Many researchers have argued that proving is essential to doing and knowing mathematics because it is the basis of mathematical understanding (Cirillo & Herbst, 2012; Hanna & Jahnke, 1996). However, research studies conducted on students' performance with constructing proofs has found that the majority of high school geometry students are unable to construct valid proofs (McCrone & Martin, 2004; Senk, 1985). The present study contributes to this literature and focuses on the formal proofs that use triangle congruence postulates, which students construct in high school geometry. This qualitative study utilizes a psychological constructivist perspective to investigate how students construct and reason about geometric proofs. The following research questions are answered. What are the different ways that students think and reason while attempting formal geometry proofs that use triangle congruence postulates? How can the information gained from answering the first research question be used to start developing a first draft of a learning progression for geometry proofs? The study presents detailed descriptions of the cognitive processes that students use to construct and reason about geometric proofs as well as describes some of the errors and difficulties students exhibit when doing proofs. The data was collected by administering a series of one-on-one semi-structured task-based interviews to six high school geometry students who were asked to complete a series of proof problems. Students were interviewed for four to five one-hour sessions in which they "thought aloud" as they worked on twelve proof problems. All interviews were video recorded and later transcribed. Data analysis methods implemented were the constant comparative method and retrospective analysis. Findings suggest that most of the proofs that students wrote were not formally correct, but that many students wrote proofs that were not reflective of the often sound proof reasoning they stated verbally during the interviews. Thus it seems like several of the students had developed sound intuitive ideas for proofs, but that they could not write proofs that were rigorous enough to stand up to axiomatic-based scrutiny. Furthermore, the results suggest that classifying students' written proofs as either correct or incorrect did not seem to capture the nuances of the intuitive ideas students verbalized in their proof plans. Consequently, a new system was developed for classifying students' proof reasoning that considered both what students stated during their plans and what they actually wrote in their formal proofs. The present study also describes several facets of the use of diagrams in students' reasoning and construction of geometric proofs. It describes how students used diagrams to aid them in forming their deductions for proof as well as describes the difficulties students had with locating relevant embedded/overlapping figures needed to complete their proofs. It also provides new ways to categorize, analyze, and investigate students' proof reasoning. In fact, insights derived in the data analysis was used to develop a tentative first draft of a learning progression of students' reasoning with geometric proofs, which extends Battista's (2012) learning progression of geometric reasoning into Level 4 or formal deductive proofs. [The dissertation citations contained here are published with the permission of ProQuest LLC. Further reproduction is prohibited without permission. Copies of dissertations may be obtained by Telephone (800) 1-800-521-0600. Web page: http://bibliotheek.ehb.be:2222/en-US/products/dissertations/individuals.shtml.]