In parallelogram PQRS, diagonals PR and QS intrsect at M. Which is always true?
(1) PR is perpendicular to QS
(2) PM is congruent to RM
(3) PQ is congruent to QR
(4) triangle PMS is congruent to triangle PMQ
To determine which of the given statements is always true in parallelogram PQRS, let's analyze each option:
(1) PR is perpendicular to QS: This statement is not always true in a parallelogram. The diagonals PR and QS may not be perpendicular to each other. Therefore, option (1) is not always true.
(2) PM is congruent to RM: This statement is also not always true. The lengths of PM and RM are not necessarily equal in a parallelogram. Thus, option (2) is not always true.
(3) PQ is congruent to QR: This statement is true in a parallelogram. In a parallelogram, opposite sides are congruent. Therefore, option (3) is always true.
(4) Triangle PMS is congruent to triangle PMQ: This statement is not always true. The triangles PMS and PMQ may not be congruent in a parallelogram. The only time these triangles will be congruent is when PM and MS are congruent to each other and PM and MQ are congruent to each other. This condition is not always satisfied, so option (4) is not always true.
In conclusion, option (3) is always true in a parallelogram PQRS: PQ is congruent to QR.