Advertisements
Advertisements
Question
Two lines AB and CD intersect at O such that BC is equal and parallel to AD. Prove that the lines AB and CD bisect at O.
Advertisements
Solution
It is given that
BC = AD
BC || AD
We have to prove that AB and CD bisect at O.
If we prove that ΔAOD ≅ ΔBOC , then
We can prove AB and CDbisects atO.
Now in ΔAOD and ΔBOC
AD = BC(Given)
∠OBC =∠OAD (Since AD || BC and AB is transversal)
And ∠OCB = ∠ODA(since AD || BC and CD is transversal)
So by ASAcongruence criterion we have,
ΔAOD ≅ ΔBOC, so
OA = OB
OD = OC
Hence ABand CD bisect each other at O.
APPEARS IN
RELATED QUESTIONS
ABC and DBC are two isosceles triangles on the same base BC (see the given figure). Show that ∠ABD = ∠ACD.
Show that the angles of an equilateral triangle are 60° each.
In Fig. 10.40, it is given that RT = TS, ∠1 = 2∠2 and ∠4 = 2∠3. Prove that ΔRBT ≅ ΔSAT
Prove that the medians of an equilateral triangle are equal.
In Figure 10.24, AB = AC and ∠ACD =105°, find ∠BAC.
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 triangle in which ∠B = 2 ∠C. D is a point on BC such that AD bisects ∠BAC and AB = CD.
Prove that ∠BAC = 72°.
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):
Angles opposite to equal sides of a triangle are equal
Which of the following statements are true (T) and which are false (F) :
If the altitude from one vertex of a triangle bisects the opposite side, then the triangle may be isosceles.
Which of the following statements are true (T) and which are false (F):
The bisectors of two equal angles of a triangle are equal
O is any point in the interior of ΔABC. Prove that
(i) AB + AC > OB + OC
(ii) AB + BC + CA > OA + QB + OC
(iii) OA + OB + OC >` 1/2`(AB + BC + CA)
Fill in the blank to make the following statement true.
The sum of any two sides of a triangle is .... than the third side.
Fill in the blank to make the following statement true.
Difference of any two sides of a triangle is........ than the third side.
Line segments AB and CD intersect at O such that AC || DB. If ∠CAB = 45° and ∠CDB = 55°, then ∠BOD =
In the given figure, for which value of x is l1 || l2?
In the given figure, if BP || CQ and AC = BC, then the measure of x is
If the bisectors of the acute angles of a right triangle meet at O, then the angle at Obetween the two bisectors is
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.
