Video: AQA GCSE Mathematics Higher Tier Pack 4 โ€ข Paper 3 โ€ข Question 8

The shape is made of four congruent trapeziums and two triangles. ๐ด๐ถ and ๐ป๐ฝ are straight lines. For each statement, tick the correct box. a) The triangles are isosceles. [A] must be true [B] could be true [C] must be false b) The triangles are congruent. [A] must be true [B] could be true [C] must be false

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Video Transcript

The shape is made of four congruent trapeziums and two triangles. ๐ด๐ถ and ๐ป๐ฝ are straight lines. For each statement, tick the correct box. a) The triangles are isosceles: must be true, could be true, must be false. b) The triangles are congruent: must be true, could be true, must be false.

Before we answer this question, letโ€™s analyse the information weโ€™ve been given. The first word weโ€™re interested in is the word โ€œcongruent.โ€ Two congruent shapes are shapes that are completely identical. They must have the same-size sides and the same-sized angles.

Weโ€™re told that the congruent shapes in our diagram are trapeziums. These are four-sided shapes which have two parallel sides only. The other two sides are never parallel. But sometimes we do have something called a right trapezium. And this is a trapezium for which two of its angles are 90 degrees.

For part a), weโ€™re trying to establish whether the triangles are isosceles. A triangle is isosceles if it has two equal sides and two equal angles. These are often called the base angles. Letโ€™s consider what would need to be true for triangles ๐ต๐ธ๐น and ๐ธ๐น๐ผ to be isosceles.

One way that this could be true is if the side ๐ต๐ธ was equal to the side ๐ต๐น. And this is entirely feasible. Our trapeziums are congruent. And if we imagine theyโ€™re not in the same orientation, that is, one of the trapeziums has been reflected in a line ๐ต๐ผ, we can see that the lines ๐ต๐ธ and ๐ต๐น are indeed of equal length. So this is one situation in which the triangle would be isosceles.

But what about if they were in the same orientation? Well, this time, ๐ด๐ท and ๐ต๐น would be the same length and ๐ต๐ธ and ๐ถ๐บ would be the same length. This triangle certainly doesnโ€™t look isosceles. It could be and this would be if the line ๐ธ๐น was equal to one of the other two sides, so say ๐ต๐ธ was equal to ๐ธ๐น. This would be isosceles. Thereโ€™s no way for us to know if this is true.

And one other way we could guarantee an isosceles triangle was if the trapeziums themselves were isosceles. That is to say, what we think of as almost the sloping sides were of equal length. Then ๐ด๐ท would be equal to ๐ต๐ธ, which is equal to ๐ต๐น, which is equal to ๐ถ๐บ. And we do indeed have an isosceles triangle. We have found a situation for where the triangles are isosceles and one for which they arenโ€™t. So this could be true. The triangles could be isosceles.

Now letโ€™s consider part b. The triangles are congruent. The same definition for congruency stands. They must have the same sides and the same angles. We donโ€™t need to prove it for all sides and all angles though. We can use these four conditions for congruency.

SSS stands for side side side. If all the sides are the same, then the triangles must be congruent. SAS, thatโ€™s two sides and an angle. ASA is two angles and a side. And RHS means both triangles must have a right angle and their hypotenuse on one other side must also be of equal length.

Letโ€™s look at a few scenarios. Weโ€™ll imagine that the trapeziums on the top row are a reflection of one another in a line between ๐ต and ๐ผ. Weโ€™ll then imagine that the trapeziums on the bottom row are a reflection of the ones on the top row in the line ๐ท๐บ. Itโ€™s quite easy to see that we have two sides of equal length in both triangles. ๐ต๐ธ is equal to ๐ธ๐ผ, and ๐ต๐น is equal to ๐น๐ผ.

In fact, there is another side of equal length. ๐ธ๐น is what we call a shared line. Itโ€™s a line that joins both triangles. And by definition, it must be the same size. So by the condition for congruency SSS, we can show that ๐ต๐ธ๐น and ๐ธ๐น๐ผ are congruent.

Now letโ€™s imagine the trapeziums on the top row are oriented in the same direction. So ๐ด๐ท is equal to ๐ต๐น and ๐ต๐ธ is equal to ๐ถ๐บ. And letโ€™s assume the triangles on the bottom row are also oriented in the same direction, but theyโ€™re not a reflection in the line ๐ท๐บ. This time, we can see that the lines ๐ต๐ธ and ๐น๐ผ are of equal length. And we can also see that ๐ต๐น is equal to ๐ธ๐ผ. Once again, they share that third side.

So by the condition SSS, the triangles are congruent. And if we change the trapeziums on the bottom row so that they are reflections of the ones on the top, once again, we still have three sides that are the same. This must be true then. The triangles must be congruent.

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