Video: Pack 3 โ€ข Paper 1 โ€ข Question 13

Pack 3 โ€ข Paper 1 โ€ข Question 13

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Video Transcript

The lines ๐ฟ two and ๐ฟ three are parallel. The lines ๐ฟ four and ๐ฟ five are parallel. Segment ๐ด๐บ is congruent to segment ๐ธ๐ท. Prove that triangle ๐ด๐ต๐บ is congruent to triangle ๐ถ๐ท๐ธ.

In order to prove that two triangles are congruent, we need to prove that they are exactly the same. There are four conditions for congruency. SSS stands for side, side, side. When we have two triangles with all three sides equal, those triangles must be congruent. SAS stands for side, angle, side. If we have two triangles where two of the sides are equal and the included angle thatโ€™s the angle that falls between those two sides is equal, then those triangles must also be congruent.

ASA, SAA, and AAS always have saying that the two triangles have two angles that are equal and one side that is equal. This differs to SAS, where it was important that the order mattered. The angle had to be the included angle. In this case, order is less important.

Finally, RHS stands for right angle, hypotenuse, and side. If we have two right-angled triangles with the same-length hypotenuse and the same length for one of the other sides, then those two right-angled triangles are congruent. Remember itโ€™s not enough to show that three angles are the same since a triangle thatโ€™s been enlarged will have the same angles, but different-length sides. Two triangles that have the same angles are called โ€œsimilar triangles.โ€

Letโ€™s make sure weโ€™re clear which triangles weโ€™re interested in. ๐ด๐ต๐บ and ๐ถ๐ท๐ธ are the triangles highlighted. Iโ€™ve drawn them a little bit bigger so we can follow clearly whatโ€™s happening. One of the pieces of information we have is that segment ๐ด๐บ and ๐ธ๐ท are congruent. This means theyโ€™re exactly the same. So we can write ๐ด๐บ is equal to ๐ธ๐ท.

Next, weโ€™ll use the fact that the lines ๐ฟ two and ๐ฟ three are parallel and the lines ๐ฟ four and ๐ฟ five are also parallel. Alternate angles are equal. Those are the angles that look like theyโ€™re enclosed in the letter ๐‘. That means that angle ๐บ๐ด๐ต is equal to angle ๐ถ๐ท๐ธ. Remember itโ€™s not enough just to say alternate angles or to use the letter ๐‘. We must say alternate angles are equal.

Next, letโ€™s look at angle ๐ด๐ต๐บ. ๐ด๐ต๐บ is equal to angle ๐ถ๐ต๐น because vertically opposite angles are equal. On our diagram, thatโ€™s the angles marked by the two arcs. We also know that corresponding angles are equal. Those are the angles that look like theyโ€™re enclosed in the letter ๐น. What that means is angle ๐ถ๐ต๐น is equal to angle ๐ท๐ถ๐ธ.

Since both angle ๐ด๐ต๐บ and ๐ท๐ถ๐ธ were equal to ๐ถ๐ต๐น, that must mean that angle ๐ด๐ต๐บ is equal to angle ๐ท๐ถ๐ธ. Weโ€™ve shown that both of the triangles share two angles and one side.

By the condition AAS then, the triangles ๐ด๐ต๐บ and ๐ถ๐ท๐ธ are congruent.

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