Advertisements
Advertisements
Question
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°.
Advertisements
Solution
In ΔABC,
∠B = 2∠C or, ∠B = 2y, where ∠C = y.
AD is the bisector of ∠BAC.
So, let ∠BAD = ∠CAD = x.
Let BP be the bisector of ∠ABC. Join PD.
In ΔBPC, we have
∠CBP = ∠BCP = y ⇒ BP = PC
In Δ′s ABP and DCP, we have
∠ABP = ∠DCP,
∠ABP = ∠DCP = y
BP = PC ........(As proved above)
∠ADC = ∠ABD + ∠BAD
⇒ x + 2x = 2y + x
⇒ x = y
APPEARS IN
RELATED QUESTIONS
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
Show that the angles of an equilateral triangle are 60° each.
In a ΔABC, if ∠A=l20° and AB = AC. Find ∠B and ∠C.
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.
Find the measure of each exterior angle of an equilateral triangle.
BD and CE are bisectors of ∠B and ∠C of an isosceles ΔABC with AB = AC. Prove that BD = CE.
If the bisector of the exterior vertical angle of a triangle be parallel to the base. Show that the triangle is isosceles.
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.
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.
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):
The bisectors of two equal angles of a triangle are equal
Fill the blank in the following so that the following statement is true.
In an equilateral triangle all angles are .....
Fill the blank in the following so that the following statement is true.
In a ΔABC if ∠A = ∠C , then AB = ......
In a ΔABC, if ∠B = ∠C = 45°, which is the longest side?
Which of the following statements are true (T) and which are false (F)?
Sum of the three sides of a triangle is less than the sum of its three altitudes.
In the given figure, if AB || DE and BD || FG such that ∠FGH = 125° and ∠B = 55°, find x and y.
In ΔABC, if ∠A = 100°, AD bisects ∠A and AD ⊥ BC. Then, ∠B =
In ∆PQR, if ∠R > ∠Q, then ______.
Show that in a quadrilateral ABCD, AB + BC + CD + DA < 2(BD + AC)
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.
