Sec 4 A-Math Proofs in Plane Geometry topical questions

In triangle ABC, AB = AC and angle BAC = 40°. D lies on AB such that CD is perpendicular to AB. Find angle ACD and angle DCB.

PQ and RS are parallel. A transversal meets PQ at M and RS at N, and the angle on the upper side of the transversal at M is 62°. Find the corresponding angle at N, the co-interior partner of the 62° angle at N, and the alternate angle to the 62° angle.

ABCD is a cyclic quadrilateral with angle DAB = 88° and angle ABC = 74°. Find angle BCD and angle CDA.

A, B and C lie on a circle with centre O, and angle AOB = 140°, where the angle stands on the same arc as angle ACB. Find angle ACB and angle OAB.

PT is a tangent to a circle at P, and Q, R lie on the circle on the far side of chord PQ from T. The angle between PT and chord PQ is 52°, and angle QPR = 63°. Find angle PRQ and angle PQR.

In triangle ABC, D is the midpoint of AB and E is the midpoint of AC. Given BC = 12.4 cm, find DE and state, with reasons, the relationship between DE and BC.

TA and TB are tangents to a circle with centre O from an external point T, and angle ATB = 48°. Find angle TAB, angle AOB and angle OAB.

In triangle ABC, D lies on AB with AD = 4 cm and DB = 8 cm, and E lies on AC with AE = 5 cm and EC = 10 cm. (a) Show that triangles ADE and ABC are similar. (b) Given BC = 10.8 cm, find DE. (c) Given that the area of triangle ADE is 2.7 cm², find the area of triangle ABC.

AB is a diameter of a circle and D lies on the circle with angle ABD = 34°. The tangent to the circle at D meets BA produced (beyond A) at T. Find angle BTD, showing all reasoning.

ABCD is a rhombus whose diagonals AC and BD intersect at O. Prove that AC is perpendicular to BD.