Define congruent shapes.

Two shapes are congruent if they are identical in both shape and size . One may be a reflection, rotation or translation of the other, so they need not be facing the same way.

Why is an enlargement of a shape not congruent to it?

Because congruence needs the two shapes to be identical in size as well as in shape, and an enlargement changes the size. Reflections, rotations and translations all leave the size alone, which is why they do produce congruent images.

What must you show to prove that two shapes are congruent?

That corresponding sides are the same length and that corresponding angles are the same size. You do not have to show that the two shapes are facing the same way.

How many of a triangle's six measurements must match to prove congruence?

Just three , provided they are the right three. That means matching one of the five standard tests, rather than any three measurements you happen to know.

True or False? Two triangles with all three angles equal must be congruent.

False. Equal angles make the triangles the same shape but they may be different sizes , so AAA is not a congruence test. SSA, two sides and an angle that does not lie between them, is not enough either.

Define the term similar shapes .

Two shapes are similar if they have the same shape and their corresponding sides are in proportion . One shape is an enlargement of the other.

True or False? If two triangles of different sizes have the same angles they are not similar .

False. If two triangles of different sizes have the same angles they are similar . One is an enlargement of the other.

True or False? Shapes that are not triangles can have the same angles and not be similar .

True. Some shapes that are not triangles can have the same angles and not be similar . E.g. two rectangles of different sizes will have the same angles but their corresponding sides could have different scale factors.

True or False? To show that two non-triangular shapes are similar you need to show that their corresponding sides are in proportion .

True. To show that two non-triangular shapes are similar you need to show that their corresponding sides are in proportion .

What is a scale factor , in the context of similarity?

A scale factor is the ratio of corresponding lengths in similar shapes .

True or False? In the context of similarity, a scale factor cannot be negative .

True. Although you can have a negative scale factor in general enlargement, in the context of similarity , a scale factor cannot be negative .

What does a scale factor that is greater than 0 but less than 1 imply?

A scale factor that is greater than 0 but less than 1 implies that the similar shape is smaller than the original shape.

How can you find a length scale factor ?

A length scale factor can be found by dividing the length of a side on one shape by the length of a corresponding side on the similar shape.

What is a geometrical proof ?

A geometrical proof involves using known rules about geometry to prove a new statement about geometry.

How should each step in a geometrical proof be written?

Each step should be written in the form " [fact], [mathematical reason] ". It is really important to give a reason for each fact that you state. E.g. Angle ABE = Angle CDE = 60º, vertically opposite angles.

There are a number of different rules or facts that you can use in geometrical proof . Name three types of rules.

Any of the following types of rules may be used in geometrical proof : Properties of 2D shapes (especially isosceles triangles and quadrilaterals) Basic angle properties Angles in polygons Angles in parallel lines Congruence and similarity Circle theorems Pythagoras' theorem

What does it mean when a scale for a diagram is given as a ratio ? E.g. a scale of 1 : 10,000.

When the scale is given as a ratio (e.g., 1:10,000), it means that 1 unit on the diagram represents 10,000 of the same type of units in real life. E.g. 1 cm on map = 10, 000 cm (or 100 m) in real life

How is a scale length on a map converted to a real-life length? E.g. a length of 3 cm on a map with scale 1 : 20, 000.

To convert from a scale length to a real length, multiply by the scale factor . E.g. 3 × 20 000 = 60 000 cm (600 m).

How is a real-life length converted to a scale length on a map? E.g. a real-length of 15 km using a map with scale 1 : 50, 000.

To convert from a real length to a scale length, divide by the scale factor . E.g. 15 ÷ 50, 000 = 0.000 3 km (30 cm).

True or False? It may be helpful to convert the scale to more suitable units when dealing with very small or large scales.

True. It may be helpful to convert the scale to more suitable units when dealing with very small or large scales. For example, if 1 cm on a map represents 25,000 cm in real life, it may be easier to convert the units and use 1 cm represents 250 m or 0.25 km.

True or False? Map scales are always given with units .

False. Map scales are usually given in the form 1 : n with no reference to units .

What is meant by a " scale drawing "?

A scale drawing uses a given scale (e.g. 1 : 300) to produce an accurate drawing of a real life object. They are used when designing a vehicle or a building.

IGCSEMathsEdexcelMaths BFlashcardsGeometryCongruence, Similarity & Geometrical Proof

Congruence, Similarity & Geometrical Proof (Edexcel IGCSE Maths B): Flashcards

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Define congruent shapes.

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Cards in this collection (25)

Define congruent shapes.
Two shapes are congruent if they are identical in both shape and size.
One may be a reflection, rotation or translation of the other, so they need not be facing the same way.
Why is an enlargement of a shape not congruent to it?
Because congruence needs the two shapes to be identical in size as well as in shape, and an enlargement changes the size.
Reflections, rotations and translations all leave the size alone, which is why they do produce congruent images.
What must you show to prove that two shapes are congruent?
That corresponding sides are the same length and that corresponding angles are the same size.
You do not have to show that the two shapes are facing the same way.
How many of a triangle's six measurements must match to prove congruence?
Just three, provided they are the right three.
That means matching one of the five standard tests, rather than any three measurements you happen to know.
Fill in the two missing congruence tests.
The five tests for congruent triangles are SSS, SAS, AAS, $______$ with the side lying between the two angles, and $______$ for right-angled triangles.
The completed sentence is:
The five tests for congruent triangles are SSS, SAS, AAS, ASA with the side lying between the two angles, and RHS for right-angled triangles.
RHS stands for right-angle, hypotenuse and side, and it is the only test that applies to just one kind of triangle.
True or False?
Two triangles with all three angles equal must be congruent.
False.
Equal angles make the triangles the same shape but they may be different sizes, so AAA is not a congruence test.
SSA, two sides and an angle that does not lie between them, is not enough either.
Why are the AAS and ASA congruence tests essentially equivalent?
Knowing two angles of a triangle gives you the third, because all three add up to $180 °$ .
So any AAS situation can be rewritten as ASA as soon as the missing angle is worked out.
Two triangles each have angles of $25 °$ and $90 °$ , with an equal 6 cm side lying between them. Which test proves congruence?
ASA, because two angles are equal and the side between them is equal too.
Had the equal side been somewhere else in the triangle, this would have been AAS instead.
Define the term similar shapes.
Two shapes are similar if they have the same shape and their corresponding sides are in proportion.
One shape is an enlargement of the other.
True or False?
If two triangles of different sizes have the same angles they are not similar.
False.
If two triangles of different sizes have the same angles they are similar.
One is an enlargement of the other.
True or False?
Shapes that are not triangles can have the same angles and not be similar.
True.
Some shapes that are not triangles can have the same angles and not be similar.
E.g. two rectangles of different sizes will have the same angles but their corresponding sides could have different scale factors.
True or False?
To show that two non-triangular shapes are similar you need to show that their corresponding sides are in proportion.
True.
To show that two non-triangular shapes are similar you need to show that their corresponding sides are in proportion.
What is a scale factor, in the context of similarity?
A scale factor is the ratio of corresponding lengths in similar shapes.
True or False?
In the context of similarity, a scale factor cannot be negative.
True.
Although you can have a negative scale factor in general enlargement, in the context of similarity, a scale factor cannot be negative.
What does a scale factor that is greater than 0 but less than 1 imply?
A scale factor that is greater than 0 but less than 1 implies that the similar shape is smaller than the original shape.
How can you find a length scale factor?
A length scale factor can be found by dividing the length of a side on one shape by the length of a corresponding side on the similar shape.
What is a geometrical proof?
A geometrical proof involves using known rules about geometry to prove a new statement about geometry.
How should each step in a geometrical proof be written?
Each step should be written in the form "[fact], [mathematical reason]".
It is really important to give a reason for each fact that you state.
E.g. Angle ABE = Angle CDE = 60º, vertically opposite angles.
There are a number of different rules or facts that you can use in geometrical proof.
Name three types of rules.
Any of the following types of rules may be used in geometrical proof:
- Properties of 2D shapes (especially isosceles triangles and quadrilaterals)
- Basic angle properties
- Angles in polygons
- Angles in parallel lines
- Congruence and similarity
- Circle theorems
- Pythagoras' theorem
What does it mean when a scale for a diagram is given as a ratio?
E.g. a scale of 1 : 10,000.
When the scale is given as a ratio (e.g., 1:10,000), it means that 1 unit on the diagram represents 10,000 of the same type of units in real life.
E.g. 1 cm on map = 10, 000 cm (or 100 m) in real life
How is a scale length on a map converted to a real-life length?
E.g. a length of 3 cm on a map with scale 1 : 20, 000.
To convert from a scale length to a real length, multiply by the scale factor.
E.g. 3 × 20 000 = 60 000 cm (600 m).
How is a real-life length converted to a scale length on a map?
E.g. a real-length of 15 km using a map with scale 1 : 50, 000.
To convert from a real length to a scale length, divide by the scale factor.
E.g. 15 ÷ 50, 000 = 0.000 3 km (30 cm).
True or False?
It may be helpful to convert the scale to more suitable units when dealing with very small or large scales.
True.
It may be helpful to convert the scale to more suitable units when dealing with very small or large scales.
For example, if 1 cm on a map represents 25,000 cm in real life, it may be easier to convert the units and use 1 cm represents 250 m or 0.25 km.
True or False?
Map scales are always given with units.
False.
Map scales are usually given in the form 1 : n with no reference to units.
What is meant by a "scale drawing"?
A scale drawing uses a given scale (e.g. 1 : 300) to produce an accurate drawing of a real life object.
They are used when designing a vehicle or a building.

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