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The exterior angles, obtained on producing the base of a triangle both ways are 104° and 136°. Find all the angles of the triangle.
∠ACD = ∠ABC + ∠BAC [Exterior angle property]
Now ∠ABC = 180° − 136° = 44° [Linera pair]
∠ACB = 180° − 104° = 76° [Linera pair]
Now, In ΔABC
∠A + ∠ABC + ∠ACB = 180° [Sum of all angles of a triangle]
⇒ ∠A + 44° + 76° = 180°
⇒ ∠A = 180° − 44° − 76°
⇒ ∠A = 60°
In a triangle ABC, the internal bisectors of ∠B and ∠C meet at P and the external bisectors of ∠B and ∠C meet at Q. Prove that ∠BPC + ∠BQC = 180°.
Let ∠ABD = 2x and ∠ACE = 2y
∠ABC = 180° − 2x [Linera pair]
∠ACB = 180° − 2y [Linera pair]
⇒ ∠A + 180° − 2x + 180° − 2y = 180°
⇒ −∠A + 2x + 2y = 180°
Now in ΔBQC
x + y + ∠BQC = 180° [Sum of all angles of a triangle]
Adding (i) and (ii) we get ∠BPC + ∠BQC = 180°
Hence proved.
In figure 9.30, the sides BC, CA and AB of a triangle ABC have been produced to D, E and F respectively. If ∠ACD = 105° and ∠EAF = 45°, find all the angles of the triangle ABC.
∠BAC = ∠EAF = 45° [Vertically opposite angles]
∠ABC = 105° − 45° = 60° [Exterior angle property]
∠ACD = 180° − 105° = 75° [Linear pair]
Compute the value of x in each of the following figures:
(i) ∠BAC = 180° − 120° = 60° [Linear pair]
∠ACB = 180° − 112° = 68° [Linear pair]
∴ x = 180° − ∠BAC − ∠ACB
= 180° − 60° − 68° = 52° [Sum of all angles of a triangle]
(ii) ∠ABC = 180° − 120° = 60° [Linear pair]
∠ACB = 180° − 110° = 70° [Linear pair]
∴ e∠BAC = x
= 180° − ∠ABC − ∠ACB = 180° − 60° − 70° = 50° [Sum of all angles of a triangle]
(iii) ∠BAE = ∠EDC = 52° [Alternate angles]
∴ ∠DEC = x = 180° − 40° − ∠EDC
= 180° − 40° − 52°
= 180° − 92° = 88° [Sum of all angles of a triangle]
(iv) CD is produced to meet AB at E.
∠BEC = 180° − 45° − 50° = 85° [Sum of all angles of a triangle]
∠AEC = 180° − 85° = 95° [Linear pair]
∴ x = 95° + 35° = 130° [Exterior angle property].
In figure 9.35, AB divides ∠DAC in the ratio 1: 3 and AB = DB. Determine the value of x.
Let ∠BAD = Z, ∠BAC = 3Z
⇒ ∠BDA = ∠BAD = Z (∵ AB = DB)
Now ∠BAD + ∠BAC + 108° = 180° [Linear pair]
⇒ Z + 3Z + 108° =180°
⇒ 4Z = 72°
⇒ Z = 18°
Now, In ΔADC
∠ADC + ∠ACD = 108° [Exterior angle property]
⇒ x + 18° = 180°
⇒ x = 90°
ABC is a triangle. The bisector of the exterior angle at B and the bisector of ∠C intersect each other at D. Prove that ∠D = 1/2∠A.
Let ∠ABE = 2x and ∠ACB = 2y
∠ABC = 180° − 2x [Linear pair]
∴ ∠A = 180° − ∠ABC − ∠ACB [Angle sum property]
= 180° − 180° + 2x + 2y
= 2(x − y) ..... (i)
Now, ∠D = 180° − ∠DBC − ∠DCB
⇒ ∠D = 180° − (x + 180° − 2x) − y
⇒ ∠D = 180° − x − 180° + 2x − y
= (x − y)
In figure 9.36, AC ⊥ CE and ∠A:∠B: ∠C = 3: 2: 1, find
∠A: ∠B: ∠C = 3: 2: 1
Let the angles be 3x, 2x and x
⇒ 3x + 2x + x = 180° [Angle sum property]
⇒ 6x = 180°
⇒ x = 30° = ∠ACB
∴ ∠ECD = 180° − ∠ACB − 90° [Linear pair]
= 180° − 30° − 90°
= 60°
∴ ∠ECD = 60°
In figure 9.37, AM ⊥ BC and AN is the bisector of ∠A. If ∠B = 65° and ∠C = 33°, find ∠MAN.
Let ∠BAN = ∠NAC = x [∵ AN bisects ∠A]
∴ ∠ANM = x + 33° [Exterior angle property]
In ΔAMB
∠BAM = 90° − 65° = 25° [Exterior angle property]
∴ ∠MAN = ∠BAN − ∠BAM = (x − 25)°
Now in ΔMAN,
(x - 25)° + (x + 33)° + 90° = 180° [Angle sum property]
⇒ 2x + 8° = 90°
⇒ 2x = 82°
⇒ x = 41°
∴ MAN = x − 25°
= 41° − 25°
= 16°
In a triangle ABC, AD bisects ∠A and ∠C > ∠B. Prove that ∠ADB > ∠ADC.
∵ ∠C > ∠B [Given]
⇒ ∠C + x > ∠B + x [Adding x on both sides]
⇒ 180° - ∠ ADC > 180° - ∠ADB
⇒ - ∠ ADC > - ∠ADB
⇒ ∠ADB > ∠ADC
In triangle ABC, BD ⊥ AC and CE ⊥ AB. If BD and CE intersect at O, prove that ∠BOC = 180° - ∠A.
In quadrilateral AEOD
∠A + ∠AEO + ∠EOD + ∠ADO = 360°
⇒ ∠A + 90° + 90° + ∠EOD = 360°
⇒ ∠A + ∠BOC = 180° [∵ ∠EOD = ∠BOC vertically opposite angles]
⇒ ∠BOC = 180° − ∠A
In figure 9.38, AE bisects ∠CAD and ∠B = ∠C. Prove that AE ∥ BC.
Let ∠B = ∠C = x
Then,
∠CAD = ∠B + ∠C = 2x (exterior angle)
⇒1/2∠CAD = x
⇒ ∠EAC = x
⇒ ∠EAC = ∠C
These are alternate interior angles for the lines AE and BC
∴ AE ∥ BC
In figure 9.39, AB ∥ DE. Find ∠ACD.
Since AB ∥ DE
∴ ∠ABC = ∠CDE = 40° [Alternate angles]
∴ ∠ACB = 180° − ∠ABC − ∠BAC
=180° − 40° − 30°
= 110°
∴ ∠ACD = 180° − 110° [Linear pair]
= 70°
Which of the following statements are true (T) and which are false (F):
(i) Sum of the three angles of a triangle is 180°.
(ii) A triangle can have two right angles.
(iii) All the angles of a triangle can be less than 60°.
(iv) All the angles of a triangle can be greater than 60°.
(v) All the angles of a triangle can be equal to 60°.
(vi) A triangle can have two obtuse angles.
(vii) A triangle can have at most one obtuse angles.
(viii) If one angle of a triangle is obtuse, then it cannot be a right angled triangle.
(ix) An exterior angle of a triangle is less than either of its interior opposite angles.
(x) An exterior angle of a triangle is equal to the sum of the two interior opposite angles.
(xi) An exterior angle of a triangle is greater than the opposite interior angles.
(i) T
(ii) F
(iii) F
(iv) F
(v) T
(vi) F
(vii) T
(viii) T
(ix) F
(x) T
(xi) T
Fill in the blanks to make the following statements true:
(i) Sum of the angles of a triangle is _______ .
(ii) An exterior angle of a triangle is equal to the two ________ opposite angles.
(iii) An exterior angle of a triangle is always ________ than either of the interior opposite angles.
(iv) A triangle cannot have more than _______ right angles.
(v) A triangles cannot have more than _______ obtuse angles.
(i) 180°
(ii) Interior
(iii) Greater
(iv) One
(v) One
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Chapter 9: Triangle and its Angles Exercise...