Proofs of Important Theorems
16.1 Introduction
The theorems proved in this chapter are considered the most important in the logical sequence of geometry. They are as follows:
1. If two sides of a triangle are congruent, the angles opposite these sides are congruent. (Base angles of an isosceles triangle are congruent.)
2. The sum of the measures of the angles in a triangle equals 1808.
3. If two angles of a triangle are congruent, the sides opposite these angles are congruent.
4. Two right triangles are congruent if the hypotenuse and a leg of one are congruent to the corresponding parts of the other.
5. A diameter perpendicular to a chord bisects the chord and its arcs.
6. An angle inscribed in a circle is measured by one-half its intercepted arc.
7. An angle formed by two chords intersecting inside a circle is measured by one-half the sum of the intercepted arcs.
8a. An angle formed by two secants intersecting outside a circle is measured by one-half the difference of its intercepted arcs.
8b. An angle formed by a tangent and a secant intersecting outside a circle is measured by one-half the difference of its intercepted arcs.
8c. An angle formed by two tangents intersecting outside a circle is measured by one-half the difference of its intercepted arcs.
9. If three angles of one triangle are congruent to three angles of another triangle, the triangles are similar.
10. If the altitude is drawn to the hypotenuse of a right triangle, then (a) the two triangles thus formed are similar to the given triangle and to each other, and (b) each leg of the given triangle is the mean proportional between the hypotenuse and the projection of that leg upon the hypotenuse.
11. The square of the length of the hypotenuse of a right triangle equals the sum of the squares of the lengths of the other two sides.
12. The area of a parallelogram equals the product of the length of one side and the length of the altitude to that side.
13. The area of a triangle is equal to one-half the product of the length of one side and the length of the altitude to that side.
14. The area of a trapezoid is equal to one-half the product of the length of the altitude and the sum of the lengths of the bases.
15. The area of a regular polygon is equal to one-half the product of its perimeter and the length of its apothem.
16.2 The Proofs
1. If two sides of a triangle are congruent, the angles opposite these sides are congruent. (Base angles of an isosceles triangle are congruent.)
PROOF:
2. The sum of the measures of the angles in a triangle equals 1808.
PROOF:
3. If two angles of a triangle are congruent, the sides opposite these angles are congruent.
PROOF:
4. Two right triangles are congruent if the hypotenuse and a leg of one are congruent to the corresponding parts of the other.
PROOF:
5. A diameter perpendicular to a chord bisects the chord and its arcs.
PROOF:
6. An angle inscribed in a circle is measured by one-half its intercepted arc.
Case I: The center of the circle is on one side of the angle.
PROOF:
Case II: The center is inside the angle.
PROOF:
Case III: The center is outside the angle.
PROOF:
7. An angle formed by two chords intersecting inside a circle is measured by one-half the sum of the intercepted arcs.
PROOF:
8a. An angle formed by two secants intersecting outside a circle is measured by one-half the difference of its intercepted arcs.
PROOF:
8b. An angle formed by a secant and a tangent intersecting outside a circle is measured by one-half the difference of its intercepted arcs.
PROOF:
8c. An angle formed by two tangents intersecting outside a circle is measured by one-half the difference of its intercepted arcs.
PROOF:
9. If three angles of one triangle are congruent to three angles of another triangle, the triangles are similar.
PROOF:
10. If the altitude is drawn to the hypotenuse of a right triangle, then (a) the two triangles thus formed are similar to the given triangle and to each other, and (b) each leg of the given triangle is the mean proportional between the hypotenuse and the projection of that leg upon the hypotenuse.
PROOF:
11. The square of the length of the hypotenuse of a right triangle equals the sum of the squares of the lengths of the other two sides.
PROOF:
12. The area of a parallelogram equals the product of the length of one side and the length of the altitude to that side.
PROOF:
13. The area of a triangle is equal to one-half the product of the length of one side and the length of the altitude to that side.
PROOF:
14. The area of a trapezoid is equal to one-half the product of the length of the altitude and the sum of the lengths of the bases.
PROOF:
15. The area of a regular polygon is equal to one-half the product of its perimeter and apothem.
PROOF:
