**Exercise 14.1**

**Question 1: Three angles of a quadrilateral are respectively equal to 110°, 50° and 40°. Find its fourth angle.**

**Solution:**

**Question 2: In a quadrilateral ABCD, the angles A, B, C and D are in the ratio of 1:2:4:5. Find the measure of each angles of the quadrilateral.**

**Solution:**

**Question 3: In a quadrilateral ABCD, CO and DO are the bisectors of ∠C and ∠D respectively. Prove that ∠COD = 1/2 (∠A + ∠B).**

**Solution:**

**Question 4: The angles of a quadrilateral are in the ratio 3:5:9:13. Find all the angles of the quadrilateral.**

**Solution:**

**Exercise 14.2**

**Question 1: Two opposite angles of a parallelogram are (3x – 2)° and (50 – x)°. Find the measure of each angle of the parallelogram.**

**Solution:**

**Question 2: If an angle of a parallelogram is two-third of its adjacent angle, find the angles of the parallelogram.**

**Solution:**

**Question 3: Find the measure of all the angles of a parallelogram, if one angle is 24° less than twice the smallest angle.**

**Solution:**

**Question 4: The perimeter of a parallelogram is 22cm. If the longer side measures 6.5cm what is the measure of the shorter side?**

**Solution:**

**Question 5: In a parallelogram ABCD, ∠D = 135°. Determine the measures of ∠A and ∠B.**

**Solution:**

**Question 6: ABCD is a parallelogram in which ∠A = 70°. Compute ∠B, ∠C and ∠D.**

**Solution:**

**Question 7: In Figure, ABCD is a parallelogram in which ∠A = 60°. If the bisectors of ∠A, and ∠B meet at P, prove that AD = DP, PC = BC and DC = 2AD.**

**Solution:**

**Question 8: In figure, ABCD is a parallelogram in which ∠DAB = 75° and ∠DBC = 60°. Compute ∠CDB, and ∠ADB.**

**Solution:**

**Question 9: In figure, ABCD is a parallelogram and E is the mid-point of side BC. If DE and AB when produced meet at F, prove that AF = 2AB.**

**Solution:**

**Question 10: Which of the following statements are true (T) and which are false (F)?**

**(i) In a parallelogram, the diagonals are equal.**

**(ii) In a parallelogram, the diagonals bisect each other.**

**(iii) In a parallelogram, the diagonals intersect each other at right angles.**

**(iv) In any quadrilateral, if a pair of opposite sides is equal, it is a parallelogram.**

**(v) If all the angles of a quadrilateral are equal, it is a parallelogram.**

**(vi) If three sides of a quadrilateral are equal, it is a parallelogram.**

**(vii) If three angles of a quadrilateral are equal, it is a parallelogram.**

**(viii) If all the sides of a quadrilateral are equal, it is a parallelogram.**

**Solution:**

**Exercise 14.3**

**Question 1: In a parallelogram ABCD, determine the sum of angles ∠C and ∠D.**

**Solution:**

**Question 2: In a parallelogram ABCD, if ∠B = 135°, determine the measures of its other angles.**

**Solution:**

**Question 3: ABCD is a square. AC and BD intersect at O. State the measure of ∠AOB.**

**Solution:**

**Question 4: ABCD is a rectangle with ∠ABD = 40°. Determine ∠DBC.**

**Solution:**

**Question 5: The sides AB and CD of a parallelogram ABCD are bisected at E and F. Prove that EBFD is a parallelogram.**

**Solution:**

**Question 6: P and Q are the points of trisection of the diagonal BD of a parallelogram ABCD. Prove that CQ is parallel to AP. Prove also that AC bisects PQ.**

**Solution:**

**Question 7: ABCD is a square. E, F, G and H are points on AB, BC, CD and DA respectively, such that AE = BF = CG = DH. Prove that EFGH is a square.**

**Solution:**

**Question 8: ABCD is a rhombus, EAFB is a straight line such that EA = AB = BF. Prove that ED and FC when produced meet at right angles.**

**Solution:**

**Question 9: ABCD is a parallelogram, AD is produced to E so that DE = DC and EC produced meets AB produced in F. Prove that BF = BC.**

**Solution:**

**Exercise 14.4**

**Question 1: In a ΔABC, D, E and F are, respectively, the mid points of BC, CA and AB. If the lengths of sides AB, BC and CA are 7 cm, 8 cm and 9 cm, respectively, find the perimeter of ΔDEF.**

**Solution:**

**Question 2: In a ΔABC, ∠A = 50°, ∠B = 60° and ∠C = 70°. Find the measures of the angles of the triangle formed by joining the mid-points of the sides of this triangle.**

**Solution:**

**Question 3: In a triangle, P, Q and R are the mid points of sides BC, CA and AB respectively. If AC = 21 cm, BC = 29 cm and AB = 30 cm, find the perimeter of the quadrilateral ARPQ.**

**Solution:**

**Question 4: In a ΔABC median AD is produced to X such that AD = DX. Prove that ABXC is a parallelogram.**

**Solution:**

**Question 5: In a ΔABC, E and F are the mid-points of AC and AB respectively. The altitude AP to BC intersects FE at Q. Prove that AQ = QP.**

**Solution:**

**Question 6: In a ΔABC, BM and CN are perpendiculars from B and C respectively on any line passing through A. If L is the mid-point of BC, prove that ML = NL.**

**Solution:**

**Question 7: In figure, triangle ABC is a right-angled triangle at B. Given that AB = 9 cm, AC = 15 cm and D, E are the mid-points of the sides AB and AC respectively, calculate**

**(i) The length of BC**

**(ii) The area of ΔADE.**

**Solution:**

^{2}= AB

^{2}+ BC

^{2}(By using Pythagoras theorem)

^{2}= 9

^{2}+ AC

^{2}

^{2}= 225 – 81 = 144

^{2}

**Question 8: In figure, M, N and P are mid-points of AB, AC and BC respectively. If MN = 3 cm, NP = 3.5 cm and MP = 2.5 cm, calculate BC, AB and AC.**

**Solution:**

**Question 9: ABC is a triangle and through A, B, C lines are drawn parallel to BC, CA and AB respectively intersecting at P, Q and R. Prove that the perimeter of ΔPQR is double the perimeter of ΔABC.**

**Solution:**

**Question 10: In figure, BE ⊥ AC, AD is any line from A to BC intersecting BE in H. P, Q and R are respectively the mid-points of AH, AB and BC. Prove that ∠PQR = 90°.**

**Solution:**

**Question 11: In figure, AB = AC and CP ∥ BA and AP is the bisector of exterior ∠CAD of ∆ABC. Prove that**

**(i) ∠PAC = ∠BCA.**

**(ii) ABCP is a parallelogram.**

**Solution:**

**Question 12: ABCD is a kite having AB = AD and BC = CD. Prove that the figure found by joining the mid points of the sides, in order, is a rectangle.**

**Solution:**

**Question 13: Let ABC be an isosceles triangle in which AB = AC. If D, E, F be the mid points of the, sides BC, CA and AB respectively, show that the segment AD and EF bisect each other at right angles.**

**Solution:**

**Question 14: ABC is a triangle. D is a point on AB such that AD = 1/4 AB and E is a point on AC such that AE = 1/4 AC. Prove that DE = 1/4BC.**

**Solution:**

**Question 15: In Figure, ABCD is a parallelogram in which P is the mid-point of DC and Q is a point on AC such that CQ = (1 )/2 AC. If PQ produced meets BC at R, prove that R is a mid-point of BC.**

**Solution:**

**Question 16: In figure, ABCD and PQRC are rectangles and Q is the mid-point of AC. Prove that**

**(i) DP = PC**

**(ii) PR = 1/2 AC**

**Solution:**

**Question 17: ABCD is a parallelogram; E and f are the mid-points of AB and CD respectively. GH is any line intersecting AD, EF and BC at G, P and H respectively. Prove that GP = PH.**

**Solution:**

**Question 18: BM and CN are perpendiculars to a line passing through the vertex A of triangle ABC. If L is the mid-point of BC, prove that LM = LN.**

**Solution:**

**Question 19: Show that, the line segments joining the mid-points of opposite sides of a quadrilateral bisects each other.**

**Solution:**

**Question 20: Fill in the blanks to make the following statements correct:**

**(i) The triangle formed by joining the mid-points of the sides of an isosceles triangle is ____________.**

**(ii) The triangle formed by joining the mid-points of the sides of a right triangle is ____________.**

**(iii) The figure formed by joining the mid-points of consecutive sides of a quadrilateral is ____________.**

**Solution:**

**Exercise VSAQs**

**Question 1: In a parallelogram ABCD, write the sum of angles A and B.**

**Solution:**

**Question 2: In a parallelogram ABCD, if ∠D = 115°, then write the measure of ∠A.**

**Solution:**

**Question 3: PQRS is a square such that PR and SQ intersect at O. State the measure of ∠POQ.**

**Solution:**

**Question 4: In a quadrilateral ABCD, bisectors of angles A and B intersect at O such that ∠AOB = 75°, then write the value of ∠C + ∠D.**

**Solution:**

**Question 5: The diagonals of a rectangle ABCD meet at O. If ∠BOC = 44°, find ∠OAD.**

**Solution:**

**Question 6: If PQRS is a square, then write the measure of ∠SRP.**

**Solution:**

**Question 7: If ABCD is a rectangle with ∠BAC = 32°, find the measure of ∠DBC.**

**Solution:**

**Question 8: If ABCD is a rhombus with ∠ABC = 56°, find the measure of ∠ACD.**

**Solution:**

**Question 9: The perimeter of a parallelogram is 22 cm. If the longer side measure 6.5 cm, what is the measure of shorter side?**

**Solution:**

**Question 10: If the angles of a quadrilateral are in the ratio 3:5:9:13, then find the measure of the smallest angle.**

**Solution:**