Advertisements
Advertisements
Question
In the given figure, the sides BC, CA and AB of a Δ ABC have been produced to D, E and F respectively. If ∠ACD = 105° and ∠EAF = 45°, find all the angles of the Δ ABC.
Advertisements
Solution
In the given ΔABC, ∠ACD = 105°and ∠EAF = 45°. We need to find ∠ABC, ∠ACB, and ∠BAC.
Here, ∠EAF and ∠BAC are vertically opposite angles. So, using the property, “vertically opposite angles are equal”, we get,
∠EAF = ∠BAC
∠BAC = 45°
Further, BCD is a straight line. So, using linear pair property, we get,
∠ACB + ∠ACD = 180°
∠ACB + 105° = 180°
∠ACB = 180° - 105°
∠ACB = 75°
Now, in ΔABC, using “the angle sum property”, we get,
∠ABC + ∠ACB + ∠BAC = 180°
45° + 75° + ∠ABC = 180°
∠BAC = 180°
∠BAC = 180° - 120°
∠BAC = 60
Therefore,∠ACB = 75° , ∠BAC = 45°,∠ABC = 60°.
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 and DBC are two isosceles triangles on the same base BC (see the given figure). Show that ∠ABD = ∠ACD.
In Figure AB = AC and ∠ACD =105°, find ∠BAC.
AB is a line seg P and Q are points on opposite sides of AB such that each of them is equidistant from the points A and B (See Fig. 10.26). Show that the line PQ is perpendicular bisector of AB.
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):
Angles opposite to equal sides of a triangle are equal
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)?
Difference of any two sides of a triangle is equal to the third side.
Fill in the blank to make the following statement true.
The sum of any two sides of a triangle is .... than the third side.
If the angles of a triangle are in the ratio 2 : 1 : 3, then find the measure of smallest angle.
In a triangle ABC, if AB = AC and AB is produced to D such that BD = BC, find ∠ACD: ∠ADC.
In the given figure, x + y =
In the given figure, for which value of x is l1 || l2?
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
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 = ______.
Which of the following correctly describes the given triangle?
CDE is an equilateral triangle formed on a side CD of a square ABCD (Figure). Show that ∆ADE ≅ ∆BCE.
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
