Answer:
m∠P = 70°, m∠T = 20°, m∠SKP = 40° , and m∠MKT = 70°
Explanation:
* Lets explain how to solve the problem
- In Δ PST
∵ m∠S = 90°
∴ m∠T + m∠P = 90° ⇒ interior angles of a triangle
∵ m∠SPK/m∠KPT = 5/2
- The ratio between the two angles are 5 : 2 , multiply the parts of the
ratio by x, where x is a real number
∴ m∠SPK = 5x
∴ m∠KPT = 2x
∵ m∠SPK + m∠KPT = m∠P
∴ m∠P = 5x + 2x = 7x
- In ΔPKT
∵ KM ⊥ PT
∵ MP = Mt
∴ KM is perpendicular bisector of PT
∴ ΔPKT is an isosceles triangle with KP = KT
∵ KP = KT
∴ m∠KPT = m∠T
∵ m∠KPT = 2x
∴ m∠T = 2x
∵ m∠T + m∠P = 90°
∵ m∠P = 7x
∵ m∠T = 2x
∴ 2x + 7x = 90 ⇒ solve for x
∴ 9x = 90 ⇒ divide both sides by 9
∴ x = 10
∵ m∠P = 7x
∴ m∠P = 7(10) = 70°
∴ m∠P = 70°
∵ m∠T = 2x
∴ m∠T = 2(10) = 20°
∴ m∠T = 20°
- In ΔSKP
∵ m∠S = 90°
∵ m∠SPK = 5x = 5(10) = 50°
∴ m∠SKP = 180° - (90° + 50°) = 180° - 140° = 40° ⇒ interior angles of a Δ
∴ m∠SKP = 40°
- In Δ KMT
∵ m∠KMT = 90°
∵ m∠T = 20°
∴ m∠MKT = 180° - (90° + 20°) = 180° - 110° = 70° ⇒ interior angles of a Δ
∴ m∠MKT = 70°