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# Part 6

 Theorem 7.10 A parallelogram with a diagonal that bisects opposite angles is a rhombus. Theorem 7.11 A quadrilateral with four congruent sides is a rhombus. Theorem 7.12 All altitudes of a trapezoid are congruent. Theorem 7.13 The median of a trapezoid is parallel to its bases. Its length is one-half the sum of the lengths of the two bases. Theorem 7.14 The base angles of an isosceles trapezoid are congruent. Theorem 7.15 If the base angles of a trapezoid are congruent, then the trapezoid is isosceles. Theorem 7.16 The diagonals of an isosceles trapezoid are congruent. Theorem 7.17 If the diagonals of a trapezoid are congruent, then the trapezoid is isosceles. Theorem 8.1 In a proportion, the product of the extremes equals the product of the means. Corollary If the product of the extremes equals the product of the means, then a proportion exists. Theorem 8.2 If a/b = c/d, then: b/d = d/c; a/c = b/d; a+b/b = c+d/d; a-b/b = c-d/d; a/b = a+c/b+d. Theorem 8.3 Congruent triangles are similar. Theorem 8.4 Transitive Property of Triangle Similarity: If Triangle ABC – Triangle DEF and Triangle DEF – Triangle GHI, then Triangle ABC – Triangle GHI. Postulate 18 AA Similarity Postulate: If two angles of a triangle are congruent to two angles of another triangle, then the two triangles are similar. Theorem 8.5 Triangle Proportionality Theorem: If a line is parallel to one side of a triangle and intersects the other two sides, then it divides the two sides proportionally. Theorem 8.6 If a line divides two sides of a triangle proportionally, then the line is parallel to the third side of the triangle. Theorem 8.7 SAS Similarity Theorem: If an angle of one triangle is congruent to an angle of another triangle and the corresponding sides that include these angles are proportional, then the triangles are similar. Theorem 8.8 SSS Similarity Theorem: If all three pairs of corresponding sides of two triangles are proportional, then the two triangles are similar.

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Note: The concepts in this collection may not be entirely accurate.
They are for reference only.