Groupwork Assignments

2/15 1. Prove that a quadrilateral is a parallelogram if and only if the diagonals bisect each other.

2/20 2. Prove that if one pair of opposite sides in a quadrilateral is both parallel and congruent, then the figure is a parallelogram.

3. Prove that a parallelogram is a rectangle if and only if the diagonals are congruent.

2/22 4. Prove that if two angles in a triangle are congruent, then the sides opposite them are congruent.

5. If ABC is an isosceles triangle where AB = AC, then the following are equivalent.

1. D is the midpoint of BC
2. AD is perpendicular to BC
3. AD bisects the angle at A.

6. 2/26 1. Prove that if the line from a vertex of a triangle perpendicular to the other side meets the other side at its midpoint, then the triangle is isosceles.

7. Prove that if the bisector of the angle of a triangle is perpendicular to the other side, then the triangle is isosceles.

8. (Extra Credit) If the bisector of an angle of a triangle meets the opposite side at it's midpoint, then the triangle is isosceles.

3/6 9. Prove that a point is equidistant from two given points if and only if it is on the perpendicular bisector of the line segment joining the two given points.

3/8 10. The feet in the mirror question: If you are looking at yourself in a mirror and you want to see your feet but the image in the mirror doesn't quite go down to your feet, should you move closer to the mirror or back up?

3/13 11. Prove Thales of Miletus' construction for finding the closest distance from the ship to the shore.

3/15 12. Prove that if two triangles are congruent, their corresponding altitudes are congruent.

13. Prove that a quadrilateral is a rhombus if and only if the diagonals bisect all the angles.

3/20 14. Prove that a point is equidistant from two given points if and only if it is on the perpendicular bisector of the line segment joining the two points.

15. Prove that a point is on the bisector of an angle if and only if its perpendicular distances to the two arms of the angle are the same.

3/22 16. Prove the construction for copying an angle.

17. Prove the construction for bisecting an angle.

18. Prove the construction for erecting a perpendicular from a point on a line.

19. Prove the construction for dropping a perpendicular from a point to a line.

20. Prove the construction of the perpendicular bisector of a line segment.

3/29 21. Fill in all the angles in a pentagram.

4/19 22. Complete the problem set on equilateral triangles.

4/24 23. Make up a table showing the number of vertices, edges, and faces of a tetrahedron, octahedron, cube, icosahedron, and dodecahedron.

4/26. Find the volume and surface area of a regular tetrahedron whose edges are all 1m. in length.