B Nirmala Shastry solutions for Mathematics [English] Class 9 ICSE chapter 8 - Triangles [Latest edition]

In ΔABC, AB = AC, D is a point inside the triangle such that ∠DBC = ∠DCB.

Prove that ΔBAD ≅ ΔCAD.

In ΔPQR, PQ = PR, QN = RM. Prove that ∠QPM = ∠RPN.

In ΔABC, AB = AC and CP and BQ are altitudes. Prove that CP = BQ.

In the quadrilateral ABCD, AD = CD and ∠A = 90° = ∠C.

Prove that AB = BC.

In the given triangles, AC = DF, BD = CE and ∠ACB = ∠FDE. Prove that ∠A = ∠F.

In the given figure, ΔABC is right angled at B. ACDE and BCGF are squares.

Prove that

1. ΔBCD ≅ ΔACG
2. AG = BD

Diagonal AC is the perpendicular bisector of diagonal BD in the quadrilateral ABCD.

Prove that

1. AB = AD 
2. BC = DC

PS bisects ∠QPR and PS ⊥ QR. If PQ = 2x units, PR = (3y + 8) units, QS = x units and SR = 2y units. Find the values of x and y.

In ΔAOB and ΔCOD, ∠B = ∠C and O is the midpoint of BC. Find the values of x and y if AB = 3x units, CD = y + 2 units, AO = x + 2 units, DO = y units.

Prove that ΔABD = ΔCBD. Find the values of x and y, if ∠ABD = 35°, ∠CBD = (3x + 5)°, ∠ADB = (y – 3)°, ∠CDB = 25°.

In the square PQRS, equilateral ΔOPQ is drawn. Prove that ΔOPS ≅ ΔOQR.

In the adjoining figure, QX and RX are the bisectors of the angles Q and R respectively of the triangle PQR.
If XS ⊥ QR and XT ⊥  PQ;

Prove that:

1. ΔXTQ ≅ ΔXSQ.
2. PX bisects angle P.

In the given figure, AD = BC, AC = BD. Prove that ∠ADB = ∠ACB.

ABCD is a square. P and Q are points on AB and BC such that AQ = DP. Prove that ΔAPD ≅ ΔBQA.

ABCD is a parallelogram, X is the midpoint of BC. AX is produced to meet DC produced at Q. ABPQ is a parallelogram.

Prove that

1. ΔABX ≅ ΔQCX
2. DC = CQ = QP

In ΔABC, AB = AC and AP = AQ. Prove that CP = BQ.

In ΔABC, ∠A – ∠B = 24° and ∠B – ∠C = 30°. Find angle A.

In the given figure, AB = BC = CD = AC and AD = DE. Find ∠BAE.

In ΔABC, AC = BC. ∠BAC is bisected by AD and AD = AB. Find ∠ACB.

In ΔABC, AC = BC. DE is drawn parallel to BC through A. ∠EAC = 42°. Find angle x.

In ΔPQR, PQ = PR, ∠P = (2x + 20)° and ∠R = (x + 10)°. Find the value of x. Assign a special name to the triangle according to the angles.

In the given figure, AB = AC, AD ⊥ BC, ∠PAC = 102°. Find the values of x and y.

In the given figure, PQ ⊥ AB, AQ = QB, ∠PAC = 42°, ∠C = 68°. Find ∠ABC.

In the given figure, AB = AD, ∠PAD = 70° and ∠DBC = 90°. Find x, ∠BDQ and ∠BCD if AB || DC.

In the given figure, PQ = PR = PS. If ∠Q = 38°, find the value of x.

In the given figure, AB || FC, ∠B = 70° and ∠FDE = 148°. Find the values of a and b.

∠ECF = 3∠ACE and ∠ADB = 72°. Find the value of x.

∠PBC = 70°, ∠CND = 36° and PQ || RS. Find the value of x.

Find y in the given diagram if AB || CD, EF || AC and ∠ADC = 58°.

In the given figure, BE || CD, ∠ABC = 120° ∠C = 80°, ∠CDE = 110° and BC = CD. Find ∠BAE, ∠CBE and ∠BDE.

In the figure ABCD, BD = BC = AD and ∠ACD = 37°. Find ∠ADB.

In ΔABC, BC is produced to D. ∠A = x + 30°, ∠B = 2x + 25° and ∠ACD = 5x – 5°. Find x and ∠B.

QI and RI bisect exterior angles RQT and QRS. Prove that `∠QIR = 90^circ - 1/2 ∠P`.

In the given figure, AB || CD. PA and PC are bisectors of ∠BAC and ∠ACD. Find ∠APC.

In ΔABC, ∠B = 90° and BD ⊥ AC. CE bisects ∠C. If ∠A = 48°, find ∠DMC and ∠BEM.

In ΔPQR, PQ = QR and ΔPSR is equilateral. If ∠Q = 26°, find y.

Find the values of x and y in the given triangles where ∠A = 70°, ∠B = 36°, ∠D = 68° and ∠F = 56°.

In the given figure, BE || CD, ∠DCG = 85°, ∠EBF = 10° and AC = BC. Find ∠FAC and ∠ACB.

In the given figure, ΔABC is equilateral. ∠P = 40° and ∠Q = 30°. Find the values of x, y and ∠PAQ if PBCQ is a straight line.

In the given figure, AD bisects ∠BAC and DE bisects ∠BDA. Find ∠BED.

In ΔPQR, PQ = PR. S is a point on PQ such that SR = QR = SP. Find ∠P.

In ΔABC, AB = AC and ∠A : ∠B = 4 : 3. Find the measure of ∠C.

AB = BC = AC = AD, l || m, ∠BAE = x, ∠ADF = 5x. Find the measure of angle x.

In the figure in ΔABC, AD is ⊥ BC and BD = DC.

∴ ΔABD ≅ ΔACD by ______.

In ΔABC, ∠A = ∠C, AB = 5 cm, AC = 6 cm

∴ BC is ______.

In ΔABC, BC = AC and ∠B = 65°.

∴ ∠C = ______.

BD bisects ∠ABC. ΔABD ≅ ΔCBD by ______.

AO = OD. ΔABO ≅ ΔDCO by ______.

PQ = SR, ∠P = ∠S, ΔPOQ ≅ ΔSOR by ______.

In ||^gm ABCD, BQ and DP are ⊥ AC. ΔADP ≅ ΔCBQ by ______.

In ΔABC, AB = AC, BP ⊥ AC and CQ ⊥ AB. ΔABP ≅ ΔACQ by ______.

PQ = PR, PS || QR, ∠SPR = 72. ∴ find ∠QPR.

ABCD is a parallelogram. O is the mid point of CD. ΔBOC ≅ ΔEOD by ______.

In ΔABC, PQ || AB, ∠ACD = 150°.

∴ find x.

ΔPQR is equilateral. PR = RS.

∴ find ∠RPS.

AB || CD, ∠ACP = 50°.

∴ find ∠DAB.

In ΔPQR, PR = QR. ∠P = 2x – 10°, ∠R = x + 5°.

∴ The value of x is ______.

In ΔABC, AB = BC. ∠A : ∠B = 2 : 1.

∴ find ∠B.

In ΔPQR, ∠Q = 90°, ∠P = 40°, RS bisects ∠PRQ.

∴ find x.

By which test are the Δ_s congruent?

The value of x is ______.

Line l₁ || l₂, ∠ABC = 90°. x = ______.

Assertion: In ΔABC, D is the mid point of BC. DP ⊥ AB and DQ ⊥ AC, DP = DQ. ΔBPD ≅ ΔCQD.

Reason: These triangles are congruent by SAS rule if 2 sides and included angle of one triangle are equal to 2 sides and included angle of the other triangle.

Assertion: In the figure line `l ||` line `m`, AB = BC, ∠DAE = 130° then x = 115°.

Reason: Equal sides of a triangle have equal angles opposite to them.

Assertion: In ΔABC, AB ⊥ BC ∠EAC = 2x, ∠BCF = 3x then x = 54°.

Reason: Exterior angle of a triangle = sum of interior opposite angles.

Assertion: In the figure AD ⊥ BC and BD = DC. ∴ ΔABD = ΔACD by RHS test of congruency.

Reason: Two right angled Δs are congruent if their hypotenuses are equal and one side of one Δ is equal to one side of the other.

Assertion: In the figure ∠A = 2x, ∠B = 3x, ∠E = 75° and CD || EF. ∴ x = 15°.

Reason: Co-interior ∠S of || lines are equal.

Assertion: In the figure AD = DC, ∠B = 56° and ∠ADB = 80°. ∴ AB > DC.

Reason: In a triangle, greater angle has longer side opposite to it.

Classify the following triangles:

1. In ΔABC, AB² + BC² = AC². What type of triangle is ABC? 
2. The circumcentre of the triangle lies at the ‘midpoint of one side.
3. In ΔPQR, all altitudes are equal.
4. The circumcentre of the triangle lies outside the triangle.
   

AD and BC are equal perpendiculars drawn on AB. Prove that DC bisects AB.

In ΔPQR, QR is produced to S so that ∠PRS = 130°. T is a point on QR so that PT = QT = TR. Find ∠QPR.

In ABC, PQ || BC. If PA = PC, ∠B = 74° and ∠PCB = 56°, find angles x and y.

In ΔABC, AB = AD = DC and AB is extended to E. ∠DAC = 22° and ∠E = 24°. Find angles x and y.

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

1. ΔCAQ ≅ ΔBAP 
2. ΔBQC ≅ ΔCPB

ABCD is a quadrilateral in which AP and CQ are perpendicular to diagonal BD and AP = CQ. Prove that BD bisects AC.

Prove that if altitudes from two vertices of a triangle to the opposite sides are equal, then the triangle is isosceles.

PQR is an equilateral triangle. QM and RN are medians. Prove that QM = RN.

ABCD is a parallelogram. P and Q are points on diagonal DB such that DP = QB. Prove that ΔAPB ≅ ΔCQD.

ABCD is a parallelogram. M is the midpoint of BC. Show that DC = CP. [Hint: Prove that ΔABM ≅ ΔPCM]

In ΔABD, AC = AD = BC. Exterior ∠DAE = 102°, find the angle x.