Advertisements
Advertisements
प्रश्न
ABCD is a quadrilateral in which AB = BC and AD = CD. Show that BD bisects both the angles ABC and ADC.
Advertisements
उत्तर
Given: ABCD is a quadrilateral in which AB = BC and AD = CD.
To show: BD bisects both the angles ABC and ADC.
Proof: Since, AB = BC ...(Given)
∴ ∠2 = ∠1 ...(i) [Angles opposite to equal sides are equal]
And AD = CD ...[Given]
⇒ ∠4 = ∠3 ...(ii) [Angles opposite to equal sides are equal]
On adding equations (i) and (ii), we get
∠2 + ∠4 = ∠1 + ∠3
⇒ ∠BCD = ∠BAD ...(iii)
In ΔBAD and ΔBCD,
AB = BC ...[Given]
∠BAD = ∠BCD ...[From equation (iii)]
And AD = CD ...[Given]
∴ ΔBAD ≅ ΔBCD ...[By SAS congruence rule]
Hence, ∠ABD = ∠CBD and ∠ADB = ∠CDB i.e., BD bisects the angles ABC and ADC. ...[By CPCT]
APPEARS IN
संबंधित प्रश्न
If ΔDEF ≅ ΔBCA, write the part(s) of ΔBCA that correspond to ∠E
In a squared sheet, draw two triangles of equal areas such that
The triangles are congruent.
What can you say about their perimeters?
In Fig. 10.22, the sides BA and CA have been produced such that: BA = AD and CA = AE.
Prove that segment DE || BC.
In the given figure, if AB = AC and ∠B = ∠C. Prove that BQ = CP.
In the pair of triangles given below, the parts shown by identical marks are congruent. State the test and the one-to-one correspondence of vertices by which the triangles in the pair are congruent, the remaining congruent parts.
In the pair of triangles given below, the parts shown by identical marks are congruent. State the test and the one-to-one correspondence of vertices by which the triangles in the pair are congruent, the remaining congruent parts.
The following figure has shown a triangle ABC in which AB = AC. M is a point on AB and N is a point on AC such that BM = CN.
Prove that: (i) BN = CM (ii) ΔBMC ≅ ΔCNB
Which of the following pairs of triangles are congruent? Give reasons
ΔABC;(BC = 5cm,AC = 6cm,∠C = 80°);
ΔXYZ;(XZ = 6cm,XY = 5cm,∠X = 70°).
In the figure, RT = TS, ∠1 = 2∠2 and ∠4 = 2∠3. Prove that ΔRBT ≅ ΔSAT.
Is it possible to construct a triangle with lengths of its sides as 4 cm, 3 cm and 7 cm? Give reason for your answer.
