Congruency and Similarity

thisQUEENslayz

I don’t even know

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Reply 1

Notnek

Original post by thisQUEENslayz

I don’t even know

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Part a)? Do you know anything about congruent triangles e.g. the ASA, SAS rules etc? Please post what you've tried so far.

Reply 2

thisQUEENslayz

Original post by Sir Cumference

Part a)? Do you know anything about congruent triangles e.g. the ASA, SAS rules etc? Please post what you've tried so far.

This is a topic that I struggle with so I don’t know where to even start. I know the rules but I don’t know how to apply them to exam questions

Reply 3

Notnek

Original post by thisQUEENslayz

This is a topic that I struggle with so I don’t know where to even start. I know the rules but I don’t know how to apply them to exam questions

To start a congruent triangles question you need to find any side/angle that is the same in both triangles. Can you see any?

Reply 4

thisQUEENslayz

Original post by Sir Cumference

To start a congruent triangles question you need to find any side/angle that is the same in both triangles. Can you see any?

does (PQ and SR) and (PS and QR) count as similar sides

Reply 5

Notnek

Original post by thisQUEENslayz

does (PQ and SR) and (PS and QR) count as similar sides

Yes that's correct and you always have to give a reason even if it seems obvious. The reason here is "opposite sides in a parallelogram are equal".

Can you see any other pairs of sides/angles that are the same? You'll need at least 1 more.

Reply 6

thisQUEENslayz

Original post by Sir Cumference

The angles P and R ?

Reply 7

Notnek

Original post by thisQUEENslayz

The angles P and R ?

Yes that's right. Any thoughts on the reason why they must be equal?

Reply 8

thisQUEENslayz

Original post by Sir Cumference

Yes that's right. Any thoughts on the reason why they must be equal?

Thanks and I’m not sure on how to explain why

Reply 9

Notnek

Original post by thisQUEENslayz

Thanks and I’m not sure on how to explain why

Again it's to do with the fact that its a parallelogram. Opposite angles in a parallelogram are equal. Try to remember these facts about parallelograms because they're common in congruence questions.

OK so your congruency proof so far looks like this:

PQ = SR (opposite sides in a parallelogram are equal)
PS = QR (opposite sides in a parallelogram are equal)
Angle P angle R (opposite angles in a parallelogram are equal)

So two pairs of sides are the same in the two triangles and so is the angle between these two sides. Any ideas how to finish the proof?

Reply 10

thisQUEENslayz

Original post by Sir Cumference

Sorry these are late I’m just trying to think- But im not sure. I understand everything so far

Reply 11

Notnek

Original post by thisQUEENslayz

Sorry these are late I’m just trying to think- But im not sure. I understand everything so far

You have shown that one side is the same in both triangles (S)
You have shown that a second side is the same in both triangles (S)
You have shown that an angle is the same in both triangles and it is between the two sides (A)

So you have the SAS rule. To finish the proof you just need to say something like

"Therefore the triangles must be congruent by the SAS rule"

Remember that congruence means that all of the sides and angles are the same. So once you've proved that two triangles are congruent (e.g. by the SAS rule) then you now know that all of the sides/angles are equal and this information could be useful in follow up questions.

Reply 12

thisQUEENslayz

Original post by Sir Cumference

Okay thank you. That helps me with my understanding of the rules. What happens if it is not a parallelogram? There are a few questions with shapes such as squares and circles

Reply 13

Notnek

Original post by thisQUEENslayz

Okay thank you. That helps me with my understanding of the rules. What happens if it is not a parallelogram? There are a few questions with shapes such as squares and circles

Ok sticking with this question, there are other ways to prove that the two triangles are congruent. E.g. angle PQS = angle QSR. Do you know why they're equal?

You need to try lots of questions to get used to the different ways that two sides/angles can be equal. Normally its something to do with parallel lines. I recommend trying another congruence question and then post it here if you get stuck.

Reply 14

thisQUEENslayz

Original post by Sir Cumference

Are they equal due to them being alternate angles and I’ll post another question

Original post by thisQUEENslayz

Are they equal due to them being alternate angles and I’ll post another question

Yes and there is another pair like that. You could have actually done this question using SAS, ASA or SSS depending on which sides/angles you shoe are equal.

Reply 17

Notnek

Original post by thisQUEENslayz

To prove two triangles are similar you have to show AAA i.e. you have to show that there are 3 pairs of equal angles in both triangles. Can you see any angles that are the same? There should be one obvious pair without doing any thinking.

Reply 18

thisQUEENslayz

Original post by Sir Cumference

Is it the right angle in ADB with ADC and since it’s an equilateral triangle - the three other angles would be the same (ABC, ACB and BAC)

Reply 19

Notnek

Original post by thisQUEENslayz

Is it the right angle in ADB with ADC and since it’s an equilateral triangle - the three other angles would be the same (ABC, ACB and BAC)

Yes ADB = ADC is the first fact. I think the easiest way to do this question is to work out the actual angles. Since it's an equilateral triangle, what are the sizes of the three angles in the main triangle? And so what's the size of angle XBD?