triangle PQR is isosceles.S and T are the mid points of the equal sides PQ and PR respectively.prove that triangle QTR=triangle RSQ.
SR = TQ midpoints of equal lengths
< SRQ = < TQR base angles of isoceles triangle
RQ = QR same line segment
so
side, angle, side
To prove that triangle QTR is congruent to triangle RSQ, we need to show that their corresponding sides and angles are equal.
First, let's analyze the given information. Triangle PQR is isosceles, which means that two sides of the triangle are equal in length. Therefore, PQ = PR.
We also know that S and T are the midpoints of PQ and PR, respectively. That means that ST is parallel to QR and halves its length. Therefore, ST = 1/2 * QR.
Now, let's compare the two triangles:
1. Side QT and SR: Since S and T are midpoints, we know that ST is half the length of QR. Therefore, by substitution, QT = SR.
2. Side QR: We know that PQ = PR because triangle PQR is isosceles. Since S and T are midpoints, we can substitute PQ for ST + QT and PR for ST + SR. Therefore, QR = (ST + QT) + (ST + SR), which simplifies to QR = 2(ST + QT).
3. Lastly, angle QTR and angle RSQ: These angles are equal since they are vertical angles formed by transversals QT and SR intersecting the parallel lines QR and ST.
By proving that the corresponding sides and angles of triangles QTR and RSQ are equal, we have shown that the two triangles are congruent.