Answer:
From TM = MS, MS = MR, and TM = MR
QT = TM = MR Transitive property
Explanation:
ΔPQR is an equilateral triangle Given
∠PQR = ∠RPQ = ∠PRQ = 60° Definition of equilateral triangle
∠PQR = ∠SQT + ∠SQP = 60° Addition property of angles
∠PRQ = ∠SRM + ∠SRP = 60° Addition property of angles
In triangle PQR, ∠SQT = ∠SQP = 30° = ∠SRM = ∠SRP Angles bisected by angle bisector
∠STM = ∠PQR - 60° Angle of same side of two parallel lines crossing the same transversal
∠PRQ = ∠SMT = 60° Angle of same side of two parallel lines crossing the same transversal
∠MST = 180 - ∠SMT - ∠STM = 180 - 60 - 60 = 60°
∠MST = ∠SMT = ∠STM = 60°, ΔMST is an equilateral triangle, all angles equal
ST = MS = TM definition of equilateral triangle
∠STQ = 180 - ∠STM = 180° - 60° = 120° (∠STQ and ∠STM are supplementary angles)
∠QST = 180 - ∠SQT - ∠STQ = 180 - 30 - 120 = 30 (Interior angles in the same triangle)
ΔQTS is an isosceles triangle, base angles are equal
∴ ST = QT Definition of equilateral triangle
Similarly
∠SMR = 180 - ∠SMT = 180° - 60° = 120° ∠SMR and ∠SMT are supplementary angles
∠MSR = 180 - ∠SRM - ∠SMR = 180 - 30 - 120 = 30 (Angles in the same triangle)
ΔRMS is an isosceles triangle, base angles are equal
∴ MS = MR Definition of equilateral triangle
Since ST = QT, and, ST = TM, QT = TM Transitive property
Also since TM = MS, and MS = MR, TM = MR Transitive property
Therefore;
QT = TM = MR Transitive property