(a) Prove that the sum of the angles in a triangle is two right angles.
(b) In a triangle LMN, the side NM is produced to P and the bisector of < LNP meets ML produced at Q. If < LMN = 46°, and < MLN = 80°, calculate < LQN, stating clearly your reasins.
Explanation
(a) Given any (Delta) ABC,
Required: That < A + < B + < C = 180°.
Construction : Draw CP // BA and produce BC to Q.
Proof: Using the above figure,
(a_{1} = a_{2}) (alternate angles)
(b_{1} = b_{2}) (corresponding angles)
(a_{1} + b_{1} + c = 180°) (angles on a straight line)
(therefore a_{2} + b_{2} + c = 180°)
(< A + < B + < C = 180°) (proved)
(b)
< LNP = 46° + 80° = 126° (exterior angles of triangle LMN)
(theta = frac{126°}{2} = 63°) (bisector divides into 2 equal halves)
(< MLN = 180° – 80° = 100°) (angles on a straight line)
(< LQN = 180° – (63° + 100°) = 17°)