Page 4: Advanced Applications of Segment Proofs
This page presents more complex applications of segment proofs, incorporating various geometric properties and postulates.
The tenth proof demonstrates how to prove that Q is the midpoint of PR given that 2PQ = PR. This proof uses the segment addition postulate and the subtraction property to establish that PQ = QR, which defines Q as the midpoint of PR.
The eleventh proof shows how to prove that AD = CE given that AB ≅ CD and BD ≅ DE. This proof combines the segment addition postulate with the transitive property to establish the equality of AD and CE.
Example: In the proof AB + BD = AD, CD + DE = CE, the segment addition postulate is applied twice to show that the sum of two adjacent segments is equal to the whole segment on both sides of the equation.
The twelfth proof demonstrates how to prove that HI ≅ JK given that GI ≅ JL and GH ≅ KL. This proof employs the segment addition property, substitution property, and subtraction property of equality to establish the congruence of HI and JK.
Vocabulary: The subtraction property of equality states that if you subtract the same quantity from both sides of an equation, the equation remains true.
These advanced proofs reinforce the importance of logical reasoning and the application of multiple geometric properties in solving complex segment problems.