In like manner it may be proved that plane AC coincides with plane A'C'. Hence, diedral angles AB and A'B' coincide. Ex. 1047. Vertical diedral angles are equal. Q.E.D. Ex. 1048. If two parallel planes are intersected by a third one, (a) the corresponding diedral angles are equal, (b) the alternate diedral angles are equal. Ex. 1049. State and prove the converse of Ex. 1048. PROPOSITION XVI. THEOREM 500. Two diedral angles have the same ratio as their plane angles. Hyp. ABC and A'B'C' are respectively the plane angles of diedral angles ABDC and A'B'D'C'. Proof. CASE I. Angles ABC and A'B'C' are commensurable. Apply a common measure P to these angles and let P be contained in ABC m times, and in A'B'C' n times. Through the lines of division draw planes which contain BD and B'D', respectively. These planes divide ABDC into m and A'B'C'D' into n CASE II. ABDC LABC = ZA'B'C Q.E.D. Angles ABC and A'B'C' are incommensurable. Divide ABC into any number of equal parts and apply one of the parts to ▲ A'B'C' as often as possible. [To be completed by the student, who may compare 280, Bk. III.] 501. COR. A diedral angle is measured by its plane angle. PROPOSITION XVII. THEOREM 502. If a straight line is perpendicular to a plane, every plane passed through this line is perpendicular to the plane. M B Hyp. AB is to plane MN. E N PROPOSITION XVIII. THEOREM 503. If two planes are perpendicular to each other, a straight line drawn in one of them, perpendicular to their intersection, is perpendicular to the other. Hyp. Plane CE is 1 to plane MN, and in CE line BA is 1 to CD, the intersection of the two planes. HINT. At B construct the plane angle and prove that AB is to two lines drawn through its foot. 504. COR. 1. If two planes are perpendicular to each other, a perpendicular to one of these, at any point of their intersection, lies in the other. [Prove by indirect method.] 505. COR. 2. If two planes are perpendicular to each other, a straight line drawn from any point of one, perpendicular to the other, lies in the first. [Prove by indirect method.] Ex. 1050. If the opposite sides of a quadrilateral in space are equal, the opposite plane angles are equal. Ex. 1051. Three points not in a straight line each equidistant from the ends of a given line determine a perpendicular bisecting plane of the given line. PROPOSITION XIX. THEOREM 506. If two intersecting planes are each perpendicular to a third plane, their intersection is perpendicular to that plane. P R M N Hyp. The planes PQ and RA, intersecting in AB, are perpendicular to plane MN. Proof. At A draw a straight line to MN. This line must lie in plane PQ and in plane RA. (504) Or AB is to plane MN. Q.E.D. 507. COR. If a plane is perpendicular to two other planes, perpendicular to each other, the intersection of any two planes is perpendicular to the third plane and perpendicular to the other intersections. Ex. 1052. If a line is parallel to a plane, any other plane perpendicular to the line is perpendicular to the plane. Ex. 1053. If two parallel planes are intersected by a third one, the interior angles on the same side of the third plane are supplementary. S PROPOSITION XX. THEOREM 508. Every point in a plane bisecting a diedral angle is equidistant from the faces of the angle. A H E Hyp. Plane CB bisects diedral / ABED, and FG and FH are the respective distances of a point F in BC, from AB Proof. Through FG and FH pass a plane intersecting the faces in EG and EH. .. GEF and HEF are the plane angles of diedral angles ABEC and DBEC. [To be completed by the student. Compare Remark 73.] Ex. 1054. The locus of a point equidistant from two intersecting planes consists of the two planes bisecting the diedral angles formed by the planes. Ex. 1055. Find the locus of a point equidistant from two given intersecting lines (in space). |