Advertisements
Advertisements
प्रश्न
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
उत्तर
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
संबंधित प्रश्न
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
In ΔABC, AD is the perpendicular bisector of BC (see the given figure). Show that ΔABC is an isosceles triangle in which AB = AC.
In figure, AB = AC and DB = DC, find the ratio ∠ABD : ∠ACD
Prove that the medians of an equilateral triangle are equal.
If the base of an isosceles triangle is produced on both sides, prove that the exterior angles so formed are equal to each other.
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°.
Which of the following statements are true (T) and which are false (F):
Angles opposite to equal sides of a triangle are equal
If the angles A, B and C of ΔABC satisfy the relation B − A = C − B, then find the measure of ∠B.
In the given figure, if AB ⊥ BC. then x =
In the given figure, what is y in terms of x?
In ∆ABC, AB = AC and ∠B = 50°. Then ∠C is equal to ______.
In ∆ABC, BC = AB and ∠B = 80°. Then ∠A is equal to ______.
It is given that ∆ABC ≅ ∆FDE and AB = 5 cm, ∠B = 40° and ∠A = 80°. Then which of the following is true?
AD is a median of the triangle ABC. Is it true that AB + BC + CA > 2AD? 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.
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
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].
