AMSCO'S Geometry. New York - Rye High School

432 The Geometry of Three DimensionsExercisesWriting About Mathematics1. Carmen said if two planes intersect to form four dihedral angles that have equal measures,then the planes are perpendicular to each other. Do you agree with Carmen? Explain whyor why not.2. Each of three lines is perpendicular to the plane determined by the other two.a. Is each line perpendicular to each of the other two lines? Justify your answer.b. Name a physical object that justifies your answer.Developing SkillsIn 3–11, state whether each of the statements is true or false. If it is true, state a postulate or theoremthat supports your answer. If it is false, describe or draw a counterexample.3. At a given point on a given line, only one line is perpendicular to the line.4. If A is a point in plane p and B is a point not in p, then no other point on AB gis in plane p.5. A line perpendicular to a plane is perpendicular to every line in the plane.6. A line and a plane perpendicular to the same line at two different points have no points incommon.7. Two intersecting planes that are each perpendicular to a third plane are perpendicular toeach other.8. If AB gis perpendicular to plane p at A and AB gis in plane q, then p ⊥ q.9. At a given point on a given plane, only one plane is perpendicular to the given plane.10. If a plane is perpendicular to one of two intersecting lines, it is perpendicular to the other.11. If a line is perpendicular to one of two intersecting planes, it is perpendicular to the other.Applying Skills12. Prove step 1 of Theorem 11.4.13. Prove step 3 of Theorem 11.4.14. Prove step 5 of Theorem 11.4.15. Prove that if a line segment is perpendicular to a plane at themidpoint of the line segment, then every point in the plane isequidistant from the endpoints of the line segment.Given: AB ⊥ plane p at M, the midpoint of AB, and R is anypoint in plane p.Prove: AR BRppMARRB