Advertisements
Advertisements
प्रश्न
ABC is a triangle in which altitudes BE and CF to sides AC and AB are equal (see the given figure). Show that
- ΔABE ≅ ΔACF
- AB = AC, i.e., ABC is an isosceles triangle.
Advertisements
उत्तर
i. In △ABE and △ACF, we have
∠AEB = ∠AFC ...[Each = 90° as BE ⊥ AC and CF ⊥ AB]
∠A = ∠A ...[Common]
BE = CF ...[Given]
∴ △ABE ≌ △ACF ...[By AAS congruence rule]
ii. Since, △ABE ≌ △ACF
∴ AB = AC ...[By Corresponding parts of congruent triangles]
⇒ ABC is an isosceles triangle.
APPEARS IN
संबंधित प्रश्न
Show that the angles of an equilateral triangle are 60° each.
In Fig. 10.40, it is given that RT = TS, ∠1 = 2∠2 and ∠4 = 2∠3. Prove that ΔRBT ≅ ΔSAT
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.
Prove that each angle of an equilateral triangle is 60°.
ABC is a triangle and D is the mid-point of BC. The perpendiculars from D to AB and AC are equal. Prove that the triangle is isosceles.
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
Which of the following statements are true (T) and which are false (F):
The measure of each angle of an equilateral triangle is 60°
Which of the following statements are true (T) and which are false (F) :
If the altitude from one vertex of a triangle bisects the opposite side, then the triangle may be isosceles.
In ΔABC, if ∠A = 40° and ∠B = 60°. Determine the longest and shortest sides of the triangle.
Prove that the perimeter of a triangle is greater than the sum of its altitudes.
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.
Write the sum of the angles of an obtuse triangle.
In the given figure, if AB || DE and BD || FG such that ∠FGH = 125° and ∠B = 55°, find x and y.
The side BC of ΔABC is produced to a point D. The bisector of ∠A meets side BC in L. If ∠ABC = 30° and ∠ACD = 115°, then ∠ALC = ______.
In ∆PQR, ∠R = ∠P and QR = 4 cm and PR = 5 cm. Then the length of PQ is ______.
M is a point on side BC of a triangle ABC such that AM is the bisector of ∠BAC. Is it true to say that perimeter of the triangle is greater than 2 AM? Give reason for your answer.
Is it possible to construct a triangle with lengths of its sides as 9 cm, 7 cm and 17 cm? Give reason for your answer.
In a triangle ABC, D is the mid-point of side AC such that BD = `1/2` AC. Show that ∠ABC is a right angle.
