle equals the sum of the two opposite interior angles.
Match corresponding sides and angles to name a congruency condition.
Set up a proportion between corresponding sides of similar triangles.

Traps that cost marks

Using 360 degrees as the angle sum of a triangle. A triangle's three angles always add up to 180 degrees, never 360.
Pairing up sides and angles that are not actually corresponding. Match vertices in the order the triangles are named, even if one is rotated or flipped.
Assuming similar triangles must also have equal sides. Similar triangles have equal angles, with sides in the same proportion, not equal sides.

Worked example

In triangle ABC, angle A = 50° and angle B = 65°. Side BC is extended to point D, so that angle ACD is an exterior angle. Calculate angle ACD.

The exterior angle of a triangle equals the sum of the two interior angles not next to it.
Those two angles here are angle A and angle B.
Angle ACD = 50° + 65° = 115°.

Triangles and quadrilaterals

The names of the shapes, what each name promises, and how the families fit inside one another.

isosceles: A triangle with at least two equal sides. The angles opposite those sides are equal too.
scalene: A triangle with no two sides equal, so no two angles are equal either.
quadrilateral: Any closed shape with four straight sides.
parallelogram: A quadrilateral with both pairs of opposite sides parallel.
diagonal: A straight line joining two corners that are not next to each other.
bisect: To cut exactly in half.

Triangles are named by their sides or by their angles. By sides: equilateral, isosceles or scalene. By angles: acute, right-angled or obtuse. A triangle has one name from each list.

Isosceles means AT LEAST two equal sides. An equilateral triangle has three, so it is isosceles as well. The names describe the shape; they do not compete for it.

A trapezium has at least one pair of parallel sides, so a parallelogram is one too. But CAPS and most exam papers say exactly one pair. If a question asks you to pick out the trapezium, they mean the shape with only one pair. A kite has two pairs of equal ADJACENT sides, next to each other, not opposite.

The diagonals are what tell the families apart. In every parallelogram the diagonals bisect each other. In a rectangle they are also equal. In a rhombus they also cross at right angles.

The families sit inside one another. A square is a rectangle, and it is also a rhombus, and all three are parallelograms. Naming a shape as fully as possible means giving the most detailed name that is true.

Examples

A B C D

has

A B = C D = 5

cm,

B C = D A = 11

cm and all four angles

9 0^{\circ}

. Name it as fully as possible.

Worked answer

Opposite sides are equal, so it is a parallelogram.
All four angles are right angles, so it is a rectangle.
The sides are not all equal, since $5$ and $11$ differ, so it is not a square.
The fullest true name is rectangle.

Answer: a rectangle

A rectangle IS a parallelogram. The more detailed name is the one that scores.

A quadrilateral has diagonals that bisect each other and cross at right angles, but are not equal. Name it.

Worked answer

Diagonals that bisect each other make it a parallelogram.
Crossing at right angles makes it a rhombus.
In a square the diagonals would also be equal, and here they are not.
So it is a rhombus and not a square.

Answer: a rhombus

Each extra property on the diagonals moves you one step further into the family.

A triangle has two sides of 7 cm and one of 7 cm. Is it isosceles?

Worked answer

All three sides are equal, so it is equilateral.
Isosceles asks for at least two equal sides.
It has three, which is more than two.
So yes, it is isosceles as well as equilateral.

Answer: yes, it is both

Reading isosceles as EXACTLY two would wrongly rule the equilateral triangle out.

Traps

Saying an equilateral triangle cannot be isosceles because isosceles means exactly two equal sides. Isosceles means at least two. An equilateral triangle has three, so it qualifies.
Thinking a trapezium must have two pairs of parallel sides. It needs at least one. Two pairs makes it a parallelogram, which is one special kind of trapezium.
Assuming the diagonals of every parallelogram are equal because they bisect each other. Bisecting and being equal are different properties. Only in a rectangle are they also equal.
Treating the families as exclusive, so a square cannot be called a rectangle. A square is a rectangle and a rhombus. Give the most detailed name, not the only one.

Learn and practise “Triangles and quadrilaterals” in the app →

Congruent and similar triangles

How few measurements you need before two triangles must be copies, and what equal angles alone can promise.

congruent: Exactly the same shape and the same size. One would fit right on top of the other.
similar: The same shape at a different size. Equal angles, and sides in the same ratio.
included angle: The angle lying between the two sides you have named.
corresponding sides: Sides in matching positions in the two triangles. Match the letters in order.
hypotenuse: The longest side of a right-angled triangle, opposite the right angle.

Congruent triangles are exact copies. Similar triangles are the same shape at a different size. Congruent is the stronger word: congruent triangles are similar too.

You do not need all six measurements to prove two triangles congruent. Three matching facts are enough. The four sets that work are SSS, SAS, AAS and RHS.

SSS is three equal sides. SAS is two sides and the included angle. AAS is two angles and a matching side. RHS is a right angle, the hypotenuse and one more side.

SAS needs the INCLUDED angle, between the two sides. AAA is not on the list. Three equal angles fix the shape but not the size.

For similar triangles, divide a side of the one by the matching side of the other. Do that for all three pairs of corresponding sides. Every pair must give the same number.

Rules to remember

congruent: SSS, SAS, AAS or RHS

Examples

Triangle

A B C

has

A B = 6

cm,

B C = 8

cm and angle

B = 4 0^{\circ}

. Triangle

P QR

has

P Q = 6

cm,

QR = 8

cm and angle

Q = 4 0^{\circ}

. Are they congruent?

Worked answer

Two sides of the one match two sides of the other.
Angle $B$ lies between $A B$ and $B C$ , so it is the included angle.
Angle $Q$ lies between $P Q$ and $QR$ in the same way.
Two sides and the included angle match, so this is SAS.

Answer: yes, congruent by SAS

The equal angle sits between the two equal sides, as SAS asks.

Two triangles each have angles of

5 0^{\circ}

6 0^{\circ}

and

7 0^{\circ}

. Must they be congruent?

Worked answer

All three angles match, so the triangles are similar.
Equal angles fix the shape but not the size.
One could have sides twice as long as the other.
So they need not be congruent. AAA is not a condition.

Answer: no, only similar

Angles carry no size, so AAA can never prove congruency.

Triangle

D E F

has sides

3

cm,

4

cm and

5

cm. Triangle

K L M

has sides

9

cm,

12

cm and

15

cm. Are they similar?

Worked answer

Pair the shortest side with the shortest, and so on.
$9 \div 3 = 3$ and $12 \div 4 = 3$ .
$15 \div 5 = 3$ as well.
All three pairs give $3$ , so the triangles are similar.

Answer: yes, similar

Every pair of corresponding sides had to give one same number.

Traps

Using AAA as a condition for congruency. Three equal angles only prove similarity. AAA leaves the size free.
Treating two sides and any angle at all as SAS. SAS needs the angle BETWEEN the two sides. Any other angle will not do.
Pairing up sides and angles that are not actually corresponding. Match vertices in the order the triangles are named, even if one is rotated or flipped.
Assuming similar triangles must also have equal sides. Similar triangles have equal angles, with sides in the same proportion, not equal sides.

Learn and practise “Congruent and similar triangles” in the app →

Unknown angles and sides

Using angle sums, congruency and similarity to work out what a diagram does not tell you.

interior angle: An angle inside the shape, at one of its corners.
exterior angle: The angle made outside the shape when one side is carried on past a corner.
scale factor: The number every side is multiplied by to get from one similar shape to the other.

The three interior angles of a triangle add up to $18 0^{\circ}$ . The four interior angles of a quadrilateral add up to $36 0^{\circ}$ . Mixing these two up is the most expensive mistake in this section.

An exterior angle of a triangle equals the sum of the two opposite interior angles. Only the two OPPOSITE ones, not all three.

Congruent triangles have equal matching sides and equal matching angles. Match the letters in order: if triangle ABC is congruent to triangle PQR, then A goes with P, B with Q and C with R.

Similar triangles have equal angles and sides in the same ratio. Find the scale factor by dividing a side of one by the matching side of the other, then multiply.

Sides in similar triangles are MULTIPLIED by the scale factor, never added to. If one triangle is twice as big, every side doubles.

Rules to remember

triangle: $a + b + c = 18 0^{\circ}$
quadrilateral: $a + b + c + d = 36 0^{\circ}$

Examples

Two angles of a triangle are

5 5^{\circ}

and

6 5^{\circ}

. Find the third.

Worked answer

The three angles add up to $180$ .
The two known angles give $55 + 65 = 120$ .
So the third is $180 - 120$ .
$180 - 120 = 60$ , so the third angle is $6 0^{\circ}$ .

Answer: $6 0^{\circ}$

Using $360$ here would give $24 0^{\circ}$ , bigger than the whole triangle.

Three angles of a quadrilateral are

8 0^{\circ}

9 5^{\circ}

and

11 0^{\circ}

. Find the fourth.

Worked answer

The four angles add up to $360$ .
The three known angles give $80 + 95 + 110 = 285$ .
So the fourth is $360 - 285$ .
$360 - 285 = 75$ , so the fourth angle is $7 5^{\circ}$ .

Answer: $7 5^{\circ}$

A quadrilateral is two triangles, so its angle sum is twice a triangle's.

Triangle

A B C

is similar to triangle

P QR

A B = 4

cm,

P Q = 12

cm and

B C = 5

cm. Find

QR

Worked answer

$A B$ matches $P Q$ , so the scale factor is $12 \div 4$ .
$12 \div 4 = 3$ , so every side of $P QR$ is $3$ times the matching side.
$B C$ matches $QR$ , so multiply: $5 \times 3$ .
$5 \times 3 = 15$ , so $QR$ is 15 cm.

Answer: 15 cm

Adding the difference of $8$ instead would give $13$ cm, and the shapes would not match.

Traps

Using $36 0^{\circ}$ as the angle sum of a triangle. A triangle is $18 0^{\circ}$ . Only a quadrilateral adds to $36 0^{\circ}$ .
Using $18 0^{\circ}$ as the angle sum of a quadrilateral. A quadrilateral splits into two triangles, so its angles add to $36 0^{\circ}$ .
Copying a value onto the wrong corner because the letters were not matched in order. Read the naming carefully. In triangle ABC congruent to triangle PQR, B matches Q.
Adding the difference between two matching sides to every other side. Similar shapes are scaled, not shifted. Divide to find the factor, then multiply.

Learn and practise “Unknown angles and sides” in the app →

Grade 9 Geometry of 2D shapes

What gets asked

You must be able to

Traps that cost marks

Worked example

Triangles and quadrilaterals

Examples

Traps

Congruent and similar triangles

Rules to remember

Examples

Traps

Unknown angles and sides

Rules to remember

Examples

Traps

About this material