Advertisements
Advertisements
Question
In a triangle ABC, if AB = AC and AB is produced to D such that BD = BC, find ∠ACD: ∠ADC.
Advertisements
Solution
In the given ΔABC,AB = ACand AB is produced to D such that BD = BC
We need to find ∠ACD : ∠ADC
Now, using the property, “angles opposite to equal sides are equal”
As AB = AC
∠6 = ∠4 ........(1)
Similarly,
As BD = BC
∠1 = ∠2 ........(2)
Also, using the property, “an exterior angle of the triangle is equal to the sum of the two opposite interior angle”
In ΔBDC
ext. ∠6 = ∠1 + ∠2
ext. ∠6 = ∠1 + ∠1 (Using 2)
ext. ∠6 = 2∠1
From (1), we get
ext. ∠4 = ∠2 .......(3)
Now, we need to find ∠ACD : ∠ADC
That is,
(∠4 + ∠2): ∠1
(2∠1 + ∠2) : ∠1 (Using 3)
(2∠1 + ∠1) : ∠1(Using 2)
3∠1 :∠1
Eliminating ∠1from both the sides, we get 3:1
Thus, the ratio of ∠ACD :∠ADC is 3 :1
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
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.
In Fig. 10.40, it is given that RT = TS, ∠1 = 2∠2 and ∠4 = 2∠3. Prove that ΔRBT ≅ ΔSAT
The vertical angle of an isosceles triangle is 100°. Find its base angles.
In Figure 10.24, AB = AC and ∠ACD =105°, find ∠BAC.
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°.
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.
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.
Fill the blank in the following so that the following statement is true.
If altitudes CE and BF of a triangle ABC are equal, then AB = ....
Fill the blank in the following so that the following statement is true.
In right triangles ABC and DEF, if hypotenuse AB = EF and side AC = DE, then ΔABC ≅ Δ ……
In a ΔABC, if ∠B = ∠C = 45°, which is the longest side?
In the given figure, if AB || DE and BD || FG such that ∠FGH = 125° and ∠B = 55°, find x and y.
In the given figure, if l1 || l2, the value of x is
In ∆ABC, BC = AB and ∠B = 80°. Then ∠A is equal to ______.
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 8 cm, 7 cm and 4 cm? Give reason for your answer.
Find all the angles of an equilateral triangle.
ABC is an isosceles triangle with AB = AC and D is a point on BC such that AD ⊥ BC (Figure). To prove that ∠BAD = ∠CAD, a student proceeded as follows:
In ∆ABD and ∆ACD,
AB = AC (Given)
∠B = ∠C (Because AB = AC)
and ∠ADB = ∠ADC
Therefore, ∆ABD ≅ ∆ACD (AAS)
So, ∠BAD = ∠CAD (CPCT)
What is the defect in the above arguments?
[Hint: Recall how ∠B = ∠C is proved when AB = AC].
