mathematics

Semester 2 Research Project: ProofsStudent number:Overview: The purpose of this project is to prove a few geometric theorems. The project isdivided into two activities, each requiring one proof. The proofs will relate to topics that you’llcover in future chapters. The first proof will be a three-part, two-column proof. The next will bea paragraph proof.Your online textbook will be an invaluable reference for this project. In each activity,the research section will identify the portion of your textbook most applicable to the requiredproof.Instructions: To complete the project, you’ll fill in the text boxes (for example,????? ) withyour answers. This file is set up as a reader-enabled form. This means you can only entercontent into the required fields. To navigate through the file, hit tab or click in the text boxes toenter your answers. Hitting tab will take you to each of the fields you need to complete for theproject. Often, before entering your answers in the text boxes, you’ll need to do some work onscratch paper.Once you have filled in all your answers, choose “Save As” from the File menu. Include yourstudent number in the file name before you upload your assignment to Penn Foster. Forexample, the file you downloaded the file named student-number_0236B12S.pdf. When thewindow appears to “Save As,” include your student number in the file name(12345678_0236B12S.pdf), where 12345678 is your eight-digit student number).Course title and number: MA02B01Assignment number: 0236B12SPage 1 of 4Activity 1: Proof of the SSS Similarity TheoremTheorem 8.3.2: If the three sides in one triangle are proportional to the three sides in anothertriangle, then the triangles are similar.Setup: On scratch paper, draw two triangles with one larger than the other and the sides of onetriangle proportional to the other. Label the larger triangle ABC and the smaller triangle DEF sothatGiven: The sides of triangle ABC are proportional to the sides of triangle DEF so thatProve: Triangle ABC is similar to triangle DEF.Research: In your online textbook, study Chapter 8 to understand properties of similarity. Ifnecessary, review reasoning and proof in Chapter 2, properties of parallel lines in Chapter 3, andtriangle congruence in Chapter 4. To complete this proof, you may use any definition, postulate,or theorem in your online textbook on or before page 517.Statements1. Segment GH is parallel to segment BC.2. Segment AB and AC are tosegments GH and BC.3. Angle AGH is to angle ABC,and angle AHG is congruent to angle ACB.4. Triangle AGH is to triangle ABC.Reasons1. By2. Definition of a3. Angles Postulate4. AA PropertyPage 2 of 4Proof:Part 1: Construct segment GH in triangle ABC so that G is between A and B, AG = DE, andsegment GH is parallel to segment BC. (Hint: You should actually do this on your setupfigure.) Show that triangle AGH is similar to triangle ABC.Part 2: Show that triangle AGH is congruent to triangle DEF.Page 3 of 4Statements1. Triangle AGH is to triangle ABC.2. The sides of triangle AGH areto the sides of triangle ABC.3. The sides of triangle ABC are proportional tothe sides of triangle DEF.4. The sides of triangle AGH areto the sides of triangle DEF.5.6. AG = DE7. GH = EF and HA = FD8. Triangle AGH is to triangle DEF.Reasons1. Result from Part 12. Polygon Postulate3.4. Property5. Definition of sides6. By7. Transitive Property8. Congruency PostulatePart 3: Show the required result.Statements1. Triangle AGH is to triangle DEF.2. Angle AGH is to angle DEF, andangle GAH is congruent to angle EDF.3. Angle AGH is congruent to angle ABC, andangle AHG is to angle ACB.4. Angle ABC is to angle DEF, andangle ACD is congruent to angle EDF.5. Triangle ABC is to triangle DEF.Reasons1. Result from Part 22.3. Repeat of statement shown in Part 14. Property5. AA PropertyNote: You can prove the SAS Similarity Theorem in like fashion.Activity 2: Proof of the Converse of the Chords and Arcs TheoremTheorem 9.1.6: In a circle or in congruent circles, the chords of congruent arcs are congruent.Setup: On scratch paper, construct congruent circles with centers at P and M. Then constructcongruent arcs QR on circle P and NO on circle M. Finally, draw triangles PQR and MNO.Research: In your online textbook, study Chapter 9 to understand the properties of arcs andcircles in general. If necessary, review reasoning and proof in Chapter 2 and triangle congruencein Chapter 4. To complete this proof, you may use any postulate or theorem on or before 568 inyour online textbook.Proof: Since QR and NO are , angle QPR is congruent to angle NMO by theof the degree measure of arcs.Segments PQ, PR, MN, and MO are all of congruent circles, so they are all .In particular, segment PQ is to segment MN and segment PR is congruent toMO.Therefore, triangle PQR is to triangle MNO by . Consequently, segment QRis congruent to segment NO by , which proves that, in aor in congruent circles, the of congruent are congruent.