### Groupwork

### Spring 2012

1. Prove that in an isosceles triangle, a line from the vertex to the base has one of the following properties, if and only if it also has the other two.

- It hits the base at its midpoint
- It is perpendicular to the base
- It bisects the vertex angle.

2. Prove that if a line from the vertex of a triangle to the base has two of the properties listed above, the triangle is isosceles and it also has the other property.

3. Thales of Miletus, Feet in the Mirror.

4. Snell's Law.

5. A point is on the perpendicular bisector of a line segment if and only if it is equidistant from the endpoints.

6. In a parallelogram, the opposite sides are congruent. If the opposite sides of a quadrilateral are congruent, then it is a parallelogram

7. A quadilateral is a parallelogram if and only if the diagonals bisect each other. If one pair of opposite sides in a quadrilateral are both parallel and congruent, the figure is a parallelogram.

8. A quadrilateral is a rhombus if and only if the diagonals are perpendicular bisectors of each other.

9. A quadrilateral is a rhombus if and only if the diagonals bisect all the vertex angles.

10. A parallelogram is a rectangle if and only if the diagonals are congruent.

11. A line from a point outside a circle to the center of the circle bisects the angle between the tangents from the point to the circle. The line segments from a point outside a circle to the two points of tangency are congruent. A point is on the bisector of an angle if and only if it is equidistant from both of the sides of the angle.

12. Prove the constructions for copying an angle, bisecting an angle, erecting a perpendicular from a point on a line, dropping a perpendicular from a point to a line.

13.Prove the construction for constructing the perpendicular bisector of a line segment.

14. Fill in all the angles in a pentagram.