Q4 of 76 Page 11

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

Given that in figure,

AD ⊥ CD and CB ⊥ CD


And,


AQ = BP, DP = CQ


WE have to prove that,


∠DAQ = ∠CBP


Given that, DP = QC


Adding PQ on both sides, we get


DP + PQ = PQ + QC


DQ = PC (i)


Now consider , we have


∠ADQ = ∠BCP = 90o (Given)


AQ = BP (Given)


And,


DQ = PC (From i)


So, by RHS congruence rule, we have



Now,


∠DAQ = ∠CBP (By c.p.c.t)


Hence, proved


More from this chapter

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2

ABC is a triangle in which BE and CF are, respectively, the perpendiculars to the sides AC and AB. If BE=CF, prove that ΔABC is isosceles.

 3

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

 5

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.

 6

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.