WEBVTT
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The shape is made of four congruent trapeziums and two triangles.
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๐ด๐ถ and ๐ป๐ฝ are straight lines.
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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.
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Before we answer this question, letโs analyse the information weโve been given.
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The first word weโre interested in is the word โcongruent.โ
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Two congruent shapes are shapes that are completely identical.
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They must have the same-size sides and the same-sized angles.
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Weโre told that the congruent shapes in our diagram are trapeziums.
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These are four-sided shapes which have two parallel sides only.
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The other two sides are never parallel.
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But sometimes we do have something called a right trapezium.
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And this is a trapezium for which two of its angles are 90 degrees.
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For part a), weโre trying to establish whether the triangles are isosceles.
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A triangle is isosceles if it has two equal sides and two equal angles.
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These are often called the base angles.
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Letโs consider what would need to be true for triangles ๐ต๐ธ๐น and ๐ธ๐น๐ผ to be isosceles.
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One way that this could be true is if the side ๐ต๐ธ was equal to the side ๐ต๐น.
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And this is entirely feasible.
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Our trapeziums are congruent.
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And if we imagine theyโre not in the same orientation, that is, one of the trapeziums has been reflected in a line ๐ต๐ผ, we can see that the lines ๐ต๐ธ and ๐ต๐น are indeed of equal length.
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So this is one situation in which the triangle would be isosceles.
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But what about if they were in the same orientation?
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Well, this time, ๐ด๐ท and ๐ต๐น would be the same length and ๐ต๐ธ and ๐ถ๐บ would be the same length.
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This triangle certainly doesnโt look isosceles.
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It could be and this would be if the line ๐ธ๐น was equal to one of the other two sides, so say ๐ต๐ธ was equal to ๐ธ๐น.
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This would be isosceles.
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Thereโs no way for us to know if this is true.
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And one other way we could guarantee an isosceles triangle was if the trapeziums themselves were isosceles.
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That is to say, what we think of as almost the sloping sides were of equal length.
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Then ๐ด๐ท would be equal to ๐ต๐ธ, which is equal to ๐ต๐น, which is equal to ๐ถ๐บ.
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And we do indeed have an isosceles triangle.
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We have found a situation for where the triangles are isosceles and one for which they arenโt.
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So this could be true.
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The triangles could be isosceles.
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Now letโs consider part b.
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The triangles are congruent.
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The same definition for congruency stands.
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They must have the same sides and the same angles.
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We donโt need to prove it for all sides and all angles though.
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We can use these four conditions for congruency.
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SSS stands for side side side.
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If all the sides are the same, then the triangles must be congruent.
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SAS, thatโs two sides and an angle.
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ASA is two angles and a side.
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And RHS means both triangles must have a right angle and their hypotenuse on one other side must also be of equal length.
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Letโs look at a few scenarios.
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Weโll imagine that the trapeziums on the top row are a reflection of one another in a line between ๐ต and ๐ผ.
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Weโll then imagine that the trapeziums on the bottom row are a reflection of the ones on the top row in the line ๐ท๐บ.
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Itโs quite easy to see that we have two sides of equal length in both triangles.
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๐ต๐ธ is equal to ๐ธ๐ผ, and ๐ต๐น is equal to ๐น๐ผ.
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In fact, there is another side of equal length.
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๐ธ๐น is what we call a shared line.
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Itโs a line that joins both triangles.
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And by definition, it must be the same size.
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So by the condition for congruency SSS, we can show that ๐ต๐ธ๐น and ๐ธ๐น๐ผ are congruent.
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Now letโs imagine the trapeziums on the top row are oriented in the same direction.
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So ๐ด๐ท is equal to ๐ต๐น and ๐ต๐ธ is equal to ๐ถ๐บ.
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And letโs assume the triangles on the bottom row are also oriented in the same direction, but theyโre not a reflection in the line ๐ท๐บ.
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This time, we can see that the lines ๐ต๐ธ and ๐น๐ผ are of equal length.
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And we can also see that ๐ต๐น is equal to ๐ธ๐ผ.
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Once again, they share that third side.
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So by the condition SSS, the triangles are congruent.
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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.
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This must be true then.
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The triangles must be congruent.