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.