# Apply Congruence Rules

In this worksheet, students will find congruent shapes using the following rules: SAS, ASA, SSS, RHS

Key stage:  KS 4

GCSE Subjects:   Maths

GCSE Boards:   AQA, OCR, Eduqas, Pearson Edexcel

Curriculum topic:   Geometry and Measures, Basic Geometry

Curriculum subtopic:   Properties and Constructions, Angles

Difficulty level:

### QUESTION 1 of 10

I have had a great day today.

Today I saw twin pandas.  That bears repeating.

Also I found out I had a twin sister.  I am beside myself.

Oh come on, how else am I going to introduce the topic of congruency.

In maths we come across congruent triangles mainly, but sometimes other shapes.

A congruent shape is exactly the same shape and same size but it may have been rotated.

When this happens it is useful to know the rules to identify if a shape is congruent.

As you can see the shapes are all the same size, but you need to be able to identify the corresponding sides and angles, which when the shapes have been rotated can be a little tricky.

Lets have a go

What is the rule that shows that these triangles are congruent?

SAS

SSS

RHS

ASA

not congruent

What is the rule showing that these triangles are congruent?

What is the condition that shows these triangles are congruent.

 SSS SAS AAA RHS not congruent The rule

The rule to satisfy this pair of triangles is SSS, ASA, RHA, SAS, not congruent.

The rule to satisfy this pair of triangles is SSS, ASA, RHA, SAS, not congruent.

Are these triangles congruent?

If so state the rule, if not state that they are not congruent.

Not congruent

RHS

SAS

SSS

ASA

Triangle ABC is congruent to triangle PQR.

< means angle

< A = 60&deg;  < B = 80&deg; and AB = 5 cm.

What is  < P.

60°

80°

40°

90°

Triangle ABC is congruent to triangle PQR.

< means angle

< A = 60°  < B = 80° and AB = 5 cm.

What is  < R

60°

80°

40°

90°

Which of these is not a rule for finding congruency in a triangle?

RHS

SSS

AAA

SAS

ASA

ABCD is congruent to PQRS.

< A = 110&deg; < B = 55&deg; < C = 85&deg; and RS = 4 cm.

Match the following

## Column B

< P
4 cm
< Q
85°
< R
110°
< S
55°
CD
110°
• Question 1

What is the rule that shows that these triangles are congruent?

ASA
EDDIE SAYS
This could prove misleading as you only have one side given and only one angle is the same in both triangles. Here you just need to find the missing angle in one of the triangles. Look at triangle 1, 180 - 21 - 79 = 80° Which would correspond to the 80° in triangle 2.
• Question 2

What is the rule showing that these triangles are congruent?

SSS
EDDIE SAYS
You don't always need angles or side lengths given. As long as you know what the marking mean you are away. Make sure you can visualise them flipped so that one sits on top of the other correctly and that the sides are corresponding. I know it can make your eyes go funny at times.
• Question 3

What is the condition that shows these triangles are congruent.

 SSS SAS AAA RHS not congruent The rule
EDDIE SAYS
Both triangles have a right angle, an equal hypotenuse and another equal side.
• Question 4

The rule to satisfy this pair of triangles is SSS, ASA, RHA, SAS, not congruent.
EDDIE SAYS
Isn't it great to have a question where you just have to look and not work anything out. It is as long as you know what you are looking for I suppose. Here you can see there are two pairs of corresponding sides and one corresponding angle.
• Question 5

EDDIE SAYS
Are you getting used to looking at the triangles now to decide the rule of congruency? Or are your eyes still going wonky trying to decide. Just think pairs. You are looking for two corresponding angles to be the same or two corresponding sides to be the same.
• Question 6

Are these triangles congruent?

If so state the rule, if not state that they are not congruent.

ASA
EDDIE SAYS
Oh this looks a bit different. There is one corresponding side and one corresponding angle. You are supposed to look for pairs. Did you spot the second angle? Where the lines cross in the middle, vertically opposite angles are formed. From other areas of maths you may have learnt that vertically opposite angles are equal. A bit sneaky I know.
• Question 7

Triangle ABC is congruent to triangle PQR.

< means angle

< A = 60&deg;  < B = 80&deg; and AB = 5 cm.

What is  < P.

60°
EDDIE SAYS
If in doubt sketch it out. This can help visualise the problem. If you label PQR in the same way you have labeled ABC then P would correspond with A. Therefore 60°
• Question 8

Triangle ABC is congruent to triangle PQR.

< means angle

< A = 60°  < B = 80° and AB = 5 cm.

What is  < R

40°
EDDIE SAYS
No this isn't the same question You just need the added extra on one at least. If in doubt sketch it out. This can help visualise the problem. If you label PQR in the same way you have labeled ABC then Angle A and Angle P would be 60°, B and Q would be 80°. Angle R isn't mentioned in the question so use your knowledge of angles in a triangle add up to 180° 180 - 60 - 80 = 40°
• Question 9

Which of these is not a rule for finding congruency in a triangle?

AAA
EDDIE SAYS
This makes sense doesn\'t it. The only thing this proves is that all the angles add up to 180° The triangles could be different sizes. (the angle won\'t change) and when they are congruent they are exactly the same size.
• Question 10

ABCD is congruent to PQRS.

< A = 110&deg; < B = 55&deg; < C = 85&deg; and RS = 4 cm.

Match the following

## Column B

< P
110°
< Q
55°
< R
85°
< S
110°
CD
4 cm
EDDIE SAYS
You must have been getting fed up of triangles. Did you sketch out the quadrilateral and label each point corresponding? This should help with angles PQ and R. Angle S is just the angle missing from a quadrilateral which adds up to 360° 360 - 110 - 85 - 55 = 110° CD is corresponding to RS therefore 4 cm.
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