Given: ∠PQR = ∠PRQ
Prove :∠PQS=∠PRT
SQR is a line, or straight angle, so m∠SQT= 180⁰
m∠SQR= m∠PQS + m∠PQR
180⁰= m∠PQS + m∠PQR
QRT is a line, or straight angle, som∠QRT=180⁰
m∠QRT= m∠PRQ + m∠PRT
180⁰= m∠PRQ + m∠PRT
So, we have
180⁰= m∠PQS + m∠PQR and
180⁰= m∠PRQ + m∠PRT.
Left parts of these equations are equal, so right parts are equal also.
m∠PQS + m∠PQR=m∠PRQ + m∠PRT (1)
We have that ∠PQR = ∠PRQ, so m∠PQR = m∠PRQ, and we can substitute m∠PQR by m∠PRQ into (1).
m∠PQS + m∠PRQ=m∠PRQ + m∠PRT
m∠PQS + m∠PRQ -m∠PRQ =m∠PRQ -∠PRQ + m∠PRT
So,
m∠PQS = m∠PRT
If measures of angles a equal, then angles are equal (congruent) .
∠PQS = ∠PRT