POLYEDRAL ANGLES 603. Definitions. When three or more planes meet at a point they form a polyedral angle, or a solid angle. The polyedral angle is formed by a portion of the planes as shown in the figure. The point V in which all the planes meet is the vertex of the polyedral angle; the lines, AV, BV, and CV, of intersection of consecutive planes are the A edges; the portions of the planes lying between the edges are the faces; and the angles in the faces between the edges are the face angles. D E B C A polyedral angle is read by the letter at the vertex, or by this letter together with a letter on each edge. Thus, the polyedral angle in the figure is read "polyedral angle V, or V-ABC." 604. Parts of a Polyedral Angle. The face angles are AVB, BVC, and CVA. The diedral angles formed by the faces are the diedral angles of the polyedral angle. The parts of a polyedral angle are its face angles and its diedral angles. For convenience in representing a polyedral angle, a plane is often passed through it cutting all its edges. It should be noted that this plane is not a part of the polyedral angle. 605. Convex and Concave Polyedral Angles. If a plane cuts all the edges of a polyedral angle, but not through the vertex, the intersections of the faces with this plane form a Q K G JL H M N CONVEX CONCAVE In this text only convex polyedral angles will be considered, unless otherwise stated. 606. Classification. A polyedral angle that has three faces called a triedral angle. The words tetraedral, pentaedral, hexaedral, etc., may be applied when the polyedral angle has our, five, six, etc., faces. 607. Congruent Polyedral Angles. If the corresponding parts of two polyedral angles are equal and arranged in the ame order, the polyedral angles are congruent. Thus, in the figure, the face angles ure equal and arranged in the same ›rder, that is, ZAVB=ZA'V'B', ZBVC=ZB'V'C', and ZCVA = ZC'V'A'. Also in the diedral angles, ZAV=ZA'V', ZBV=ZB'V', and B <CV=ZC'V', the arrangement is in the same order. If the corresponding parts of two polyedral angles are equal and arranged in the opposite order, the polyedral angles are said to be symmetric. These will be considered later (§ 836). EXERCISES 1. Does the size of a polyedral angle depend upon the lengths of its edges? How do the number of edges, the number of diedral angles, and the number of face angles of a polyedral angle compare? 2. Construct from stiff paper a triedral angle having face angles equal to 50°, 70°, and 90°. Can you tell the number of degrees in the diedral angles? The paper may be cut as indicated in the figure, folded along the dotted lines and pasted. 3. How many degrees in each of the face angles and the diedral angles of the polyedral angle formed by the walls of a room? 4. The face angles and diedral angles of a convex polyedral angle are each less than 180°. 5. Bearing the plane geometry definitions in mind, define vertical polyedral angles. Are they congruent or symmetric? Explain how, if it is possible to have two vertical triedral angles congruent. 6. Could a triedral angle have one right diedral angle? Could it have two? Three? Give illustrations. 608. Theorem. The sum of two face angles of a triedral angle is greater than the third face angle. Given the triedral V-ABC. To prove that the sum of two face angles is greater than the third face angle. Proof. Three cases arise: (1) all the face angles equal; (2) two face angles equal; (3) no two face angles equal. Cases (1) and (2) are left to the student. In case (3) suppose ZAVB is the greatest. A B In the face AVB, construct ZAVD=ZAVC, take VD VC, and draw AB, AC, and BC. = In ACVB and DVB, BV=BV, and CV DV, but CB>DB. Then ZAVC+ZCVB><AVD+ZDVB. :. ZAVC+ZCVB>ZAVB. Or the sum of two face angles is greater than the third. EXERCISES § 247 Why? $182 § 174 $ 259 § 173 1. Any face angle of a triedral angle is greater than the difference of the other two. 2. State the theorems in plane geometry that correspond to the theorems of § 608 and Exercise 1. There is a close analogy between the plane triangle and the triedral angle. Many theorems concerning plane triangles can be changed to theorems concerning triedral angles by replacing the word side by face angle, and the word angle by diedral angle. 3. State theorems for triedral angles analogous to the theorems of plane geometry that have to do with congruent triangles. 4. Referring to the figure of § 608, ZAVC=55° and ZCVB =65°. Make a statement regarding the number of degrees in ZAVB. 609. Theorem. The sum of the face angles of any convex polyedral angle is less than four right angles. E Given the convex polyedral V-ABCDEF. To prove ZAVB+ZBVC+ZFVA <4 rt.4. Proof. Let a plane intersect the edges of the polyedral angle in A, B, C, etc., and intersect the faces in AB, BC, CD, etc. Join any point in the polygon thus formed to the vertices of the polygon. Then Similarly ZVBA+ZV BC>ZABC. ZVCB+ZVCD>ZBCD, etc. $608 Add these and the result is that the sum of all the base angles of the triangles with vertices at V is greater than the sum of all the base angles of the triangles with vertices at 0. Then, since the sum of all the angles of the triangles with vertices at V is equal to the sum of those with vertices at 0, the sum of the angles about V is less than the sum of the angles about 0. But the sum of the angles about O = 4 rt s. :. ZAVB+ZBVC+···ZFVA <4 rt.. EXERCISES § 178 Why? 1. Can a polyedral angle have for its face angles three angles from an equilateral triangle? Can it have four? Five? Six? Why? 2. Can a polyedral angle have as face angles the angles of a square? Of a regular pentagon? Of a regular hexagon? Why? 3. Any face angle of a polyedral angle is less than the sum of the others. 4. The sum of the face angles of a polyedral angle is 300°. What is the greatest value that any one of the face angles may have? 5. One of the face angles of a triedral angle is 60°. What are the limitations on the size of each of the other face angles? 610. Theorem. Two triedral angles are congruent if the three face angles of one are equal respectively to the three face angles of the other, and arranged in the same order. Proof. On the edges of triedral angles V and V' lay off the six equal segments VA, VB, VC, V'A', V'B', and V'C', and draw AB, BC, CA, A'B', B'C', and C'A'. Then there are three pairs of congruent isosceles triangles as From any point D in VA construct DE in face AVB and DF in face AVC, each 1 VA. These lines meet AB and AC respectively in E and F. Draw EF. Why? Take V'D'VD and in a similar manner construct, D'E', D'F', and E'F'. Then, since AD=A'D' and ZDAE=ZD'A'E', Similarly, prove the other two pairs of diedral angles equal. .. triedral Vtriedral ZV'. $ 607 |