Advertisements
Advertisements
Question
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].
Solution
In ∆ABC, AB = AC
⇒ ∠ACB = ∠ABC ...[Angles opposite to the equal sides are equal]
In ∆ABD and ∆ACD,
AB = AC ...[Given]
∠ABD = ∠ACD ...[Proved above]
∠ADB = ∠ADC ...[Each 90°]
∴ ∆ABD ≅ ∆ACD ...[By AAS]
So, ∠BAD = ∠CAD ...[By CPCT]
So, the defect in the given argument is that firstly prove ∠ABD = ∠ACD
Hence, ∠ABD = ∠ACD is defect.
APPEARS IN
RELATED QUESTIONS
Prove that the medians of an equilateral triangle are equal.
If the bisector of the exterior vertical angle of a triangle be parallel to the base. Show that the triangle is isosceles.
P is a point on the bisector of an angle ∠ABC. If the line through P parallel to AB meets BC at Q, prove that triangle BPQ is isosceles.
Which of the following statements are true (T) and which are false (F):
The measure of each angle of an equilateral triangle is 60°
Fill the blank in the following so that the following statement is true.
In an equilateral triangle all angles are .....
Prove that the perimeter of a triangle is greater than the sum of its altitudes.
If the angles of a triangle are in the ratio 2 : 1 : 3, then find the measure of smallest angle.
In ∆PQR, if ∠R > ∠Q, then ______.
In triangles ABC and PQR, AB = AC, ∠C = ∠P and ∠B = ∠Q. The two triangles are ______.
CDE is an equilateral triangle formed on a side CD of a square ABCD (Figure). Show that ∆ADE ≅ ∆BCE.
