Construct ΔPQR if PQ = 5 cm, m∠PQR = 105° and m∠QRP = 40°.
(Hint: Recall angle sum property of a triangle).
A rough sketch of the required ΔPQR is as follows.
In order to construct ΔPQR, the measure of ∠RPQ has to be calculated.
According to the angle sum property of triangles,
∠PQR + ∠PRQ + ∠RPQ = 180º
105º + 40º + ∠RPQ = 180º
145º + ∠RPQ = 180º
∠RPQ = 180° − 145° = 35°
The steps of construction are as follows.
(i) Draw a line segment PQ of length 5 cm.
(ii) At P, draw a ray PX making an angle of 35º with PQ.
(iii) At point Q, draw a ray QY making an angle of 105º with PQ.
(iv)Point R has to lie on both the rays, PX and QY. Therefore, R is the point of intersection of these two rays.
This is the required triangle PQR.