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The lines ๐ฟ two and ๐ฟ three are parallel.
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The lines ๐ฟ four and ๐ฟ five are parallel.
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Segment ๐ด๐บ is congruent to segment ๐ธ๐ท.
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Prove that triangle ๐ด๐ต๐บ is congruent to triangle ๐ถ๐ท๐ธ.
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In order to prove that two triangles are congruent, we need to prove that they are exactly the same.
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There are four conditions for congruency.
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SSS stands for side, side, side.
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When we have two triangles with all three sides equal, those triangles must be congruent.
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SAS stands for side, angle, side.
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If we have two triangles where two of the sides are equal and the included angle thatโs the angle that falls between those two sides is equal, then those triangles must also be congruent.
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ASA, SAA, and AAS always have saying that the two triangles have two angles that are equal and one side that is equal.
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This differs to SAS, where it was important that the order mattered.
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The angle had to be the included angle.
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In this case, order is less important.
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Finally, RHS stands for right angle, hypotenuse, and side.
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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.
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Remember itโs not enough to show that three angles are the same since a triangle thatโs been enlarged will have the same angles, but different-length sides.
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Two triangles that have the same angles are called โsimilar triangles.โ
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Letโs make sure weโre clear which triangles weโre interested in.
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๐ด๐ต๐บ and ๐ถ๐ท๐ธ are the triangles highlighted.
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Iโve drawn them a little bit bigger so we can follow clearly whatโs happening.
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One of the pieces of information we have is that segment ๐ด๐บ and ๐ธ๐ท are congruent.
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This means theyโre exactly the same.
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So we can write ๐ด๐บ is equal to ๐ธ๐ท.
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Next, weโll use the fact that the lines ๐ฟ two and ๐ฟ three are parallel and the lines ๐ฟ four and ๐ฟ five are also parallel.
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Alternate angles are equal.
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Those are the angles that look like theyโre enclosed in the letter ๐.
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That means that angle ๐บ๐ด๐ต is equal to angle ๐ถ๐ท๐ธ.
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Remember itโs not enough just to say alternate angles or to use the letter ๐.
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We must say alternate angles are equal.
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Next, letโs look at angle ๐ด๐ต๐บ.
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๐ด๐ต๐บ is equal to angle ๐ถ๐ต๐น because vertically opposite angles are equal.
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On our diagram, thatโs the angles marked by the two arcs.
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We also know that corresponding angles are equal.
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Those are the angles that look like theyโre enclosed in the letter ๐น.
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What that means is angle ๐ถ๐ต๐น is equal to angle ๐ท๐ถ๐ธ.
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Since both angle ๐ด๐ต๐บ and ๐ท๐ถ๐ธ were equal to ๐ถ๐ต๐น, that must mean that angle ๐ด๐ต๐บ is equal to angle ๐ท๐ถ๐ธ.
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Weโve shown that both of the triangles share two angles and one side.
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By the condition AAS then, the triangles ๐ด๐ต๐บ and ๐ถ๐ท๐ธ are congruent.