Theory question 110
Angles and PolygonsIn the diagram, PQT is a straight line and SQ//RT.
(a) Join QR and show that: (i) ∠RPS = ∠QRT, (ii) ∠PRS = ∠QTR
(b) ABC is a triangle. The sides AB and AC are produced to D and E respectively such that ∠DBC = 132° and ∠ECB = 96°. Show that ΔABC is isosceles.
Show the model answer
Model answer
(a)(i) ∠RPS = ∠RQS (angles in the same segment, on chord RS).
∠RQS = ∠QRT (alternate angles, SQ ∥ RT). Hence ∠RPS = ∠QRT.
(ii) ∠PRS = ∠PQS (angles in the same segment, on chord PS).
∠PQS = ∠QTR (corresponding angles, SQ ∥ RT, PQT a straight line). Hence ∠PRS = ∠QTR.
(b) ∠ABC = 180 − 132 = 48° and ∠ACB = 180 − 96 = 84° (angles on a straight line).
∠BAC = 180 − 48 − 84 = 48°. ∠BAC = ∠ABC, so CA = CB and triangle ABC is isosceles.
Explanation
Give a reason for every equal pair of angles.