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.
Show that the angles of an equilateral triangle are 60° each.
If the base of an isosceles triangle is produced on both sides, prove that the exterior angles so formed are equal to each other.
PQR is a triangle in which PQ = PR and S is any point on the side PQ. Through S, a line is drawn parallel to QR and intersecting PR at T. Prove that PS = PT.
ABC is a right angled triangle in which ∠A = 90° and AB = AC. Find ∠B and ∠C.
Fill the blank in the following so that the following statement is true.
Sides opposite to equal angles of a triangle are ......
Fill the blank in the following so that the following statement is true.
In right triangles ABC and DEF, if hypotenuse AB = EF and side AC = DE, then ΔABC ≅ Δ ……
Prove that the perimeter of a triangle is greater than the sum of its altitudes.
Which of the following statements are true (T) and which are false (F)?
Sum of any two sides of a triangle is greater than the third side.
In the given figure, if EC || AB, ∠ECD = 70° and ∠BDO = 20°, then ∠OBD is
In the given figure, if BP || CQ and AC = BC, then the measure of x is
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 = ______.
The angles of a right angled triangle are
In ∆ABC, BC = AB and ∠B = 80°. Then ∠A is equal to ______.
In ∆PQR, ∠R = ∠P and QR = 4 cm and PR = 5 cm. Then the length of PQ is ______.
It is given that ∆ABC ≅ ∆FDE and AB = 5 cm, ∠B = 40° and ∠A = 80°. Then which of the following is true?
In ∆PQR, if ∠R > ∠Q, then ______.
If ∆PQR ≅ ∆EDF, then is it true to say that PR = EF? Give reason for your answer
CDE is an equilateral triangle formed on a side CD of a square ABCD (Figure). Show that ∆ADE ≅ ∆BCE.
