In the diagram, if $\angle PQR = 48^\circ$ , what is the measure of $\angle PMN$?
Since QR = PR , △PQR is isoscelese and its base angles are congruent. So...
∠PQR = ∠RPQ = 48°
And vertical angles are congruent, so...
∠RPQ = ∠MPN = 48°
Since MP = NP , △MNP is isoscelese and its base angles are congruent. So...
∠PNM = ∠PMN
Since there are 180° in every triangle...
∠MPN + ∠PMN + ∠PNM = 180°
Plug in 48° for ∠MPN and plug in ∠PMN for ∠PNM
48° + ∠PMN + ∠PMN = 180°
Combine like terms.
48° + 2(∠PMN) = 180°
Subtract 48° from both sides.
2(∠PMN) = 132°
Divide both sides by 2 .
∠PMN = 66°