In figure 3.9, measures of some angles are given. Using the measures find the values of x, y, z.

Given ∠TEN = 100°, ∠EMR = 140°

∠NEM = y, ∠ENM = x, ∠NME = z

In a triangle ENM

The measure of an exterior angle of a triangle is equal to the sum of its remote interior angles

∠TEN and ∠EMR is an exterior angle of triangle ENM

So from theorem of remote interior angles,

∠TEN = ∠ NME + ∠ENM

⇒ 100° = z + x ……. (1)

∠EMR = ∠NEM + ∠ENM

⇒ 140° = x + y

⇒ x = 140° - y …(2)

In a triangle we know sum of interior angles is 180°

∴ x + y + z = 180 ……….(3)

Putting (1) in (3)

⇒ y + 100° = 180°

⇒ y = 180° - 100° = 80°

Putting y in (2)

∴ x = 140° - 80°

⇒ x = 60°

Putting x in (1)

∴ 60° + z = 100°

⇒ z = 100° - 60°

⇒ z = 40°

Measure of all the angles are

x = 60°, y = 80°, z = 40°