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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 180 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.



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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.





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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..

1. If two sides of a triangle are congruent, the angles opposite these sides are congruent. (Base angles of an isosceles triangle are congruent.)

Given: ^ABC, AB > BC To Prove: /A > /C Plan: When the bisector of the vertex angle is drawn, the angles to be proved congruent become corresponding angles of congruent triangles.

PROOF:

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Statements 1. 2. 3. 4. 5. 6. Draw BD bisecting /B. /l > /2 AB > BC BD > BD ^ADB > ^CDB /A > /C 1. 2. 3. 4. 5. 6.

In This :

Reasons An angle may be bisected. To bisect is to divide into two congruent parts. Given Reflexive property SAS Corresponding parts of congruent triangles are congruent.

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Given: ^ABC To Prove: m/A m/B m/C 180 Plan: When a line is drawn through one vertex parallel to the opposite side, a straight angle is formed whose parts can be proved congruent to the angles of the triangle.

PROOF:

Reasons 1. Through an external point, a line can be drawn parallel to a given line. 2. A straight angle is an angle whose measure is 180 . 3. The whole equals the sum of its parts. 4. Alternate interior angles of parallel lines are congruent. 5. Substitution Postulate

Given: ^ABC, /A > /C To Prove: AB > BC Plan: When the bisector of /B is drawn, the sides to be proved congruent become corresponding sides of congruent triangles.

PROOF:

Summary

Statements 1. 2. 3. 4. 5. 6. Draw BD bisecting /B. /1 > /2 /A > /C BD > BD ^BDA > ^BDC AB > BC 1. 2. 3. 4. 5. 6.

Reasons An angle may be bisected. To bisect is to divide into two congruent parts. Given Reflexive property SAA Corresponding parts of congruent triangles 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.

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