if in triangle JKL, JK= 9, KL = 11, and JL=12, and in triangle PQR, PQ=9, QR=11, and PR=11, explain why triangle JKL and triangle PQR are not congruent
Congruent triangles have exactly the same size and shape.
JKL has a perimeter of 32, but the perimeter of PQR is only 31. Therefore
the triangles are not the same size.
PQR has 2 equal sides. Therefore, it is
an isosceles triangle. JKL is not isosceles. So the triangles do not have
the same shape.
Triangle JKL and triangle PQR are not congruent because their corresponding sides are not equal. In order for two triangles to be congruent, all three pairs of corresponding sides must be equal in length.
In triangle JKL, JK is 9 units long, while in triangle PQR, PQ is also 9 units long, so the corresponding sides are equal in length.
However, in triangle JKL, KL is 11 units long, while in triangle PQR, QR is 11 units long. These sides are not equal in length, so the triangles cannot be congruent based on the Side-Side-Side (SSS) criterion for congruence.
Lastly, in triangle JKL, JL is 12 units long, whereas in triangle PQR, PR is 11 units long. Again, these sides are not equal in length, further confirming that the triangles are not congruent.
To determine whether two triangles are congruent, we need to compare the lengths of their sides and the measures of their angles. In this case, we are comparing triangle JKL and triangle PQR.
Looking at the given lengths of the sides:
In triangle JKL, JK = 9, KL = 11, and JL = 12.
In triangle PQR, PQ = 9, QR = 11, and PR = 11.
While two sides of triangle JKL are equal in length to two sides of triangle PQR, that's not sufficient proof of congruence. To establish congruence, all three sides of one triangle must be equal to the corresponding sides of the other triangle.
Since in triangle JKL, JL = 12, and in triangle PQR, PR = 11, the sides JL and PR are not equal. Therefore, we can conclude that triangle JKL and triangle PQR are not congruent.
It's important to note that even if some sides of two triangles are equal, they may not necessarily be congruent. In order to determine congruence, both side lengths and angle measures need to be considered.