Answer:
m∠QRS = m∠QPS = 110°
m∠PQR= m∠RSP = 70°
PQ = SR = 17.5
PS = QR = 26.04 (nearest hundredth)
ST = TQ = 18
QS = 36
PT = TR = 12.97 (nearest hundredth)
PR = 25.93 (nearest hundredth)
Explanation:
In a congruent parallelogram, opposite sides are congruent
⇒ PQ = SR = 17.5
⇒ PS = QR
In a congruent parallelogram opposite angles are congruent
⇒ m∠QRS = m∠QPS = 110°
Given:
- Sum of interior angles of a quadrilateral = 360°
- m∠PQR = m∠RSP
⇒ 2(m∠PQR) + 2(110) = 360
⇒ m∠PQR= m∠RSP = 70°
ST = TQ = 18
⇒ QS = 18 + 18 = 36
Using cosine rule to find PS:
QS² = PQ² + PS² - 2(PQ)(PS)cos(∠QPS)
⇒ 36² = 17.5² + PS² - 2(17.5)(PS)cos(110)
⇒ 1296 = 306.25 + PS² - 35(PS)cos(110)
⇒ PS² - 35cos(110)(PS) - 989.75 = 0
⇒ PS = 26.03923874 only (as measure > 0)
Using cosine rule to find PR:
PR² = SR² + PS² - 2(SR)(PS)cos(70)
⇒ PR² = 17.5² + 26.04² - 2(17.5)(26.04)cos(70)
⇒ PR² = 672.5839084
⇒ PR = 25.93422273
PT = TR = PR ÷ 2 = 25.93422273 ÷ 2 = 12.96711136