Question

ps bisects ∠qsu and ∠rpt. complete the proof that rs ≅ st.
1 ps bisects ∠qsu
2 ps bisects ∠rpt
3 ∠tsu ≅ ∠qsr
4 ∠psu ≅ ∠psq
5 ∠spt ≅ ∠rps
6 m∠pst = m∠psu + m∠tsu
7 m∠psr = m∠psq + m∠qsr
8 m∠pst = m∠psq + m∠qsr
9 m∠psr = m∠pst
10 ps ≅ ps
11 △pst ≅ △psr
12 rs ≅ st

Explanation:

Step1: Given

Statement 1: Given
Statement 2: Given

Step2: Vertical angles are congruent

Statement 3: Vertical angles are congruent

Step3: Definition of angle bisector

Statement 4: Definition of angle bisector
Statement 5: Definition of angle bisector

Step4: Angle - addition postulate

Statement 6: Angle - addition postulate
Statement 7: Angle - addition postulate

Step5: Substitution property

Statement 8: Substitution property ($m\angle PSU + m\angle TSU=m\angle PSQ + m\angle QSR$)

Step6: Substitution property

Statement 9: Substitution property (From statements 7 and 8)

Step7: Reflexive property

Statement 10: Reflexive property of congruence

Step8: ASA (Angle - Side - Angle)

Statement 11: $\triangle PST\cong\triangle PSR$ by ASA ($\angle SPT\cong\angle RPS$, $PS\cong PS$, $\angle PST\cong\angle PSR$)

Step9: CPCTC (Corresponding Parts of Congruent Triangles are Congruent)

Statement 12: $\overline{RS}\cong\overline{ST}$ because of CPCTC

Answer:

1. Given
2. Given
3. Vertical angles are congruent
4. Definition of angle bisector
5. Definition of angle bisector
6. Angle - addition postulate
7. Angle - addition postulate
8. Substitution property
9. Substitution property
10. Reflexive property of congruence
11. ASA (Angle - Side - Angle)
12. CPCTC (Corresponding Parts of Congruent Triangles are Congruent)