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Practice Sheet 3

The document presents a series of geometric problems involving triangles, quadrilaterals, and angle bisectors, requiring proofs of congruence and equality. Each problem is structured to demonstrate fundamental properties of geometric figures, such as isosceles triangles and angle bisectors. The problems are designed for practice in geometric reasoning and proof techniques.

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50 views4 pages

Practice Sheet 3

The document presents a series of geometric problems involving triangles, quadrilaterals, and angle bisectors, requiring proofs of congruence and equality. Each problem is structured to demonstrate fundamental properties of geometric figures, such as isosceles triangles and angle bisectors. The problems are designed for practice in geometric reasoning and proof techniques.

Uploaded by

dippu963
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Some Basic Problem

1. In the given figure, BE and CF are two equal altitudes of ABC . Show that

(i) ABE  ACF ,


(ii) AB = AC .

2. 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 intersect 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.

3. In the given figure, if x = y and AB = CB then prove that AE = CD.

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Some Basic Problem

4. In the given figure, line l is the bisector of an angle A and B is any point on l. If BP and B
Q 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 .

5. ABCD is a quadrilateral such that diagonal AC bisects the angles A and C . Prove that
AB = AD and CB = CD .

6. In the given figure, ABCD is a quadrilateral in which AB ‖ DC and P is the midpoint of B


C. On producing, AP and DC meet at Q . Prove that

(i) AB = CQ,
(ii) DQ = DC + AB.

7. 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.

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Some Basic Problem

8. 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.

9. 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 .

10. The bisectors of B and C of an 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 .

11. 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 .

12. 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.

13. 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.

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Some Basic Problem

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