Advertisements
Advertisements
प्रश्न
Bisectors of the angles B and C of an isosceles triangle ABC with AB = AC intersect each other at O. Show that external angle adjacent to ∠ABC is equal to ∠BOC
Advertisements
उत्तर
Given: ΔABC is an isosceles triangle in which AB = AC, BO and CO are the bisectors of ∠ABC and ∠ACB respectively intersect at O.
To show: ∠DBA = ∠BOC
Construction: Line CB produced to D.
Proof: In ΔABC, AB = AC ...[Given]
∠ACB = ∠ABC ...[Angles opposite to equal sides are equal]
⇒ `1/2 ∠ACB = 1/2 ∠ABC` ...[On dividing both sides by 2]
⇒ ∠OCB = ∠OBC ...(i) [∵ BO and CO are the bisectors of ∠ABC and ∠ACB]
In ΔBOC, ∠OBC + ∠OCB + ∠BOC = 180° ...[By angle sum property of a triangle]
⇒ ∠OBC + ∠OBC + ∠BOC = 180° ...[From equation (i)]
⇒ 2∠OBC + ∠BOC = 180°
⇒ ∠ABC + ∠BOC = 180° ...[∵ BO is the bisector of ∠ABC]
⇒ 180° – ∠DBA + ∠BOC = 180° ...[∵ DBC is a straight line]
⇒ – ∠DBA + ∠BOC = 0
⇒ ∠DBA = ∠BOC
APPEARS IN
संबंधित प्रश्न
ABC is an isosceles triangle in which altitudes BE and CF are drawn to equal sides AC and AB respectively (see the given figure). Show that these altitudes are equal.
Find the measure of each exterior angle of an equilateral triangle.
Two lines AB and CD intersect at O such that BC is equal and parallel to AD. Prove that the lines AB and CD bisect at O.
In a ΔABC, it is given that AB = AC and the bisectors of ∠B and ∠C intersect at O. If M is a point on BO produced, prove that ∠MOC = ∠ABC.
Angles A, B, C of a triangle ABC are equal to each other. Prove that ΔABC is equilateral.
ABC is a triangle in which ∠B = 2 ∠C. D is a point on BC such that AD bisects ∠BAC and AB = CD.
Prove that ∠BAC = 72°.
Which of the following statements are true (T) and which are false (F):
Angles opposite to equal sides of a triangle are equal
Fill the blank in the following so that the following statement is true.
In right triangles ABC and DEF, if hypotenuse AB = EF and side AC = DE, then ΔABC ≅ Δ ……
In Fig. 10.131, prove that: (i) CD + DA + AB + BC > 2AC (ii) CD + DA + AB > BC
Which of the following statements are true (T) and which are false (F)?
Difference of any two sides of a triangle is equal to the third side.
Fill in the blank to make the following statement true.
The sum of three altitudes of a triangle is ..... than its perimeter.
Write the sum of the angles of an obtuse triangle.
In a triangle ABC, if AB = AC and AB is produced to D such that BD = BC, find ∠ACD: ∠ADC.
In the given figure, if EC || AB, ∠ECD = 70° and ∠BDO = 20°, then ∠OBD is
In the given figure, what is y in terms of x?
In the given figure, what is the value of x?
In triangles ABC and PQR, AB = AC, ∠C = ∠P and ∠B = ∠Q. The two triangles are ______.
ABC is an isosceles triangle with AB = AC and D is a point on BC such that AD ⊥ BC (Figure). To prove that ∠BAD = ∠CAD, a student proceeded as follows:
In ∆ABD and ∆ACD,
AB = AC (Given)
∠B = ∠C (Because AB = AC)
and ∠ADB = ∠ADC
Therefore, ∆ABD ≅ ∆ACD (AAS)
So, ∠BAD = ∠CAD (CPCT)
What is the defect in the above arguments?
[Hint: Recall how ∠B = ∠C is proved when AB = AC].
