Your teacher will grade your responses to questions 6–9 to ensure that you receive proper credit for your answers.
Supply the missing reasons to complete the proof.
Given: angleQ congruent to angleT and Line segment Q Rcongruent to segment T R
Prove: Segment P R is congruent to Segment S R
An incomplete 2 column proof is shown with a figure.
Statement Reason
1. angle Q congruent to angle T Given
2. Q R congruent to T R Given
3. Angle Q R T congruent to angle T R Q Definition of congruent angles
4. Segment P R congruent to segment S R ??? (missing reason)
It seems that I cannot see the figure you mentioned. However, I can provide you with the missing reasons to complete the proof based on the given information.
Here is the completed 2-column proof:
Statements Reasons
-------------------------------------------------------------
1. angle Q congruent to angle T Given
2. Line segment Q R congruent to segment T R Given
3. angle Q congruent to angle P Corresponding angles
4. triangle P R Q congruent to triangle R Q T SAS (Side-Angle-Side)
5. angle R congruent to angle R Reflexive property
6. segment R P congruent to segment R S Corresponding parts of congruent triangles are congruent (CPCTC)
7. segment P R congruent to segment S R Symmetry of equality
Therefore, segment P R is congruent to segment S R.
To complete the proof, we can use the following reasoning:
1. Given: angle Q congruent to angle T and Line segment QR congruent to segment TR
2. The angles Q and T are congruent, which means they have the same measure.
- Reason: Given
3. The line segment QR is congruent to segment TR, which means they have the same length.
- Reason: Given
4. Let's consider triangle PQR and triangle STR.
- Reason: Chosen to compare segments PR and SR
5. By definition, triangles PQR and STR have angle Q congruent to angle T.
- Reason: Corresponding parts of congruent triangles are congruent (CPCTC)
6. By definition, triangles PQR and STR have QR congruent to TR.
- Reason: Corresponding parts of congruent triangles are congruent (CPCTC)
7. By definition, triangles PQR and STR are congruent.
- Reason: If two triangles have two angles congruent and a side between these angles congruent, the triangles are congruent (ASA Congruence)
8. Since triangles PQR and STR are congruent, their corresponding sides are congruent.
- Reason: Corresponding parts of congruent triangles are congruent (CPCTC)
9. Therefore, segment PR is congruent to segment SR.
- Reason: Corresponding sides of congruent triangles are congruent (CPCTC)
By completing the proof with the reasons given above, we have proven that segment PR is congruent to segment SR