**Exercise 9.1**

**Question 1: In a ΔABC, if ∠A = 55°, ∠B = 40°, find ∠C.**

**Solution:**

**Question 2: If the angles of a triangle are in the ratio 1:2:3, determine three angles.**

**Solution:**

**Question 3: The angles of a triangle are (x − 40)°, (x − 20)° and (1/2 x-10°). Find the value of x.**

**Solution:**

**Question 4: The angles of a triangle are arranged in ascending order of magnitude. If the difference between two consecutive angles is 10°, find the three angles.**

**Solution:**

**Question 5: Two angles of a triangle are equal and the third angle is greater than each of those angles by 30°. Determine all the angles of the triangle.**

**Solution:**

**Question 6: If one angle of a triangle is equal to the sum of the other two, show that the triangle is a right angle triangle.**

**Solution:**

**Question 7: ABC is a triangle in which ∠A = 72°, the internal bisectors of angles B and C meet in O. Find the magnitude of ∠BOC.**

**Solution:**

**Question 8: The bisectors of base angles of a triangle cannot enclose a right angle in any case.**

**Solution:**

**Question: 9 If the bisectors of the base angles of a triangle enclose an angle of 135°, prove that the triangle is a right angle.**

**Solution:**

**Question 10: In a ΔABC, ∠ABC = ∠ACB and the bisectors of ∠ABC and ∠ACB intersect at O such that ∠BOC = 120°. Show that ∠A = ∠B = ∠C = 60°.**

**Solution:**

**Question 11: Can a triangle have:-**

**(i) Two right angles?**

**(ii) Two obtuse angles?**

**(iii) Two acute angles?**

**(iv) All angles more than 60°?**

**(v) All angles less than 60°?**

**(vi) All angles equal to 60°?**

**Justify your answer in each case.**

**Solution:**

**Question 12: If each angle of a triangle is less than the sum of the other two, show that the triangle is acute angled.**

**Solution:**

**Exercise 9.2**

**Question 1: The exterior angles, obtained on producing the base of a triangle both ways are 104° and 136°. Find all the angles of the triangle.**

**Solution:**

**Question 2: 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°.**

**Solution:**

**Question 3: In figure, the sides BC, CA and AB of a △ABC have been produced to D, E and F respectively. If ∠ACD = 105° and ∠EAF = 45°, find all the angles of the △ABC.**

**Solution:**

**Question 4: Compute the value of x in each of the following figures:**

**(i)**

**Solution:**

**Solution:**

**Solution:**

**Solution:**

**Question 5: In figure, AB divides ∠DAC in the ratio 1 : 3 and AB = DB. Determine the value of x.**

**Solution:**

**Question 6: 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.**

**Solution:**

**Question 7: In figure 9.36, AC ⊥ CE and ∠A:∠B: ∠C = 3: 2: 1, find**

**Solution:**

**Question 8: In figure 9.37, AM ⊥ BC and AN is the bisector of ∠A. If ∠B = 65° and ∠C = 33°, find ∠MAN.**

**Solution:**

**Question 9: In a triangle ABC, AD bisects ∠A and ∠C > ∠B. Prove that ∠ADB > ∠ADC.**

**Solution:**

**Question 10: In triangle ABC, BD ⊥ AC and CE ⊥ AB. If BD and CE intersect at O, prove that ∠BOC = 180° - ∠A.**

**Solution:**

**Question 11: In figure 9.38, AE bisects ∠CAD and ∠B = ∠C. Prove that AE ∥ BC.**

**Solution:**

**Question 12: In figure 9.39, AB ∥ DE. Find ∠ACD.**

**Solution:**

**Question 13; 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.**

**Solution:**

**Question 14 : 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.**

**Solution:**

**Exercise VSAQs ........................**

**Question 1: Define a triangle.**

**Solution:**

**Question 2: Write the sum of the angles of an obtuse triangle.**

**Solution:**

**Question 3: In △ABC, if ∠B = 60°, ∠C = 800 and the bisectors of angles ∠ABC and ∠ACB meet at point O, then find the measure of ∠BOC.**

**Solution:**

**Question 4: If the angles of a triangle are in the ratio 2:1:3, then find the measure of smallest angle.**

**Solution:**

**Question 5: If the angles A, B and C of △ABC satisfy the relation B – A = C – B, then find the measure of ∠B.**

**Solution:**