Page 178 - ICSE Math 7

• As all the three sides of an equilateral triangle are equal, all the three A
angles are also equal and each angle is of magnitude 60°. 60°
• As all the three sides of a scalene triangle are unequal, all the angles of a
scalene triangle are also unequal. 60° 60°
• If all the angles of a triangle are equal, then it must be an equilateral triangle. B C

Property 3: In any triangle, the sum of any two sides is always greater than the third side.

Example 1: In a DPQR, ∠P = 50° and ∠Q = 55°. Find ∠R.
Solution: As the sum of all the angles of a triangle is 180°, therefore
∠P + ∠Q + ∠R = 180°
⇒ 50° + 55° + ∠R = 180°
⇒ ∠R = 180° – 105° = 75°

Example 2: Find angle x in each of the following.
P C
A 35°
46°

(a) (b) 60° E

Q R x D
x B
S T F

Solution: (a) In DPQR, PQ = PR
\ ∠PQR = ∠PRQ
Also, ∠PQR + ∠PRQ + ∠RPQ = 180° (Sum of the angles of a triangle)
⇒ ∠PQR + ∠PQR + 46° = 180°
⇒ 2∠PQR = 180° – 46° = 134°
134°
⇒ ∠PQR = = 67°
2
∠PQR + ∠RQS = 180° (Linear pair)
⇒ 67° + x = 180°
⇒ x = 180° – 67° = 113°
or, ∠RQS = ∠QPR + ∠PRQ (Exterior angle is equal to the sum of
= 113° its two interior opposite angles)
⇒ x = 46° + ∠PQR
⇒ x = 46° + 67° = 113°
(b) ∠AEB = ∠CED (Vertically opposite angles)
⇒ ∠CED = 60°
∠EDF = ∠CED + ∠DCE (Exterior angle is equal to the sum of
its two interior opposite angles)
⇒ x = 60° + 35° = 95°

Try This P

In an isosceles DPQR, PQ = PR and the bisectors of 86°
∠Q and ∠R intersect at S. Find the angle x. S
x
Q R

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