Advertisements
Advertisements
प्रश्न
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
उत्तर
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
संबंधित प्रश्न
If the base of an isosceles triangle is produced on both sides, prove that the exterior angles so formed are equal to each other.
In figure, AB = AC and DB = DC, find the ratio ∠ABD : ∠ACD
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.
PQR is a triangle in which PQ = PR and S is any point on the side PQ. Through S, a line is drawn parallel to QR and intersecting PR at T. Prove that PS = PT.
Angles A, B, C of a triangle ABC are equal to each other. Prove that ΔABC is equilateral.
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°.
ABC is a right angled triangle in which ∠A = 90° and AB = AC. Find ∠B and ∠C.
In a ΔPQR, if PQ = QR and L, M and N are the mid-points of the sides PQ, QR and RP
respectively. Prove that LN = MN.
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
Which of the following statements are true (T) and which are false (F)?
Sum of any two sides of a triangle is greater than the third side.
Fill in the blank to make the following statement true.
If two sides of a triangle are unequal, then the larger side has .... angle opposite to it.
In the given figure, if AB || DE and BD || FG such that ∠FGH = 125° and ∠B = 55°, find x and y.
If the bisectors of the acute angles of a right triangle meet at O, then the angle at Obetween the two bisectors is
Which of the following correctly describes the given triangle?
In ∆PQR, if ∠R > ∠Q, then ______.
Show that in a quadrilateral ABCD, AB + BC + CD + DA < 2(BD + AC)
