A plane figure bounded by straight lines is called a polygon. THEOREM 9. If the sides of a convex polygon are produced in order, the sum of the angles so formed is equal to four right angles. P / D fig. 95. S Data ABCDE is a convex polygon; its sides are produced in order and form the exterior angles, w, v, x, y, z. To prove that LW + LV + L x + LY + L Z = 4 rt. ≤ s. Construction Through any point O draw OP, OQ, OR, OS, OT || to and in the same sense as EA, AB, BC, CD, DE respectively. Proof Since OP, OQ are respectively || to and in the same sense as EA, AB, .. LWL POQ, I. 7. COR. The sum of the interior angles of any convex polygon together with four right angles is equal to twice as many right angles as the polygon has sides. Ex. 388. Three of the exterior angles of a quadrilateral are 79°, 117°, 65°; find the other exterior angle and all the interior angles. Ex. 389. Prove the corollary for a pentagon (i) by considering the sum of the exterior and interior angles at each corner, and the sum of all the exterior angles; (ii) by joining a point O inside the pentagon to each corner, and considering the sums of the angles of the triangles so formed and the sum of the angles at the point O. DEF. A polygon which has all its sides equal and all its angles equal is called a régular polygon. Ex. 390. What is the size of each exterior angle of a regular octagon (8-gon)? Hence find the size of each interior angle. Ex. 391. What are the exterior angles of regular polygons of 12, 10, 5, 3 sides? Hence find the interior angles of these polygons. Ex. 392. The exterior angle of a regular polygon is 60°, how many sides has the polygon? Ex. 393. How many sides have the regular polygons whose exterior angles are (i) 10°, (ii) 1°, (iii) 21°? Ex. 394. Is it possible to have regular polygons whose exterior angles are (i) 15°, (ii) 7°, (iii) 11°, (iv) 6°, (v) 5°, (vi) 4°? Ex. 395. Is it possible to have regular polygons whose exterior angles are obtuse? Ex. 396. Is it possible to have regular polygons whose interior angles are (i) 108°, (i') 120°, (iii) 130°, (iv) 144°, (v) 60°? (Think of the exterior angles.) In the cases which are possible, find the number of sides. Ex. 397. Make a table showing the exterior and interior angles of regular polygons of 3, 4, 5...10 sides. Draw a graph showing horizontally the number of sides and vertically the number of degrees in the angles. Ex. 398. Construct a regular pentagon having each side 2 in. long. (Calculate its angles, draw AB=2 in., at B make LABC= the angle of the regular pentagon, cut off BC=2 in., &c., &c.) Ex. 399. Construct a regular octagon having each side 2 in. long. Ex. 400. Construct a regular 12-gon having each side 1.5 in. long. CONGRUENT TRIANGLES. If two figures when applied to one another can be made to coincide (i.e. fit exactly) they must be equal in all respects. · This method of testing equality is known as the method of superposition. Ex. 401. How did you test the equality of two angles? (See Ex. 28.) Ex. 402. How would you test whether two cricket bats were of the same length? Figures which are equal in all respects are said to be congruent. The sign is used to denote that figures are congruent. Ex. 403. Draw a triangle DEF having DE=3in., DF=2in., ▲D=26°; on tracing paper draw a triangle ABC having AB=3 in., AC=2 in., LA=30°. Apply ▲ ABC to ▲ DEF so that A falls on D; put a pin through these two points; turn ▲ ABC round until AB falls along DE. B falls on E. Why is this? Does AC fall along DF? (Keep the AABC for the next Ex.) Ex. 404. Draw a triangle DEF having DE=3 in., DF=2 in., D=30°. Apply ▲ ABC (made in the last Ex.) to ▲ DEF so that A falls on D; put ▲ ABC round until AB falls along DE. a pin through these two points; turn THEOREM 10. If two triangles have two sides of the one equal to two sides of the other, each to each, and also the angles contained by those sides equal, the triangles are congruent. 44 fig. 96. Data ABC, DEF are two triangles which have AB = DE, AC = DF, and included BAC = included ▲ EDF. Proof Apply AABC to ▲ DEF so that A falls on D, and AB falls. along DE. ... AB = DE, .. B falls on E. Again BAC = ▲ EDF, .. AC falls along DF. And AC DF, .. C falls on F, .. AABC coincides with A DEF, .. AABC = ADEF. Q. E. D. N.B. It must be carefully noted that the congruence of the triangles cannot be inferred unless the equal angles are the angles included (or contained) by the sides which are given equal. Ex. 405. Make a list of all the equal sides and angles in ▲3 ABC and DEF of 1. 10. Say which were given equal and which were proved equal. Ex. 406. Draw two triangles PQR, XYZ and mark QR=XY, RP=YZ, and Q=4Z. Would this theorem prove the triangles congruent? Give two reasons. Ex. 407. ABCD is a square, E is the mid-point of AB; equal lengths AP and BQ are cut off from AD and BC. Join EP and EQ. Prove that ▲ AEP A BEQ. Write down all the pairs of lines and angles in these triangles which you have proved equal. Ex. 408. ABCD is a square, E is the mid-point of AB; join CE and DE. Prove that A AED A BEC. Write down all the pairs of lines and angles in these triangles which you have proved equal. Ex. 409. PQRS is a quadrilateral in which PQ=SR, 4Q=4R, and O is the mid-point of QR. Prove that OP=OS. A E B P D fig. 97. A E B D fig. 98. R [You must first join OP and OS, and mark in your figure all the parts that are given equal; you will then see that you want to prove that ▲ OQP= AORS.] Ex. 410. ABCD is a square; E, F, G are the mid-points of AB, BC, CD respectively. Join EF and FG and prove them equal. [Which are the two triangles that you must prove equal?] Ex. 411. ABC, DEF are two triangles which are equal in all respects; X is the mid-point of BC, Y is the mid-point of EF. Prove that AX=DY, and LAXB=LDYE. [You will of course have to join AX and DY.] Ex. 412. The equal sides QP, RP of an isosceles triangle PQR are produced to S, T so that PSPT; prove that TQ SR. R fig. 100. Ex. 413. D is the mid-point of the side BC of a ▲ ABC, AD is produced to E so that DEAD. Prove that ABEC and that AB, EC are parallel. [First prove a pair of triangles congruent.] B E fig. 101. |