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

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(a)

In ΔABC, if ㄥA = 50° and ㄥB = 60°, determine the shortest and the longest side of the triangle.

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In a ΔDEF, if ㄥD = 30°, ㄥE = 60° then which side of the triangle is longest and which side is shortest?

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In ΔABC, ㄥA = 60°, ㄥB = 40°, which side of this triangle is the smallest? Give reasons for your answer.

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In ΔPQR, ㄥP = 100° and ㄥR = 60°, which side of the triangle is the longest. Give reasons for your answer.

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In the figure, \(\triangle\)ABC and \(\triangle\)DBC are two isosceles triangles on the same base BC Prove that \(\angle\) ABD = \(\angle\)ACD.

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In the figure below, the diagonal AC of quadrilateral ABCD bisects \(\angle\)BAD and \(\angle\)BCD.
Prove that BC = CD.

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In the figure below, O is the mid-point of AB and CD, Prove that AC=BD.

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In the given figure, D is the mid-point of base BC DE and DF are perpendiculars to AB and AC respectively such that DE = DF. Prove that \(\angle\)B = \(\angle\)C.

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In the figure below, ABCD is a square and P is the mid-point of AD. BP and CP are joined. Prove that \(\angle\)PCB = \(\angle\)PBC.

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In the figure, OA = OB and OD = OC Show that:
(i) \(\triangle AOD\cong \triangle BOC\),
(ii) AD II BC.

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In figure \(\angle\)B = \(\angle\)E, BD = CE and \(\angle\)1 =\(\angle\)2. Show \(\triangle ABC\cong \triangle AED.\)

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In figure, AX = BY and AX II BY, prove that \(\triangle APX\cong \triangle BPY.\)

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PS is an altitude of an isosceles triangle PQR in which PQ = PR. Show that PS bisects \(\angle\)P.

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In the figure, given AC > AB and AD is the bisector of \(\angle\)A. Show that \(\angle\)ADC > \(\angle\)ADB.

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(a)