As the base or sum of the segments, is to the sum of the other two sides; so is the difference of those sides, to the difference of the segment of the bases: then add half the difference of the segments to the half sum, or the half base, for the greater segment; and subtract the same for the less segment. Hence, in each of the two right-angled triangles, there will be known two sides, and the right angle opposite to one of them, consequently the other angle will be found by the method in Case 1.
USEFUL THEOREMS, AND COROLLARIES.
1. When one line meets another, the angles, which it makes on the same side of the other, are together equal to two right angles.
2. All the angles, which can be made at any point (by any number of lines), on the same side of a right line, are, when taken all together, equal to two right angles: and, as all the angles that can be made, on the other side of the line, are also equal to two right angles; therefore all the angles that can be made quite round a point, by any number of lines, are equal to four right angles. Hence also the whole circumference of a circle, being the sum of all the angles that can be made about the centre, is the measure of four right angles.
3. When two lines intersect each other, the opposite angles are equal.
4. When one side of a triangle is produced, or extended, the outward angle is equal to the sum of the two inward opposite angles.
5. In any triangle, the sum of all the three angles is equal to two right angles (180°). Hence, if one angle of a triangle be a right angle, the sum of the other two angles will be equal to a right angle (90°).
6. In any quadrilateral, the sum of all the four inward angles is equal to four right angles.
7. In any right-angled triangle, the square of the hypothenuse (or side opposite to the right angle) is equal to the sum of the squares of the other two sides. Therefore, to find the hypothenuse, add together the squares of the other two sides, and extract the square root of that sum: and to find one of the other sides, subtract from the square of the hypothenuse the square of the other given side, and extract the square root of the remainder for the side required.
Or hypothenuse = √base2 + perpendicular2