Answer:
QR = 19
SR = 24
PT = 21
SQ = 20
m∠QRS = 74°
m∠PQS = 49°
m∠RPS = 39°
m∠PSQ = 57°
Explanation:
Attached is the complete question:
By definition, a parallelogram is a quadrilateral, which means it has 4 sides where the opposite pairs are parallel to each other. These pair of sides are congruent.
With that we can assume the following:
PQ ≅ SR
PS ≅ QR
m∠PQR ≅ m∠ PSR
m∠QRS ≅ m∠ QPS
The given states that:
PQ ≅ SR
PS ≅ QR
then:
PQ = 24
PS = 19
So we know that:
QR = 19
SR = 24
When it comes to the angles, we have the following theorems:
Opposite angles (Opposite edges) are congruent
Consecutive angles are supplementary, or they sum up to 180°
We are given the following angles:
m∠PQR = 106°
m∠QSR = 49°
m∠PRS = 35°
Based on the theorems we know the following:
m∠PQR ≅ m∠PSR
m∠PQR + m∠QRS = 180°
Then:
m∠PQR + m∠QRS = 180°
106° + m∠QRS = 180°
m∠QRS = 180° - 106°
m∠QRS = 74°
In addition, m∠PSR = 106° and m∠QSR = 49° then when we add it to m∠PSQ it should be equal to m∠PSR
m∠QSR + m∠PSQ = m∠PSR
49° + m∠PSQ = 106°
m∠PSQ = 106° - 49°
m∠PSQ = 57°
Now notice that a parallelogram when you cut them in half, it makes two congruent triangles. All angles in a triangle sum up to 180° so we can assume that:
m∠PSR + m∠PRS + m∠RPS = 180°
So:
106° + 35° + m∠RPS = 180°
141° + m∠RPS = 180°
m∠RPS = 180° - 141°
m∠RPS = 39°
Following the same logic:
m∠QPS ≅ m∠QRS
m∠QPS = 74°
m∠QPS + m∠PSQ + m∠PQS = 180°
74° + 57° + m∠PQS = 180°
131° + m∠PQS = 180°
m∠PQS = 180° - 131°
m∠PQS = 49°