PROPOSITIONS OF GEOMETRY Table 1b. Propositions of Geometry Machinery's Handbook, 31st Edition
62
In an isosceles triangle , two sides a, b are congruent, hence the two angles opposite them (the base angles) are congruent. An equilateral (equiangular) triangle is also an isosceles triangle.
a
b
B
A
60
In an equilateral (equiangular) triangle , all three sides (angles) are congruent. Since the sum of angle measures in any triangle is 180 degrees, then each angle in an equilateral (equiangular) triangle measures 60 degrees.
a
a
60
60
a
A
A line in an equilateral triangle that bisects any of the angles (that is, divides it into two 30-degree angles), also bisects the side opposite the bisected angle and is perpendicular (at right angles) to it. Thus, if line AB bisects angle CAD , it also bisects line CD into two equal parts and is perpendicular to it.
30
30
90
C
D
B
1 / 2 a
1 / 2 a
If a line in an isosceles triangle drawn from the vertex where the two congruent sides meet in such a way that is bisects the third side (or base), then it also bisects the angle at the vertex from which it is drawn.
a
b
1 / 2 B
90
B
1 / 2 b
1 / 2 b
b
In every triangle, the greatest angle is opposite the longest side. And, the longest side is opposite the greatest angle. Thus, if a is longer than b , then the measure of angle A is greater than that of angle B . And, if angle A measure is greater than B , then side a is longer than side b . According to the triangle inequality , for any triangle, the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side. Thus, a + b ≥ c . The Pythagorean theorem states that in a right-angle triangle, the square of the hypotenuse , that is, the side opposite the right angle, is equal to the sum of the squares on the two sides that form the right angle. These sides are the legs of the right triangle. Thus, a 2 + b 2 = c 2
a
b
B
A
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