Extended parallelogram
Weekly Problem 11 - 2014
The diagram shows a parallelogram and an isosceles triangle. What is the size of angle TQR?
The diagram shows a parallelogram and an isosceles triangle. What is the size of angle TQR?
Age
11 to 14
Challenge level
Problem
Image
In the diagram, PQRS is a parallelogram; ∠QRS = 50 °; ∠SPT = 62 ° and PQ = PT. What is the size of ∠TQR?
If you liked this problem, here is an NRICH task that challenges you to use similar mathematical ideas.
Student Solutions
Image
Opposite angles of a parallelogram are equal, so ∠QPS = 50 °. Therefore, ∠QPT = 112 ° and, as triangle QPT is isosceles, ∠PQT = (180 °-112 °)/2=34 °. As PQRS is a parallelogram, ∠PQR = 180 ° - 50 ° = 130 °.
So ∠TQR = 130 ° - 34 ° = 96 °.