Advertisements
Advertisements
Question
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 = ______.
Options
85°
- \[72\frac{1}{2}^\circ\]
145°
none of these
Advertisements
Solution
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 = `bbunderline(72 1/2)` °
Explanation:
In the given problem, BC of ΔABC is produced to point D. bisectors of ∠A meet side BC at L, ∠ABC = 30° and ∠ACD = 115°
Here, using the property, “exterior angle of a triangle is equal to the sum of the two opposite interior angles”, we get,
In ΔABC
∠ACD = ∠CAB + ∠CBA
115° = ∠CAB + 30°
∠CAB = 115° - 30°
∠CAB = 85°
Now, as AL is the bisector of ∠A
∠CAL = 1/2 ∠CAB
∠CAL = 1/2 (85°)
`∠CAL = 44 (1^\circ)/2`
Also, ∠ACD is the exterior angle of ΔALC
Thus,
Again, using the property, “exterior angle of a triangle is equal to the sum of the two opposite interior angles”, we get,
In ΔALC
∠ACD = ∠CAL + ∠ALC
`115= 44 (1^\circ)/2+∠ALC`
`∠ALC =115- 44 1^\circ/2`
`∠ALC = 72 (1^\circ)/2`
Thus, `∠ALC = 72 1^\circ/2 `
APPEARS IN
RELATED QUESTIONS
ABC and DBC are two isosceles triangles on the same base BC (see the given figure). Show that ∠ABD = ∠ACD.
Find the measure of each exterior angle of an equilateral triangle.
In Fig. 10.40, it is given that RT = TS, ∠1 = 2∠2 and ∠4 = 2∠3. Prove that ΔRBT ≅ ΔSAT
In a ΔABC, if AB = AC and ∠B = 70°, find ∠A.
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.
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.
Sides opposite to equal angles of a triangle are ......
In a ΔABC, if ∠B = ∠C = 45°, which is the longest side?
In Fig. 10.131, prove that: (i) CD + DA + AB + BC > 2AC (ii) CD + DA + AB > BC
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 a ΔABC, if ∠A = 60°, ∠B = 80° and the bisectors of ∠B and ∠C meet at O, then ∠BOC =
In the given figure, what is the value of x?
In the given figure, the value of x is ______.
In the given figure, AB and CD are parallel lines and transversal EF intersects them at Pand Q respectively. If ∠APR = 25°, ∠RQC = 30° and ∠CQF = 65°, then
In ∆ABC, BC = AB and ∠B = 80°. Then ∠A is equal to ______.
If ∆PQR ≅ ∆EDF, then is it true to say that PR = EF? Give reason for your answer
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.
Bisectors of the angles B and C of an isosceles triangle ABC with AB = AC intersect each other at O. Show that external angle adjacent to ∠ABC is equal to ∠BOC
