Three angles of a quadrilateral are respectively equal to 110°, 50° and 40°. Find its fourth angles.

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

In a quadrilateral ABCD, CO and DO are the bisectors of ∠C and ∠D respectively. Prove that ∠COD = (∠A+∠B).

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

Two opposite angles of a parallelogram are (3x-2)° and (50-x)°. Find the measure of each angle of the parallelogram.

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

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

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

In a parallelpgram ABCD, ∠D=135°, determine the measures of ∠A and ∠B.

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

In Fig. 14.34, 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.

In Fig. 14.35, ABCD is a parallelogram in which ∠DAB =75° and ∠DBC = 60°. Compute ∠CDB and ∠ADB.

In Fig. 14.36, 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.

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.

In a parallogram ABCD, determine the sum of angles ∠C and ∠D.

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

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

ABCD is a rectangle with ∠ABD=40°. Determine ∠DBC.

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

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

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.

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

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.

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

In a triangle ∠ABC, ∠A =50°, ∠B =60° and ∠C = 70°. Find the measure of the angles of the triangle formed by joining the mid-points of the sides of this triangle.

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.

In a ΔABC, median AD is produced to X such that AD=DX. Prove that ABXC is a parallelogram.

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

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.

In Fig. 14.95, triangle ABC is right-angled 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.

In Fig. 14.96, 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.

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.

In Fig. 14.97, 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°

In Fig. 14.98, AB=AC and CP||BA and AP is the bisector of exterior ∠CAD of ΔABC. Prove that (i) ∠PAC=∠BCA (ii) ABCP is a parallelogram.

ABCD is a kite having AB=AD and BC=CD. Prove that the figure formed by joining the mid-points of the sides, in order, is a rectangle.

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.

ABC is a triangle. D is a point on AB such that AD=AB and E is a point on AC such that AE= AC. Prove that DE=BC.

In Fig. 14.99, ABCD is a parallelogram in which P is the mid-point of DC and Q is a point on AC such that CQ= AC. If PQ produced meets BC at R, Prove that R is a mid-point of BC.

In Fig. 14.100, ABCD and PQRC are rectangles and Q is the mid-point of AC. Prove that

(i) DP = PC (ii) PR = AC.

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.

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

Show that the line segments joining the mid-points of the opposite sides of a quadrilateral bisect each other.

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……

In a parallelogram ABCD, write the sum of angles A and B.

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

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

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

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

If PQRS is a square, then write the measure of ∠SRP.

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

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

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

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

If the bisectors of two adjacent angles A and B of a quadrilateral ABCD intersect at a point O such that ∠C+∠D=k∠AOB, then find the value of k.

In a parallelogram ABCD, if ∠A=(3x-20)°, ∠B = (y+15)° and ∠C = (x+40)°, then find the values of x and y.

If measures opposite angles of a larallelogram are (60-x)° and (3x-4)°, then find the measures of angles of the parallelogram.

In a parallelogram ABCD, the bisector of ∠A also bisects BC at X. Find AB : AD.

In Fig. 14.111, PQRS is an isosceles trapezium. Find x and y.

In Fig. 14.112, ABCD is a trapezium. Find the values of x and y.

In Fig. 14.113, PQRS is a rhombus in which the diagonal PR is produced to T. If ∠SRT=152°, find x,y and z.

In Fig. 14.114, ABCD is a rectangle in which diagonal AC is produced to E. If ∠ECD=146°, find ∠AOB.

In Fig. 14.115, ANCD and AEFG are two parallelograms. If ∠C=58°, find ∠F.

Complete each of the following statements by means of one of those given in brackets against each:

(i) If one pair of opposite sides are equal and parallel, then the figure is ………. (parallelogram, rectangle, trapezium)

(ii) If in a quadrilateral only one pair of opposite sides are parallel, the quadrilateral is ………….. (square, rectangle, trapezium)

(iii) A line drawn from the mid-point of one side of a rriangle….. another side intersects the third side at its mid-point. (perpendicular to, parallel to, to meet)

(iv) If one angle of a parallelogram is a right angle, then it is necessarily a ….. (rectangle, square, rhombus)

(v) Consecutive angles of a parallelogram are ………… (Supplementary, complementary)

(vii) If opposite angles of a quadrilateral are equal, then it is necessarily a ……. (parallelogram, rhombus, rectangle)

(vi) If both pairs of opposite sides of a quadrilateral are equal, then it is necessarily a ……… (rectangle, parallelogram, rhombus)

(viii) If consecutive sides of a parallelogram are equal, then it is necessarily a …….. (kite, rhombus, square)

The opposite sides of a quadrilateral have

Two consecutive sides of a quadrilateral have

PQRS is a quadrilateral. PR and QS intersect each other at O. In which of the following cases, PQRS is a parallelogram?

Which of the following quadrilateral is not a rhombus?

Diagonals necessarily bisect opposite angles in a

The two diagonals are equal in a

We get a rhombus by joining the mid-points of the sides of a

The bisectors of any two adjacent angles of a parallelogram intersect at

The bisectors of the angle of a parallelogram enclose a

The figure formed by joining the mid-points of the adjacent sides of a quadrilateral is a

The figure formed by joining the mid-points of the adjacent sides of a rectangle is a

The figure formed by joining the mid-points of the adjacent sides of a rhombus is a

The figure formed by joining the mid-points of the adjacent sides of a square is a

The figure formed by joining the mid-points of the adjacent sides of a parallelogram is a

If one side of a parallelogram is 24° less than twice the smallest angle, then the measure of the largest angle of the parallelogram is

In a parallelogram ABCD, if ∠DAB=75° and ∠DBC=60°, then ∠BDC=

ABCD is a parallelogram and E and F are the centroids of triangles ABD and BCD respectively, then EF=

ABCD is a parallelogram M is the mid-point of BD and BM bisects ∠B. Then, ∠AMB=

ABCD is a parallelogram and E is the mid point of BC.DE and AB when produced meet at F. Then, AF=

If an angle of a parallelogram is two-third of its adjacent angle, the smallest angle of the parallelogram is

If the degree measures of the angles of quadrilateral are 4x, 7x, 9x and 10x, what is the sum of the measures of the smallest angle and largest angle?

In a quadrilateral ABCD, ∠A+∠C is 2 times ∠B+∠D. If ∠A= 140° and f∠D = 60°, then ∠B=

If the diagonals of a rhombus are 18 cm and 24 cm respectively, then its side is equal to

The diagonals AC and BD of a rectangle ABCD intersect each other at P. If ∠ABD =50°, then ∠DPC =

ABCD is a parallelogram in which diagonal AC bisects ∠BAD. If ∠BAC =35°, then ∠ABC =

In a rhombus ABCD, if ∠ACB =40°, then ∠ADB =

In ΔABC, ∠A =30°, ∠B=40° and ∠C=110°. The angles of the triangle formed by joining the mid-points of the sides of this triangle are

The diagonals of a parallelogram ABCD intersect ay O. If ∠BOC =90° and ∠BDC=50°, then ∠OAB =

ABCD is a trapezium in which AB||DC. M and N are the mid-points of AD and BC respectively, If AB=12cm, MN=14 cm, then CD=

Diagonals of a quadrilateral ABCD bisect each other. If ∠A =45°, then ∠B =

P is the mid point of side BC of a parallelogram ABCD such that ∠BAP =∠DAP. If AD=10 cm, then CD=

In ΔABC, E is the mid-point of median AD such that BE produced meets AC at F. If AC = 10.5 cm, then AF=