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Study Notes 022 Geometry 15 Theorems:
By William E. Steinman:
:
These are considered to be the most important theorems in the progression of geometry.
If two sides of a triangle are congruent, the angles opposite the sides are congruent.
The sum of the angles in a triangle equals 1800.
If two angles of a triangle are congruent, the sides opposite these angles are congruent.
Two right triangles are congruent if the hypotheses and one leg of one are congruent with the corresponding parts of the other.
A diameter perpendicular to a chord of a circle bisects the chord and its arcs.
An angle inscribes in a circle is measured by ½ of its intercepted arc.
In this example, abc is equal to ½ obc.
The proof for this comes from our postulates.
Radii of a circle are congruent. Thus oa and oc are congruent.
If two sides of a triangle are congruent the angles opposite the sides are congruent. So aoc is congruent with coa.
In a triangle the measures of an exterior angle equals the sum of the measures of the two adjacent interior angles. So aoc + coa = obc.
Since aoc = coa we can substitute and get 2 aoc = obc.
Notice that abc and aoc are the same angle.
Therfore 2 abc = obc.
Or abc ½ obc.
You can prove the rest of these by yourself if you wish.
An angle formed by two chords intersecting inside a circle is measured by ½ the sum of the intercepted arcs.
An angle formed by two secants intersecting outside a circle is measured by ½ the difference of its intercepted arcs.
Remember a secant is a straight line segment intersecting a curve at two or more points.
An angle formed by a secant and a tangent intersecting outside a circle is measured by ½ the differenced of its intercepted arcs.
An angle formed by two tangents intersecting outside a circle is measured by ½ the difference of its intercepted arcs.
If three angles of one triangle are congruent to three angles of another triangle the triangles are similar.
If the altitude is drawn to the hypotheses of a right triangle then the two angles formed are similar to the given triangle and to each other. Also, each leg of the given triangle is the mean proportional between the hypotheses and the projection of that leg upon the hypotheses.
The square of the length of the hypotheses of a right triangle equals the sum of the squares of the lengths of the two sides.
This is the well know Pythagorean Theorem. You have probably been using it and trusting it with no proof. Now you have a chance to prove it using the postulates and other proven theorems. Ain’t that swell?
The area of a parallelogram is equal to the product of the length of one side and the length of the altitude to that side.
The area of a triangle is equal to ½ the product of the length of one side and the length of the altitude to that side.
The area of a trapezoid is equal to ½ the product of the length of the altitude and the sum of the lengths of its bases.
Remember a trapezoid is a quadrilateral having two parallel sides.
The area of a regular polygon is equal to ½ the product of its perimeter and apothem.
We already did this one in SN 018.