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Triangles

Short Answer Type

21. In the figure, ∠BCD = ∠ADC and ∠ACB = ∠BDA. Prove that AD = BC and ∠A = ∠B.

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22. In figure, AX = BY and AX || BY. Prove that ∆APX ≅ ∆BPY.

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23. In figure, if ∠AEO = ∠CDO and AB = CB, prove that ∆ABD ≅ ∆CBE.

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24. In figure, if ∠ABD = ∠ACE and AB = AC. Prove that ∆ABD ≅ ∆ACE.

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25. In the figure given below, ABCD is a quadrilateral in which diagonal AC bisects ∠A and ∠C, prove that ∆ABC ≅ ∆ADC.

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26. In the figure below, the diagonal AC of quadrilateral ABCD bisects ∠BAD and ∠BCD. Prove that BC = CD.

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27. In the following figure, AD is the bisector of ∠A of ∆ABC. PQ and PR are perpendiculars from any point lying on AD, P to sides AB and AC respectively. Show that ∆PQA ≅ ∆PRA and PQ = PR.

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28.

In an isosceles triangle ABC, with AB = AC, the bisectors of ∠B and ∠C intersect each other at O. Join A to O. Show that:

(i) OB = OC
(ii) AO bisects ∠A.

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29. In ∆ ABC, AD is the perpendicular bisector of BC (see figure). Show that A ABC is an isosceles triangle in which AB = AC.

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30. ABC is an isosceles triangle in which altitudes BE and CF are drawn to sides AC and AB respectively (see figure). Show that these altitudes are equal.

Given: ABC is an isosceles triangle in which altitudes BE and CF are drawn to sides AC and AB respectively.
To Prove: BE = CF.
Proof: ∵ ABC is an isosceles triangle
∴ AB = AC
∴ ∠ABC = ∠ACB    ...(1)
| Angles opposite to equal sides of a triangle are equal
In ∆BEC and ∆CFB,
∠BEC = ∠CFB    | Each = 90°
BC = CB    | Common
∠ECB = ∠FBC    | From (1)
∴ ∆BEC ≅ ∆CFB    | By AAS Rule
∴ BE = CF.    | C.P.C.T.

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