1. ABCD, EFGH are two squares. If they be placed so that F falls on C, and FE along CD, show that FG will either fall along CB, or be in the same straight line with it. 2. If in the straight line AB, a point E be taken and two straight lines EC, ED be drawn on opposite sides of AB, making L AEC = L BED, prove that EC and ED are in the same straight line. 3. If four straight lines, AE, CE, BE, DE, meet at a point E, so that AEC = ▲ BED and 2 AED = ▲ BEC, then AE and EB are in the same straight line, and also CE and ED. 4. P is any point, and AOB a right angle; PM is drawn perpendicular to OA and produced to Q, so that QM = MP; PN is drawn perpendicular to OB and produced to R, so that RN NP. Prove that Q, O, R lie in the same straight line. = 5. If in the enunciation of the proposition the words 'on opposite sides of it' be omitted, is the proposition necessarily true? Draw a figure to illustrate your answer. PROPOSITION 15. THEOREM. If two straight lines cut one another, the vertically opposite angles shall be equal. Let AB and CD cut one another at E: it is required to prove ▲ AEC = = L BED, and ▲ BEC = L AED. Take away Hence also, 1. Prove from these equals ▲ BEC, which is common; LAEC = L BED. I. Ax. 3 AEC = ▲ BED, making AED the common angle. L BEC = L AED, 11 L BEC = L AED, L BED 3. 4. If AED is bisected by FE, and FE is produced to G, prove that EG bisects / BEC. 5. If ▲ AED is bisected by FE, and ▲ BEC bisected by GE, prove FE and GE in the same straight line. 6. If in a straight line AB, a point E be taken, and two straight lines, EC, ED, be drawn on opposite sides of AB, making LAEC = L BED, prove that EC and ED are in the same straight line. 7. ABC is a triangle, BD, CE straight lines drawn making equal angles with BC, and meeting the opposite sides in D and E and each other in F; prove that if ▲ AFE = ▲ AFD, the triangle is isosceles. PROPOSITION 16. THEOREM. If one side of a triangle be produced, the exterior angle shall be greater than either of the interior opposite angles. Let ABC be a triangle, and let BC be produced to D: it is required to prove ▲ ACD greater than ▲ BAC, and also greater than ▲ ABC. Bisect AC at E; D I. 10 join BE, and produce it to F, making EF = BE; I. 3 Const. 9. Draw three figures to show that an exterior angle of a triangle may be greater than, equal to, or less than the interior adjacent angle. 10. From a point outside a given straight line, there can be drawn to the straight line only one perpendicular. 11. ABC is a triangle whose vertical line which meets BC at D; A is bisected by a straight prove ▲ ADC greater than ▲ DAC, and ▲ ADB greater than ▲ BAD. 12. In the figure to the proposition, if AF be joined, prove: (1) AF BC. (2) Area of ▲ ABC = area of ▲ BCF. (3) Area of A ABF area of A ACF. = 13. Hence construct on the same base a series of triangles of equal area, whose vertices are equidistant. 14. To a given straight line there cannot be drawn more than two equal straight lines from a given point without it. 15. Any two exterior angles of a triangle are together greater than two right angles. PROPOSITION 17. THEOREM. The sum of any two angles of a triangle is less than two right angles. A Д B Let ABC be a triangle: -D it is required to prove the sum of any two of its angles less Now ABC and ▲ ACB are any two angles of the triangle; .. the sum of any two angles of a triangle is less than 2 rt. 4 s. 1. Prove that in any triangle there cannot be two right angles, or two obtuse angles, or one right and one obtuse angle. 2. Prove that in any triangle there must be at least two acute angles. 3. From a point outside a straight line only one perpendicular can be drawn to the straight line. 4. Prove the proposition by joining the vertex to a point inside the base. 5. The angles at the base of an isosceles triangle are both acute. 6. All the angles of an equilateral triangle are acute. 7. If two angles of a triangle be unequal, the smaller of the two must be acute. 8. The three interior angles of a triangle are together less than three right angles. 9. The three exterior angles of a triangle made by producing the sides in succession, are together greater than three right angles. Prove by indirect demonstrations the following theorems : 10. The perpendicular from the right angle of a right-angled triangle on the hypotenuse falls inside the triangle. 11. The perpendicular from the obtuse angle of an obtuse-angled triangle on the opposite side falls inside the triangle. 12. The perpendicular from any of the angles of an acute-angled triangle on the opposite side falls inside the triangle. 13. The perpendicular from any of the acute angles of an obtuseangled triangle on the opposite side falls outside the triangle. The greater side of a triangle has the greater angle opposite to it. A Let ABC be a triangle, having AC greater than AB: it is required to prove ▲ ABC greater than C. From AC cut off AD = AB, and join BD. I. 3 |