Video: Pack 1 โ€ข Paper 3 โ€ข Question 14

Pack 1 โ€ข Paper 3 โ€ข Question 14

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Video Transcript

๐ด๐ต๐ถ๐ท is a parallelogram. ๐ธ is the point where the diagonals ๐ด๐ถ and ๐ต๐ท meet. Prove that triangles ๐ด๐ต๐ธ and ๐ถ๐ท๐ธ are congruent.

Two shapes are congruent if they are exactly the same size and shape. This means that corresponding pairs of sides must always be equal in length and corresponding angles must always be the same size. There are number of different ways that we can prove that two triangles are congruent.

The first way is we can prove that all three pairs of corresponding sides are equal in length. This is referred to as SSS or Side Side Side congruency.

The second way is to prove that the two triangles have two sides and an included angle in common. An included angle is the angle between the two sides that weโ€™re working with. So it must be Side Angle Side in this order.

The third way is to prove that the two triangles have two angles and one side in common. And it can be any two angles and any side. So this is referred to as ASA for Angle Side Angle or it can be AAS or even SAA.

There is a fourth method for proving that two triangles are congruent. But this is specifically for right-angled triangles. And we donโ€™t know that the two triangles weโ€™re working with here are right angled.

Now, Iโ€™ve shaded the two triangles that weโ€™re interested in here: triangle ๐ด๐ต๐ธ in orange and triangle ๐ถ๐ท๐ธ in pink. We need to have a closer look at their sides and angles. Remember ๐ด๐ต๐ถ๐ท is a parallelogram. And a key fact about parallelograms is that opposite sides are equal in length. This means that side ๐ด๐ต is equal to side ๐ถ๐ท. And so we have our first statement about the congruency of these two triangles. And itโ€™s a statement about the sides.

Next, letโ€™s consider the angles of this parallelogram. And weโ€™ll start with the angle at the center at the point ๐ธ. Angles ๐ด๐ธ๐ต and ๐ถ๐ธ๐ท are vertically opposite one another. And a key fact about vertically opposite angles is that they are equal. This gives us our second statement about the congruency of these two triangles: angle ๐ด๐ธ๐ต is equal to angle ๐ถ๐ธ๐ท.

Finally, letโ€™s consider another angle in these two triangles. And to do so, we need to recall that ๐ด๐ต๐ถ๐ท is a parallelogram, which means that the lines ๐ด๐ต and ๐ถ๐ท are parallel. If you look at angles ๐ด๐ต๐ธ and ๐ถ๐ท๐ธ, we can see that theyโ€™re alternate angles in parallel lines. And a key fact about alternate angles is that theyโ€™re equal. This gives us our third statement about the congruency of the two triangles: angle ๐ด๐ต๐ธ is equal to angle ๐ถ๐ท๐ธ.

Now, if we look at the three statements that weโ€™ve made, we made one about a side and two about angles, which means weโ€™re using the third condition for triangle congruency โ€” Side Angle Angle. We can conclude then that triangles ๐ด๐ต๐ธ and ๐ถ๐ท๐ธ are congruent using the Side Angle Angle rule.

The statements we made about the equality of sides and angles are also an essential part of our answer to this question.

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