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Triangles

Short Answer Type

11. In figure, diagonal AC of a quadrilateral ABCD bisects the angles A and C. Prove that AB = AD and CB = CD.

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12.
AB is a line-segment. AX and BY are two equal line-segments drawn on opposite sides of line AB such that AX || BY. If AB and XY intersect each other at P. Prove that:
(i) ∆APX ≅ ∆BPY
(ii) AB and XY bisect each other at P.

(i)    ∵ AX || BY and AB intersects them
∴ ∠PAX = ∠PBY    ...(1)
| Alternate Angles
&#8757 AX || BY and XY intersects them
∴ ∠PXA = ∠PYB    ...(2)
| Alternate Angles
In ∆APX and ∆BPY,
∠PAX = ∠PBY    | From (1)
∠PXA = ∠PYB    | From (2)
AX = BY    | Given
∴ ∆APX = ∆BPY    | ASA Axiom
(ii)    ∵    AP = BP    | C.P.C.T.
and PX = PY | C.P.C.T.
⇒ AB and XY bisect each other at P.

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13. In figure, ∠QPR = ∠PQR and M and N are respectively points on sides QR and PR of ∆PQR, such that QM = PN. Prove that OP = OQ, where O is the point of intersecting of PM and QN.

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Long Answer Type

14. Prove that the medians of an equilateral triangle are equal.

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Short Answer Type

15. In figure, ∠B = ∠.E, BD = CE and ∠1 = ∠2. Show ∆ABC ≅ ∆AED.

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16.

In figure given below, AD is the median of ∆ABC.
BE ⊥ AD, CF ⊥ AD. Prove that BE = CF.

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17. In the given figure, if AB = FE, BC = ED, AB ⊥ BD and FE ⊥ EC, then prove that AD = FC.

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18.

In figure, OA = OB and OD = OC. Show that:
(i) ∆AOD ≅ ∆BOC and (ii) AD = BC.

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19. AB is a line segment and line l is its perpendicular bisector. If a point P lies on I, show that P is equidistant from A and B.

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20.

Line-segment AB is parallel to another line-segment CD. O is the mid-point of AD (see figure). Show that: (i) ∆AOB ≅ ∆DOC (ii) O is also the mid-point of BC.