Advertisements
Advertisements
प्रश्न
ABC is a triangle in which BE and CF are, respectively, the perpendiculars to the sides AC and AB. If BE = CF, prove that ΔABC is isosceles
Advertisements
उत्तर
Given that ABC is a triangle in which BE and CF are perpendicular to the sides AC and AB respectively such that BE = CF
We have to prove that ΔABC is isosceles
Now, consider ΔBCF and ΔCBE,
We have
∠BFC=CEB=90° [Given]
BC=CB [Common side]
And CF=BE [Given]
So, by RHS congruence criterion, we have ΔBFC≅CEB
Now,
∠FBC=∠EBC [∵ Incongruent triangles corresponding parts are equal]
⇒ ∠ABC=∠ACB
⇒ AC=AB [ ∵Opposite sides to equal angles are equal in a triangle]
∴ ΔABC is isosceles
APPEARS IN
संबंधित प्रश्न
In an isosceles triangle ABC, with AB = AC, the bisectors of ∠B and ∠C intersect each other at O. Join A to O. Show that:
- OB = OC
- AO bisects ∠A
ABC and DBC are two isosceles triangles on the same base BC (see the given figure). Show that ∠ABD = ∠ACD.
Show that the angles of an equilateral triangle are 60° each.
In figure, AB = AC and DB = DC, find the ratio ∠ABD : ∠ACD
Determine the measure of each of the equal angles of a right-angled isosceles triangle.
In Fig. 10.40, it is given that RT = TS, ∠1 = 2∠2 and ∠4 = 2∠3. Prove that ΔRBT ≅ ΔSAT
In Figure 10.24, AB = AC and ∠ACD =105°, find ∠BAC.
If the base of an isosceles triangle is produced on both sides, prove that the exterior angles so formed are equal to each other.
If the bisector of the exterior vertical angle of a triangle be parallel to the base. Show that the triangle is isosceles.
Which of the following statements are true (T) and which are false (F):
The bisectors of two equal angles of a triangle are equal
Fill the blank in the following so that the following statement is true.
Sides opposite to equal angles of a triangle are ......
Fill the blank in the following so that the following statement is true.
In an equilateral triangle all angles are .....
In ΔABC, if ∠A = 40° and ∠B = 60°. Determine the longest and shortest sides of the triangle.
Is it possible to draw a triangle with sides of length 2 cm, 3 cm and 7 cm?
Which of the following statements are true (T) and which are false (F)?
Sum of any two sides of a triangle is greater than twice the median drawn to the third side.
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.
Difference of any two sides of a triangle is........ than the third side.
If ∆PQR ≅ ∆EDF, then is it true to say that PR = EF? Give reason for your answer
Show that in a quadrilateral ABCD, AB + BC + CD + DA < 2(BD + AC)
