**Exercise 10.1**

**Question 1: In figure, the sides BA and CA have been produced such that BA = AD and CA = AE. Prove that segment DE ∥ BC.**

**Solution:**

**Question 2: In a PQR, if PQ = QR and L, M and N are the mid-points of the sides PQ, QR and RP respectively. Prove that LN = MN.**

**Solution:**

**Question 3: In figure, PQRS is a square and SRT is an equilateral triangle. Prove that**

**(i) PT = QT (ii) ∠ TQR = 15°**

**Solution:**

**Question 4: Prove that the medians of an equilateral triangle are equal.**

**Solution:**

**Question 5: In a Δ ABC, if ∠A = 120° and AB = AC. Find ∠B and ∠C.**

**Solution:**

**Question 6: In a Δ ABC, if AB = AC and ∠ B = 70°, find ∠ A.**

**Solution:**

**Question 7: The vertical angle of an isosceles triangle is 100°. Find its base angles.**

**Solution:**

**Question 8: In a figure AB = AC and ∠ACD = 105°. Find ∠BAC.**

**Solution:**

**Question: 9 Find the measure of each exterior angle of an equilateral triangle.**

**Solution:**

**Question: 10 If the base of an isosceles triangle is produced on both sides, prove that the exterior angles so formed are equal to each other.**

**Solution:**

**Question 11: In Figure AB = AC and DB = DC, find the ratio ∠ABD : ∠ACD.**

**Solution:**

**Question: 12 Determine the measure of each of the equal angles of a right-angled isosceles triangle.**

**OR**

**ABC is a right-angled triangle in which ∠A = 90° and AB = AC. Find ∠B and ∠C.**

**Solution:**

**Question: 13 AB is a line segment. P and Q are points on opposite sides of AB such that each of them is equidistant from the points A and B (See Fig. (10.26). Show that the line PQ is perpendicular bisector of AB.**

**Solution:**

**Exercise 10.2**

**Question 1: In figure, it is given that RT = TS, ∠ 1 = 2∠2 and ∠4 = 2∠3. Prove that ΔRBT ≅ ΔSAT.**

**Solution:**

**Question 2: Two lines AB and CD intersect at O such that BC is equal and parallel to AD. Prove that the lines AB and CD bisect at O.**

**Solution:**

**Question 3: BD and CE are bisectors of ∠B and ∠C of an isosceles Δ ABC with AB = AC. Prove that BD = CE.**

**Solution:**

**Exercise 10.3**

**Question 1: In two right triangles one side an acute angle of one are equal to the corresponding side and angle of the other. Prove that the triangles are congruent.**

**Solution:**

**Question 2: If the bisector of the exterior vertical angle of a triangle be parallel to the base. Show that the triangle is isosceles.**

**Solution:**

**Question 3: In an isosceles triangle, if the vertex angle is twice the sum of the base angles, calculate the angles of the triangle.**

**Solution:**

**Question 4: PQR is a triangle in which PQ = PR and is any point on the side PQ. Through S, a line is drawn parallel to QR and intersecting PR at T. Prove that PS = PT.**

**Solution:**

**Exercise 10.4**

**Question 1: In figure, It is given that AB = CD and AD = BC. Prove that ΔADC ≅ ΔCBA.**

**Solution:**

**Question 2: In a Δ PQR, if PQ = QR and L, M and N are the mid-points of the sides PQ, QR and RP respectively. Prove that LN = MN.**

**Solution:**

**Exercise 10.5**

**Question 1: ABC is a triangle and D is the mid-point of BC. The perpendiculars from D to AB and AC are equal. Prove that the triangle is isosceles.**

**Solution:**

**Question 2: ABC is a triangle in which BE and CF are, respectively, the perpendiculars to the sides AC and AB. If BE = CF, prove that Δ ABC is isosceles**

**Solution:**

**Question 3: If perpendiculars from any point within an angle on its arms are congruent. Prove that it lies on the bisector of that angle.**

**Solution:**

**Question 4: In figure, AD ⊥ CD and CB ⊥ CD. If AQ = BP and DP = CQ, prove that ∠DAQ = ∠CBP.**

**AD ⊥ CD and CB ⊥ CD.**

**Solution:**

**Question 5: ABCD is a square, X and Y are points on sides AD and BC respectively such that AY = BX. Prove that BY = AX and ∠BAY = ∠ABX.**

**Solution:**

**Question 6: Which of the following statements are true (T) and which are false (F):**

**(i) Sides opposite to equal angles of a triangle may be unequal.**

**(ii) Angles opposite to equal sides of a triangle are equal**

**(iii) The measure of each angle of an equilateral triangle is 60**

**(iv) If the altitude from one vertex of a triangle bisects the opposite side, then the triangle may be isosceles.**

**(v) The bisectors of two equal angles of a triangle are equal.**

**(vi) If the bisector of the vertical angle of a triangle bisects the base, then the triangle may be isosceles.**

**(vii) The two altitudes corresponding to two equal sides of a triangle need not be equal.**

**(viii) If any two sides of a right triangle are respectively equal to two sides of other right triangle, then the two triangles are congruent.**

**(ix) Two right-angled triangles are congruent if hypotenuse and a side of one triangle are respectively equal to the hypotenuse and a side of the other triangle.**

**Solution:**

**Question 7: Fill the blanks.**

**In the following so that each of the following statements is true.**

**(i) Sides opposite to equal angles of a triangle are _________**

**(ii) Angle opposite to equal sides of a triangle are _________**

**(iii) In an equilateral triangle all angles are _________**

**(iv) In ΔABC, if ∠A = ∠C, then AB = _________**

**(v) If altitudes CE and BF of a triangle ABC are equal, then AB _________**

**(vi) In an isosceles triangle ABC with AB = AC, if BD and CE are its altitudes, then BD is _________ CE.**

**(vii) In right triangles ABC and DEF, if hypotenuse AB = EF and side AC = DE, then ΔABC ≅ Δ _________**

**Solution:**

**Exercise 10.6**

**Question 1: In Δ ABC, if ∠A = 40° and ∠B = 60°. Determine the longest and shortest sides of the triangle.**

**Solution:**

**Question 2: In a Δ ABC, if ∠ B = ∠ C = 45°, which is the longest side?**

**Solution:**

**Question 3: In Δ ABC, side AB is produced to D so that BD = BC. If ∠ B = 60° and ∠ A = 70°.**

**Prove that: (i) AD > CD (ii) AD > AC**

**Solution:**

**Question 4: Is it possible to draw a triangle with sides of length 2 cm, 3 cm and 7 cm?**

**Solution:**

**Question 5: O is any point in the interior of ΔABC. Prove that**

**(i) AB + AC > OB + OC**

**(ii) AB + BC + CA > OA + QB + OC**

**(iii) OA + OB + OC > (1/2) (AB + BC +CA)**

**Solution:**

**Question 6: Prove that the perimeter of a triangle is greater than the sum of its altitudes.**

**Solution:**

**Question 7: In Fig., prove that:**

**(i) CD + DA + AB + BC > 2AC**

**(ii) CD + DA + AB > BC**

**Solution:**

**Question 8: Which of the following statements are true (T) and which are false (F)?**

**(i) Sum of the three sides of a triangle is less than the sum of its three altitudes.**

**(ii) Sum of any two sides of a triangle is greater than twice the median drown to the third side**

**(iii) Sum of any two sides of a triangle is greater than the third side.**

**(iv) Difference of any two sides of a triangle is equal to the third side.**

**(v) If two angles of a triangle are unequal, then the greater angle has the larger side opposite to it**

**(vi) Of all the line segments that can be drawn from a point to a line not containing it, the perpendicular line segment is the shortest one.**

**Solution:**

**Question 9: Fill in the blanks to make the following statements true.**

**(i) In a right triangle the hypotenuse is the ___ side.**

**(ii) The sum of three altitudes of a triangle is ___ than its perimeter.**

**(iii) The sum of any two sides of a triangle is ___ than the third side.**

**(iv) If two angles of a triangle are unequal, then the smaller angle has the ___ side opposite to it.**

**(v) Difference of any two sides of a triangle is ___ than the third side.**

**(vi) If two sides of a triangle are unequal, then the larger side has ___ angle opposite to it.**

**Solution:**

**Exercise VSAQs**

**Question 1: In two congruent triangles ABC and DEF, if AB = DE and BC = EF. Name the pairs of equal angles.**

**Solution:**

**Question 2: In two triangles ABC and DEF, it is given that ∠A = ∠D, ∠B = ∠ E and ∠ C = ∠F. Are the two triangles necessarily congruent?**

**Solution:**

**Question 3: If ABC and DEF are two triangles such that AC = 2.5 cm, BC = 5 cm, C = 75°, DE = 2.5 cm, DF = 5 cm and D = 75°. Are two triangles congruent?**

**Solution:**

**Question 4: In two triangles ABC and ADC, if AB = AD and BC = CD. Are they congruent?**

**Solution:**

**Question 5: In triangles ABC and CDE, if AC = CE, BC = CD, ∠A = 60°, ∠C = 30° and ∠D = 90°. Are two triangles congruent?**

**Solution:**

**Question 6: ABC is an isosceles triangle in which AB = AC. BE and CF are its two medians. Show that BE = CF.**

**Solution:**