Video: Proving Results Using Triangle Congruence

In the given quadrilateral, ๐ด๐น and ๐ต๐น have the same length and ๐ธ๐น and ๐ถ๐น have the same length. Which angle has the same measure as โˆ ๐ด๐น๐ธ? Hence, are triangles ๐ด๐น๐ธ and ๐ต๐น๐ถ congruent? If yes, state which congruence criterion proves this.

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Video Transcript

In the given quadrilateral, ๐ด๐น and ๐ต๐น have the same length and ๐ธ๐น and ๐ถ๐น have the same length. Which angle has the same measure as angle ๐ด๐น๐ธ? Hence, are triangles ๐ด๐น๐ธ and ๐ต๐น๐ถ congruent? If yes, state which congruence criterion proves this.

In this diagram, we can see that thereโ€™s a quadrilateral which has some triangles within it. Weโ€™re told that there are some line segments which have the same length. So itโ€™s always worthwhile putting this onto a diagram if theyโ€™re not already marked. ๐ด๐น and ๐ต๐น are the same length and ๐ธ๐น and ๐ถ๐น are the same length. In the second question, weโ€™ll look at congruency. But the first question asks us about angle measures. Which angle would be the same as angle ๐ด๐น๐ธ? Weโ€™re not given any angle measurements in this diagram, but we should recall that angles which are vertically opposite will be equal. So angle ๐ต๐น๐ถ would also be the same measurement. And thatโ€™s our answer for the first part of the question.

In the second part of the question, we need to check if triangles ๐ด๐น๐ธ and ๐ต๐น๐ถ are congruent. So letโ€™s note down any sides or angles that we know are congruent. We were told in the question that ๐ด๐น and ๐ต๐น are the same length. We have shown in the first part of the question that we have two congruent angles, angle ๐ด๐น๐ธ and angle ๐ต๐น๐ถ. And we were told that sides ๐ธ๐น and ๐ถ๐น are the same length.

And so, we have two pairs of congruent sides equal and a pair of congruent angles. Importantly, the angle is included between the two sides, which means that we can use the SAS congruency criterion. If the angle wasnโ€™t included between the two sides, then it wouldnโ€™t be sufficient to show congruence. So our answer for this part of the question is, yes, these two triangles are congruent, and we use the SAS criterion.

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