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Home / Geometry / In the figure, PM⊥QR, ∠QPM = ∠RPM and ∠QPR =900.

# (c) If PQ = 6 cm. Find the measure of PR.

Solution: Given that, PM⊥QR, ∠QPM = ∠RPM and ∠QPR =900.

(a) ∠QPM + ∠RPM = ∠QPR

⟹∠QPM + ∠QPM = ∠QPR     [∵∠QPM = ∠RPM]

⟹ 2∠QPM = 900 [∵∠QPR =900]

⟹ ∠QPM = 900/2

⟹ ∠QPM = 450

## Therefore the value of ∠QPM is 450.

(b) Since PM⊥QR, so, ∠PMQ = ∠PMR = 1 right angle = 900

In right angle triangle PQM

∠PQM + ∠QPM = 900

⟹ ∠PQM + 450 = 900       [From (a)]

⟹ ∠PQM = 900 – 450

∴ ∠PQM = 450.

Similarly, in right angle triangle RPM

∠RPM + ∠PRM = 900

⟹ ∠PQM + ∠PRM = 900               [∵∠QPM = ∠RPM]

⟹ 450 + ∠PRM = 900

⟹ ∠PRM = 900 – 450

∴ ∠PRM = 450.

Therefore ∠PQM = 450 and ∠PRM = 450.

(c) Since ∠PQM = 450 and ∠PRM = 450

∴ ∠PQM = ∠PRM

⟹ ∠PQR = ∠PRQ

Now in triangle PQR

∠PQR = ∠PRQ

⟹ PR = PQ

⟹ PR = 6             [∵ PQ = 6]

Therefore PR = 6 cm.

## If two triangles have the three sides of the one equal to the three sides of the other, each to each, then they are equal in all respects.

If two triangles have the three sides of the one equal to the three sides ...