In the given figure, AB || CD and O is the midpoint of AD. Show that
(i) △ AOB ≅ △ DOC
(ii) O is the midpoint of BC.
In the given figure, AD and BC are equal perpendiculars to a line segment AB. Show that CD bisects AB.
In the given figure, two parallel lines l and m are intersected by two parallel lines p and q. Show that △ ABC ≅ △ CDA.
AD is an altitude of an isosceles △ ABC in which AB = AC. Show that
(i) AD bisects BC,
(ii) AD bisects ∠ A.
In the given figure, BE and CF are two equal altitudes of △ ABC. Show that
(i) △ ABE ≅ △ ACF,
(ii) AB = AC.
△ ABC and △ DBC are two isosceles triangles on the same base BC and vertices A and D are on the
same side of BC. If AD is extended to interest BC at E, show that
(i) △ ABD ≅ △ ACD
(ii) △ ABE ≅ △ ACE
(iii) AE bisects ∠ A as well as ∠ D
(iv) AE is the perpendicular bisector of BC.
In the given figure, line l is the bisector of an angle ∠ A and B is any point on l. If BP and BQ are
perpendiculars from B to the arms of ∠ A, show that
(i) △ APB ≅ △ AQB
(ii) BP = BQ, i.e., B is equidistant from the arms of ∠ A.
ABCD is a quadrilateral such that diagonal AC bisects the angles ∠ A and ∠ C. Prove that AB = AD and CB = CD.
△ ABC is a right triangle right angled at A such that AB = AC and bisector of ∠ C intersects the side AB at D. Prove that AC + AD = BC.
In the given figure, ABC is an equilateral triangle; PQ || AC and AC is produced to R such that CR= BP. Prove that QR bisects PC.
In the given figure, ABCD is a quadrilateral in which AB || DC and P is the midpoint of BC. On
producing, AP and DC meet at Q. Prove that
(i) AB = CQ,
(ii) DQ = DC + AB.
In the given figure, ABCD is a square and P is a point inside it such that PB = PD. Prove that CPA is a straight line.
In the given figure, O is a point in the interior of square ABCD such that △ OAB is an equilateral
triangle. Show that △ OCD is an isosceles triangle.
In the adjoining figure, X and Y are respectively two points on equal sides AB and AC of △ ABC such that AX = AY. Prove that CX = BY.
In △ ABC, AB = AC and the bisectors ∠ B and ∠ C meet at a point O. Prove that BO = CO and the ray AO is the bisector of ∠ A.
The line segments joining the midpoints M and N are parallel sides AB and DC respectively of a trapezium ABCD is perpendicular to both the sides AB and DC. Prove that AD = BC.
The bisectors ∠ B and ∠ C of and isosceles triangle with AB = AC intersect each other at a point O. BO is produced to meet AC at a point M. Prove that ∠ MOC = ∠ ABC.
The bisectors of ∠ B and ∠ C of an isosceles △ ABC with AB = AC intersect each other at a point O. Show that the exterior angle adjacent to ∠ ABC is equal to ∠ BOC.
P is a point on the bisector of ∠ ABC. If the line through P, parallel to BA meets BC at Q, prove that △ BPQ is an isosceles triangle.
The image of an object placed at a point A before a plane mirror LM is seen at the point B by an observer at D, as shown in the figure. Prove that the image is as far behind the mirror as the object is in front of the mirror.
“If two sides and an angle of one triangle are equal to two sides and an angle of another triangle then the two triangles must be congruent.” Is the statement true? Why?
Is it possible to construct a triangle with lengths of its sides as given below? Give reason for your answer.
(i) 5cm, 4cm, 9cm
(ii) 8cm, 7cm, 4cm
(iii) 10cm, 5cm, 6cm
(iv) 2.5cm, 5cm, 7cm
(v) 3cm, 4cm, 8cm
(i) In △ ABC, ∠ A = 90. What is the longest side?
(ii) In △ ABC, ∠ A = ∠ B = 45. Which is its longest side?
(iii) In △ ABC, ∠ A = 100 and ∠ C = 50. Which is its shortest side?
In △ ABC, side AB is produced to D such that BD = BC. If ∠ A = 70 and ∠ B = 60, prove that
(i) AD > CD
(ii) AD > AC.
AB and CD are respectively the smallest and largest sides of a quadrilateral ABCD. Show that ∠ A > ∠ C and ∠ B > ∠ D.
In △ ABC, ∠ B = 35o, ∠ C = 65o and the bisector of ∠ BAC meets BC in X. Arrange AX, BX and CX in descending order.
In the given figure, PQ > PR and QS and RS are the bisectors of ∠ Q and ∠ R respectively. Show that SQ > SR.
Prove that in a triangle, other than an equilateral triangle, angle opposite the longest side is greater 2/3 of a right angle.
If O is a point within △ ABC, show that
(i) AB + AC > OB + OC
(ii) AB + BC + CA > OA + OB + OC
(iii) OA + OB + OC > ½ (AB + BC + CA)
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