Let's begin by highlighting the given information in the diagram.
By applying the Isosceles Triangle Theorem we obtain that ∠PRQ ≅ ∠PQR, which implies that m∠PRQ = m∠PQR. Also, by the Angle Addition Postulate we can write the following two equations.
m∠PRQ = m ∠1 + m ∠4
m∠PQR = m ∠2 + m ∠3Since m∠PRQ = m∠PQR we can equate both equations above.
m ∠1 + m ∠4 = m ∠2 + m ∠3
Remember that SQ > SR, which implies that m ∠4 > m ∠3 (Theorem 5.9). By the definition of inequality, we can write the equation below.
m ∠4 = m ∠3 + x , for somex > 0
Next, we will subtract the latter two equations.
m ∠1 + m ∠4 &= m ∠2 + m ∠3
^- m ∠4 &= m ∠3 + x
m ∠1 &= m ∠2 - x
From the resulting equation we obtain that m ∠1 + x = m ∠2, and by definition of inequality it means that m ∠1 < m ∠2.
Two-Column Proof Table
In the table below, we will summarize the proof we just did before.
