Advertisements
Advertisements
Question
ΔABC is an isosceles triangle in which AB = AC. Side BA is produced to D such that AD = AB (see the given figure). Show that ∠BCD is a right angle.
Advertisements
Solution
AB = AC ...[Given] ...(1)
AB = AD ...[Given] ...(2)
From (1) and (2), we have
AC = AD
Now, in ΔABC, we have
∠ABC + ∠ACB + ∠BAC = 180° ...[Angle sum property of a Δ]
⇒ 2∠ACB + ∠BAC = 180° ...(3) ...[∠ABC = ∠ACB (Angles opposite to equal sides of a Δ are equal)]
Similarly, in ΔACD,
∠ADC + ∠ACD + ∠CAD = 180°
⇒ 2∠ACD + ∠CAD = 180° ...(4) ...[∠ADC = ∠ACD (Angles opposite to equal sides of a Δ are equal)]
Adding (3) and (4), we have
2∠ACB + ∠BAC + 2∠ACD + ∠CAD = 180° +180°
⇒ 2[∠ACB + ∠ACD] + [∠BAC + ∠CAD] = 360°
⇒ 2∠BCD + 180° = 360° ...[∠BAC and ∠CAD form a linear pair]
⇒ 2∠BCD = 360° − 180°
⇒ 2∠BCD = 180°
⇒ ∠BCD = `(180°)/2`
Thus, ∠BCD = 90°
APPEARS IN
RELATED QUESTIONS
In ΔABC, AD is the perpendicular bisector of BC (see the given figure). Show that ΔABC is an isosceles triangle in which AB = AC.
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.
ABC and DBC are two isosceles triangles on the same base BC (see the given figure). Show that ∠ABD = ∠ACD.
Determine the measure of each of the equal angles of a right-angled isosceles triangle.
In a ΔABC, if AB = AC and ∠B = 70°, find ∠A.
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 an isosceles triangle, if the vertex angle is twice the sum of the base angles, calculate the angles of the triangle.
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.
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.
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
Fill the blank in the following so that the following statement is true.
In an isosceles triangle ABC with AB = AC, if BD and CE are its altitudes, then BD is …… CE.
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.
Which of the following statements are true (T) and which are false (F)?
If two angles of a triangle are unequal, then the greater angle has the larger side opposite to it.
In a triangle ABC, if AB = AC and AB is produced to D such that BD = BC, find ∠ACD: ∠ADC.
In ΔABC, if ∠A = 100°, AD bisects ∠A and AD ⊥ BC. Then, ∠B =
Line segments AB and CD intersect at O such that AC || DB. If ∠CAB = 45° and ∠CDB = 55°, then ∠BOD =
If the measures of angles of a triangle are in the ratio of 3 : 4 : 5, what is the measure of the smallest angle of the triangle?
In the given figure, what is y in terms of x?
In the following figure, D and E are points on side BC of a ∆ABC such that BD = CE and AD = AE. Show that ∆ABD ≅ ∆ACE.
