Nootan solutions for Mathematics [English] Class 9 ICSE chapter 8 - Triangles [Latest edition]

If ΔABC ≅ ΔPQR, is it true to say that AB = PR? Why?

In the adjoining figure, AB = AC, AP = AQ. Prove that :

1. ΔAPC ≅ ΔAQB
2. CP = BQ
3. ∠APC ≅ ∠AQB

AB is a line segment and line l is its perpendicular bisector. If P is a point on line l, show that P is equidistant from A and B.

In the adjoining figure, OA = OB and OC = OD. Show that :

1. ΔOAD ≅ ΔOBC
2. AD || CB

In quadrilateral ABCD, AB = BC and AD = CD. Show that BD bisects ∠ABC and ∠ADC both.

From the adjoining figure, find the values of x and y.

In the adjoining figure, AB = PQ, BR = CQ, ∠ABC = ∠PQR = 90°. Prove that AC = PR.

In the adjoining figure, AB = AC and AP is the bisector of ∠A. Prove that:

1. P is the mid-point of BC 
2. AP is perpendicular to BC

In the adjoining figure, ABCD is a quadrilateral in which AD = BC and ∠DAB = ∠CBA. Prove that:

1. ΔABD ≅ ΔBAC
2. BD = AC
3. ∠ABD = ∠BAC

In the adjoining figure, OA = OB, OC = OD and ∠AOB = ∠COD. Prove that: AC = BD.

In the adjoining figure, AB = AC and D is the mid-point of BC. Use SSS criteria of congruency and prove that:

1. ΔABD ≅ ΔACD
2. AD bisects ∠A.
3. AD is perpendicular to BC.

In a rhombus ABCD, M is any point in its interior as shown in the adjoining figure, such that MA = MC. Prove that B, M, D are collinear.

Line segment joining the mid-points M and N of parallel sides AB and DC, respectively of a trapezium ABCD is perpendicular to both the sides AB and DC. Prove that AD = BC.

In the adjoining figure, 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. Show that the line PQ is the perpendicular bisector of AB.

In ΔABC and ΔPQR, ∠A = ∠R and ∠B = ∠P. Which side of ΔABC should be equal to ΔPQR, so that the two triangles are congruent. Give reason.

In ΔABC and ΔPQR, AB = PQ and AC = PR. Which angle of ΔABC should be equal to ΔPQR, so that the two triangles are congruent. Give reason.

In the adjoining figure, BE = CD and ∠BEA = ∠CDA. Prove that AE = AD.

P is a point equidistant from two lines m and n intersecting at a point M. Show that the line MP bisects the angle between the lines m and n.

In the adjoining figure, l and m are two parallel lines intersected by another pair of parallel lines p and q. Show that : ΔABC ≅ ΔCDA.

In the adjoining figure, AD and BC are equal perpendiculars to the line segment AB. Show that : CD bisects AB.

In quadrilateral ABCD, the diagonal AC bisects ∠A and ∠C both. Prove that AB = AD and CB = CD.

In a rectangle ABCD, X and Y are points on the sides AD and BC respectively such that AY = BX. Prove that: BY = AX and ∠BAY = ∠ABX.

ABC is an isosceles triangle with AB = AC. If AP ⊥ BC, then show that: ∠B = ∠C.

In the adjoining figure, ∠ABC = ∠ACB. M and N are points on the sides AC and AB respectively such that BN = CM. Prove that :

1. ΔNBC ≅ ΔMCB 
2. ΔPNB ≅ ΔPMC 
3. PB = РС.

In the adjoining figure, BA ⊥ AC, PQ ⊥ PR, such that BA = PQ and BR = CQ. Show that : ΔΑΒC ≅ ΔΡQR.

From the adjoining figure, find the value of x and y.

In the adjoining figure, AB || DC and ∠C = ∠D. Prove that:

1. AD = BC
2. AC = BD.

In the adjoining figure, ABCD is a squre and R is the mid-point of AB. PQ is any line segment passing through R which meets AD at P and CB produced at Q. Prove that R is the midpoint of PQ.

In the adjoining figure, AB = CD and ∠ABC = ∠BCD. Prove that : 

1. AC = BD 
2. BP = CP.

In the adjoining figure, two sides AB, AC and altitude AM of ΔABC are respectively equal to two sides PQ, PR and altitude PN of ΔPQR. Prove that ΔABC ≅ ΔPQR.

In the adjoining figure, AB || FD, AC || GE and BD = EC. Prove that:

1. BG = DF
2. EG = CF.

AD is an altitude of an isosceles triangles ABC in which AB = AC. Show that

1. AD bisects BC
2. AD bisects ∠A

In ΔАВC, AB = AC and ∠BAC = 30°. Find the value of x.

In the adjoining figure, AC = CD = BD and ∠CBD = 40°. Find ∠ACD.

In the adjoining figure, AC = BC = CD and ∠ADC = 40°. Find ∠BAC.

In the adjoining figure, AC = BC and CD = AD. If ∠ADC = 60°. Find ∠BAC.

Show that the angles of an equilateral triangle are 60° each.

In the adjoining figure, BC = CD. Find ∠ACB.

In the adjoining figure, ABC and DBC are two isosceles triangles on the same base BC. Prove that ∠ABD = ∠ACD.

In the adjoining figure, ABC is an isosceles triangle in which AB = AC. Its side BA is produced to D such that AD = AB. Show that ∠BCD = 90°.

In the adjoining figure, D is the mid-point of BC. DE and DF are perpendiculars to AB and AC respectively, such that DE = DF. Prove that ΔABC is an isosceles triangle.

In the adjoining figure, ABC is an isosceles triangle in which AB = AC and PQ is parallel to BC. If ∠A = 40°, find ∠PQC.

In the adjoining figure, AB = AC and BE, CF are the bisectors of ∠B, ∠C respectively. Prove that:

1. ΔΕΒC ≅ ΔFCB
2. BE = CF

In the adjoining figure, ABC is a right-angled triangle, right-angled at A such that AB = AC. Bisector of ∠A meets BC at D. Prove that BC = 2AD.

In the adjoining figure, ΔCDE is an equilateral triangle and ABCD is a square. Show that:

1. ΔΑDE ≅ ΔBCE
2. ΔАЕВ is an isosceles triangle.

In a triangle ABC, AB = AC, D and E are points on the sides AB and AC, respectively such that BD = CE. Show that:

1. ΔDBC ≅ ΔЕСВ
2. ΔDCB = ΔЕВС
3. OB = OC, where O is the point of intersection of BE and CD.

If the bisector of an angle of a triangle also bisects the opposite side, prove that the triangle is isosceles.

In the adjoining figure, AB = BC and AC = CD. Prove that ∠BAD : ∠ADB = 3 : 1.

In the adjoining figure, ABCD is a square and ADE is an equilateral triangle. Prove that `∠ACE = 1/2 ∠ DAE`.

In ΔАВC, ∠A = 30°, ∠B = 90°. Find the longest side of ΔАВС.

In ∆АВC, ∠A = 68°, AB = AC. Find the longest side of ∆AВС.

Show that in a right angled triangle, the hypotenuse is the longest side.

In ΔАВC, AB = 4 cm, BC = 5 cm and CA = 6 cm. Find the longest angle of ΔABC.

From the adjoining figure, find:

1. The longest side of ABC
2. The smallest side of ΔABC.

From the adjoining figure, find:

1. The longest side of ΔABC
2. The nature of ∠ВАС.

In the adjoining figure ∠DBC > ∠BCE. Prove that AB > AC.

In a quadrilateral ABCD. Prove that:

1. BC + CD + DA > AB 
2. AB + BC + CD + DA > 2AC 
3. AB + BC + CD + DA > 2BD

In ΔABC, if AB < AC < BC, then what is the relation between ∠A and ∠C? 

In ΔABC, if AB < AC < BC, then what is the relation between the values of (AB + AC) and BC?

In ΔABC, if AB < AC < BC, then what is the relation between the values of (BC – AB) and AC?

In ΔABC, if AB < AC < BC, then which is the largest angle of ΔABC?

Let ‘O’ be any point in the interior of ΔACB. Prove that:

1. AB + AC > OB + ОС 
2. AB + BC + CA > (OA + OB + OC) 
3. AB + BC + CA < 2(OA + OB + ОС)

In the adjoining figure, AD = DE. Prove that AB + BC > CЕ.

Prove that the sum of three medians of a triangle is less than the perimeter of the triangle.

If P is any interior point of ΔABC, prove that ∠BPC > ∠BAC.

In the adjacent figure, BE = DE. Prove that AC + BC > CD.

In the adjoining figure, AB = AD. Prove that BC > CD.

In the adjoining figure, x > y > z. Write the sides of ΔABC in ascending order.

In the adjoining figure AB > AC. If OB and OC are the bisectors of ∠ABC and ∠ACB, respectively, then prove that OB > ОС.

In the adjoining figure OB and OC are the bisectors of ∠DBC and ∠ECB, respectively. If AC > AB prove that OB > OC.

In the adjoining figure, AC > AB and AM is the bisector of ∠BAC. Prove that ∠AMC > ∠AMB.

In ΔPQR, ∠R < ∠Q, then ______.

In ΔABC, AB = AC and ∠A = 100°, then ∠C is equal to ______.

In ΔАВC, the correct relation is ______.

If ΔАВC ≅ ΔDEF, then the correct statement is ______.

In the adjoining figure, AD = CD = BD. ∠ABC is equal to:

In a right-angled triangle, the longest side is ______.

In a ΔABC, right-angled at A, it is given that AB = AC, ∠C is equal to ______.