Advertisements
Advertisements
Question
If the bisectors of the acute angles of a right triangle meet at O, then the angle at Obetween the two bisectors is
Options
45°
95°
135°
90°
Advertisements
Solution
In the given problem, bisectors of the acute angles of a right angled triangle meet at O. We need to find ∠AOC.
Now, using the angle sum property of a triangle
In ΔABC
∠A + ∠B + ∠C = 180°
90° + ∠A+ ∠C = 180°
∠A + ∠C = 90° ..............(1)
Now, further multiplying each of the term by 1/2in (1)
`1/2 ∠A + 1/2 ∠C = 1/2 90°`
∠OAC + ∠ACO = 45°
Also, applying angle sum property of a triangle
In ΔAOC
∠OAC + ∠ACO + ∠AOC = 180°
45° + ∠AOC = 180°
∠AOC = 180° - 45°
= 135°
Thus, ∠AOC = 135°
APPEARS IN
RELATED QUESTIONS
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 Figure AB = AC and ∠ACD =105°, find ∠BAC.
Prove that the medians of an equilateral triangle are equal.
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.
ABC is a right angled triangle in which ∠A = 90° and AB = AC. Find ∠B and ∠C.
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
Which of the following statements are true (T) and which are false (F):
The two altitudes corresponding to two equal sides of a triangle need not be equal.
Fill the blank in the following so that the following statement is true.
Angle opposite to equal sides of a triangle are .....
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 in the blank to make the following statement true.
If two angles of a triangle are unequal, then the smaller angle has the........ side opposite to it.
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.
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, what is the value of x?
In the given figure, the value of x is ______.
In a ΔABC, ∠A = 50° and BC is produced to a point D. If the bisectors of ∠ABC and ∠ACDmeet at E, then ∠E =
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 = ______.
In the given figure, if l1 || l2, the value of x is
In triangles ABC and PQR, AB = AC, ∠C = ∠P and ∠B = ∠Q. The two triangles are ______.
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.
Show that in a quadrilateral ABCD, AB + BC + CD + DA > AC + BD
