Answer to Question #97947 in Geometry for Rachel

Question

Pick one of the proofs for VAT for both the plane and the sphere.

Answer 1

A plane angle is the inclination to one another of two lines in a plane, which meet one another, but do not lie in a straight line.

A pair of opposite angles formed by two intersecting straight lines are congruent. [Note: Angles such as a and b are called vertical angles.

There are at least three different perspectives from which one can define "angle."

• a dynamic notion of angle — angle as movement;
• angle as measure; and,
• angle as geometric shape.

Each of them, separately or together, might help you prove the Vertical Angle Theorem. A dynamic notion of angle involves an action: a rotation, a turning point, or a change in direction between two lines. Angle as measure may be thought of as the arc length of a circular sector or the ratio between areas of circular sectors. Thought of as a geometric shape, an angle may be seen as the delineation of space by two intersecting lines. Each of these perspectives carries with it methods for checking angle congruency. 

1st proof:

Each line is a 180° angle. Thus, a + g = b + g.

Therefore we can conclude that a @ b.It is always true that if one subtracts a given angle from two 180° angles then the remaining angles are congruent.

2nd proof: Consider two overlapping lines and choose any point on them. Rotate one of the lines, maintaining the point of intersection and making sure that the other line remains fixed.

a. (ITT) Given a triangle with two of its sides congruent, then the two angles opposite those sides are also congruent. Look at this on both the plane and the sphere.

Looking for symmetries is very helpful.

In your proof of Part a, try to see that you have also proved the following:

b. Corollary. The bisector of the top angle of an isosceles triangle is also the perpendicular bisector of the base of that triangle.

c.Converse of ITT. For all triangles on the plane and spheres, if two angles are congruent, then the sides opposite the angles are also congruent.