Congruence and triangles (Part 1)

Key Stage 3

Maths

Constructions, congruence, and loci

﻿Hello and welcome to this lesson on congruence and triangles part one with me, Miss Oreyomi. For today's lesson, you'll be needing a paper and a pen or something that you can write on and with. If you need to put your phone on silent to avoid distraction, then please do so. Also, if you can get into a space with less noise and distractions, again, that will be great for you learning. Please do so. If at any point during the lesson, you wish to pause the video just so you can take a longer period just to understand what has just been said, then, again, please do so. When I tell you to pause and attempt a question or a task, it'd be great if you do pause as that would aid your understanding further. So now, pause the video should you need to go get your equipment or if you need to put yourself in a space with less noise and distraction and press resume when you're ready to carry on with our lesson. Okay, let's look at our try this task. Are these triangles the same or are they different? Are they congruent or not? Any triangle with one vertex, vertex meaning the point at A, so take, for example, over here, and then one vertex on each circle will be congruent. Do you agree? Can you find any examples or counterexamples? Okay, I believe this and this are congruent and these two triangles are congruent. Let's do another example. If I'm going from my vertex A over here and then I join my circle. I join one end of my-- I join one end of my vertex to there and then I join the other end to this end of the circle. It's looking congruent, assuming this is a straight line. It is looking congruent and if I do the same thing for here, it is also looking congruent. Let's try another one. If I do something like this vertex to here and then I take my other vertex over there and I connect it up. What if I do the same and I take this one here and then this all the way to the end and then I connect it up like so. Again, it is looking the same. So I have managed to come up with examples. Did you manage to come up with counterexamples? How does our try this task link with today's lesson? Well, we are learning about congruent triangles. Two triangles are congruent if they have exactly three sides that are the same and exactly three angles that are the same. However, we don't need to know all six information. We don't need to know all six, all three sides, all three angles. As long we know three of this information, we can determine if two triangles are congruent or not. The first rule, so to speak, to determine if triangles are congruent is if the three sides are the same. So take, for example, these two triangles. My side here, seven, is the same as my side here. So notice that although my triangle has been flipped, the sides are still the same. Four is still the same and five is still the same. Therefore, the first rule, if three sides of one triangle are equal to three sides of another triangle, then the triangles are congruent. Take a moment to process this. If three sides of say triangle A is the same as three sides of triangle B, then the triangles are congruent, that's the first one. Secondly, triangles are congruent if the side, the included angle and another side are equal to the side, the included angle and another side of a second triangle. A picture usually works better. So take here, for example, I've got my sides, side of seven. Then I've got my included angle, included meaning it is in between the two sides that I'm looking at. So, I've got seven here and thirty degrees is included and then I've got four. So, seven, 30 degrees and four centimetre and if I come over here, I've got seven, 30 degrees and four centimetre. So I could say these two triangles are congruent. So, it's side-angle-side. Usually, you may see something like this. SAS. Which means side-angle-side and for the other one, you can see SSS. Which means side-side-side. So the second rule then is if two sides and the included angle, remember we said 30's included between the two sides, are equal to the corresponding sides and angle of the other triangle, then the triangles are congruent. Again, take a moment to process this. Another rule, another way for which we can determine if triangles are congruent, is if we've got an angle, a side and another angle. So, take this one for example, I've got an angle, I've got a side and then I've got another angle as well. So, if two angles and the included side, so 7.9 is between 27 degrees and 57 degrees, if they are equal to the corresponding angle, then you can say these two triangles are congruent. So triangle one and triangle two are congruent because we're starting with 27 over here, then we have 7.9 and then we have 57 degrees. So, again, the triangles may have been rotated or flipped but as long as the angles, the side and another angle are the same, then we can say the two triangles are congruent. How about you have a go then? Can you draw a corresponding triangle congruent to this triangle on your screen? Draw a corresponding triangle congruent to this triangle on your screen. Pause your video now and attempt this task. Okay, hope-- Yours could have been any orientation as long as it has the same size and angles, so normally I would use a ruler for this but assuming this is my right angled triangle, I've got four centimetre here, this would be 90 and then this would be 57. What rule is this-- What rule makes these two triangles congruent? It is A-S-A, isn't it? Angle, side, angle. Angle, side, angle, included side, angle, included side, angle. Another question: Can you draw a corresponding triangle congruent to the triangle on your screen? So pause the video again and draw a congruent, a congruent triangle-- A triangle, a corresponding triangle that is congruent to the triangle on your screen. Okay, hopefully you had a go at that. So for mine, I-- You could have had something like this. So over here would be 8 centimetre, over here would be 5.6 and over here would be 3.2. Okay? So, side-side-side is what make these two triangles congruent. It is-- Okay, let's look at question two. We have six triangles that are not drawn accurately. Which two triangles are congruent to shape A? Shape A we have 4.6 centimetre, we 64 degrees and 71 degrees. Very quickly, just so we can help ourselves, what would be the missing angle here? It would 180 take away 71 + 64 and that is 45 degrees. So, here we have 45 degrees. Now, let's see if that helps us. So, I've got 4.6 here, 45 degree here and 64 degrees here. Is this the same? Well, no because my 4.6 is meant to be between my 64 and my 45, so B is not congruent to A. C then. I've got 64 degrees, 71 degrees. Here would be 45 degrees and then I've got 4.6. This is looking the same 'cause here I've got 4.6, 64 and 71 so C is congruent to A. Let's look at D. D, I've got 4.6, 45, 64 and 71 degrees. That looks right. 4.6 here, 64 over there, 71 there and 45 degrees here. So, C and D are congruent to A. For each pair below, state the condition that makes them congruent. So, I've got-- For the first one, it would be side, angle and side. Side, angle, side. And for the second one, it will be side, side, side. Okay, I have done a prototype here in advance. Mary and Beth each draw a triangle with one side of nine centimetre, one angle of 35 degree and one angle of 65 degree. Mary says their triangles are congruent. Explain why Mary is incorrect. Well, Mary is incorrect because we don't have enough information 'cause I could have drawn my triangle like this and put up my nine centimetre here and then have my 65 degrees and my 35 degrees. And Beth could have drawn her triangle as 65 degrees, nine centimetre and 35 degrees. So, Mary-- Beth could have gotten angle-side-angle and then Mary could have gotten angle-angle-side so it's not quite the same. So it's not congruent. It is now time for your explore task. Consider each set of conditions in the table below. Which of them would produce two congruent triangles? So, for example, looking at this one, I've got one side in common so if I have a triangle with one side in common, would I produce two congruent triangles? Do the same for each one and see which of them would produce two congruent triangles. So, pause the video now and attempt this task and then once you're done, resume to carry on with the lesson. We've now reached the end of today's lesson. A very big well done for staying all the way through and completing your task. Before you go today, make sure you complete the quiz as that shows you what you've learned today and also helps you to further understand today's lesson. So, until next time, goodbye.

Congruence and triangles (Part 1)

Congruence and triangles (Part 1)

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Licence

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UnitMaths / Constructions, congruence, and loci

Maths