Mathematics

Semester 2 Research Project: Proofs

Student number:

Overview: The purpose of this project is to prove a few geometric theorems. The project is

divided into two activities, each requiring one proof. The proofs will relate to topics that you’ll

cover in future chapters. The first proof will be a three-part, two-column proof. The next will be

a 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 required

proof.

Instructions: To complete the project, you’ll fill in the text boxes (for example, ) with

your answers. This file is set up as a reader-enabled form. This means you can only enter

content into the required fields. To navigate through the file, hit tab or click in the text boxes to

enter your answers. Hitting tab will take you to each of the fields you need to complete for the

project. Often, before entering your answers in the text boxes, you’ll need to do some work on

scratch paper.

Once you have filled in all your answers, choose Save As from the File menu. Include your

student number in the file name before you upload your assignment to Penn Foster. For

example, the file you downloaded the file named student-number_0236B12S.pdf. When the

window 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: MA02B01

Assignment number: 0236B12S

Page 1 of 4

Activity 1: Proof of the SSS Similarity Theorem

Theorem 8.3.2: If the three sides in one triangle are proportional to the three sides in another

triangle, then the triangles are similar.

Setup: On scratch paper, draw two triangles with one larger than the other and the sides of one

triangle proportional to the other. Label the larger triangle ABC and the smaller triangle DEF so

that

Given: The sides of triangle ABC are proportional to the sides of triangle DEF so that

Prove: Triangle ABC is similar to triangle DEF.

Research: In your online textbook, study Chapter 8 to understand properties of similarity. If

necessary, review reasoning and proof in Chapter 2, properties of parallel lines in Chapter 3, and

triangle 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.

Statements

1. Segment GH is parallel to segment BC.

2. Segment AB and AC are to

segments 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.

Reasons

1. By

2. Definition of a

3. Angles Postulate

4. AA Property

Page 2 of 4

Proof:

Part 1: Construct segment GH in triangle ABC so that G is between A and B, AG = DE, and

segment GH is parallel to segment BC. (Hint: You should actually do this on your setup

figure.) Show that triangle AGH is similar to triangle ABC.

Part 2: Show that triangle AGH is congruent to triangle DEF.

Page 3 of 4

Statements

1. Triangle AGH is to triangle ABC.

2. The sides of triangle AGH are

to the sides of triangle ABC.

3. The sides of triangle ABC are proportional to

the sides of triangle DEF.

4. The sides of triangle AGH are

to the sides of triangle DEF.

5.

6. AG = DE

7. GH = EF and HA = FD

8. Triangle AGH is to triangle DEF.

Reasons

1. Result from Part 1

2. Polygon Postulate

3.

4. Property

5. Definition of sides

6. By

7. Transitive Property

8. Congruency Postulate

Part 3: Show the required result.

Statements

1. Triangle AGH is to triangle DEF.

2. Angle AGH is to angle DEF, and

angle GAH is congruent to angle EDF.

3. Angle AGH is congruent to angle ABC, and

angle AHG is to angle ACB.

4. Angle ABC is to angle DEF, and

angle ACD is congruent to angle EDF.

5. Triangle ABC is to triangle DEF.

Reasons

1. Result from Part 2

2.

3. Repeat of statement shown in Part 1

4. Property

5. AA Property

Note: You can prove the SAS Similarity Theorem in like fashion.

Activity 2: Proof of the Converse of the Chords and Arcs Theorem

Theorem 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 construct

congruent 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 and

circles in general. If necessary, review reasoning and proof in Chapter 2 and triangle congruence

in Chapter 4. To complete this proof, you may use any postulate or theorem on or before 568 in

your online textbook.

Proof: Since QR and NO are , angle QPR is congruent to angle NMO by the

 of 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 to

 MO.

Therefore, triangle PQR is to triangle MNO by . Consequently, segment QR

is congruent to segment NO by , which proves that, in a

 or in congruent circles, the of congruent are congruent.