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
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.
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.
Determine the measure of each of the equal angles of a right-angled isosceles triangle.
Prove that the medians of an equilateral triangle are equal.
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.
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.
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.
If altitudes CE and BF of a triangle ABC are equal, then AB = ....
In Fig. 10.131, prove that: (i) CD + DA + AB + BC > 2AC (ii) CD + DA + AB > BC
ABC is a triangle. The bisector of the exterior angle at B and the bisector of ∠C intersect each other at D. Prove that ∠D = \[\frac{1}{2}\] ∠A.
In a triangle ABC, if AB = AC and AB is produced to D such that BD = BC, find ∠ACD: ∠ADC.
D is a point on the side BC of a ∆ABC such that AD bisects ∠BAC. Then ______.
In triangles ABC and PQR, AB = AC, ∠C = ∠P and ∠B = ∠Q. The two triangles are ______.
If ∆PQR ≅ ∆EDF, then is it true to say that PR = EF? Give reason for your answer
AD is a median of the triangle ABC. Is it true that AB + BC + CA > 2AD? 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.
Is it possible to construct a triangle with lengths of its sides as 9 cm, 7 cm and 17 cm? Give reason for your answer.
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.
