ABCD is a square, X and Y are points on sides AD and BC respectively such that AY = BX. Prove that BY = AX and ∠BAY = ∠ABX.

If perpendiculars from any point within an angle on its arms are congruent, prove that it lies on the bisector of that angle.

In Fig. 10.99, AD ⊥ CD and CB ⊥ CD. If AQ=BP and DP =CQ, prove that ∠DAQ = ∠CBP.

Which of the following statements are true (T) and which are false (F):

(i) Sides opposite to equal angles of a triangle may be unequal.

(ii) Angles opposite to equal sides of a triangle are equal.

(iii) The measure of each angle of an equilateral triangle is 60°.

(iv) If the altitude from one vertex of a triangle bisects the opposite side, then the triangle may be isosceles.

(v) The bisectors of two equal angles of a triangle are equal.

(vi) If the bisector of the vertical angle of a triangle bisects the base, then the triangle may be isosceles.

(vii) The two altitudes corresponding to two equal sides of a triangle need not be equal.

(viii) If any two sides of a right triangle are respectively equal to two sides of other right triangle, then the two triangles are congruent.

(ix) Two right triangles are congruent if hypotenuse and a side of one triangle are respectively equal to the hypotenuse and a side of the other triangle.

Fill in the blanks in the following so that each of the following statements is true.

(i) Sides opposite to equal angles of a triangle are ………

(i) Sides opposite to equal angles of a triangle are …………..

(iii) In an equilateral triangle all angles are ……….

(iv) In a Δ ABC, if ∠A=∠C, then AB = ………

(v) If altitudes CE and BF of a triangle ABC are equal, then, AB = ……..

(vi) In an isosceles triangle ABC with AB=AC, if BD and CE are its altitudes, then BD is ….CE.

(vii) In right triangles ABC and DEF, if hypotenuse AB=EF and side AC=DE, then Δ ABC ≅ Δ …….