Question

Prove by SAS Congreuency rule

AB is a line-segment. P and Q are

points on opposite sides of AB such that each of them

is equidistant from the points A and B (see Fig.).

Show that the line PQ is the perpendicular bisector

of AB.

Remeber-use congruency rule-SAS.

Answer 1

QUESTION-:

 AB is a line-segment. P and Q are  points on opposite sides of AB such that each of them  is equidistant from the points A and B .Show that the line PQ is the perpendicular bisector  of AB.

EXPLANATION-:

Here first let's prove ΔPAQ≅ΔBPQ

In ΔPAQ and ΔPBQ

AP=BP                                 (Given)

AQ=BQ                               (Given)

PQ=PQ                                (Common)

By SSS rule we have -:

ΔPAQ≅ΔBPQ

Now-:

∠APQ=BPQ                        (CPCT)

Now in ΔPAC and ΔPBC

AP=BP                                (Given)

∠APC=∠BPC                      (Proved above)

PC=PC                                 (Common)

By SAS -:

ΔPAC≅ΔPBC

Now

AC=BC                               (CPCT)

 AND

∠ACP=∠PCB                     (CPCT)

Now-:

∠ACP+∠BCP=180°

2∠ACP=180°

∠ACP=90°

So we can say PQ is perpendicular bisector of AB

HENCE PROVED.

Answer 2

Step-by-step explanation:

To find - line PQ is the perpendicular bisector of AB.

(By congruency rule-SAS.)

To show Line PQ perpendicular bisector we are supposed to show AC=CB and angle ACP =angle PCB = 90 °

thus, we have to show AQP and triangle PQB congurent first to get the required part ..

AP = BP (GIVEN)

AQ=BQ (GIVEN)

PQ = QP (GIVEN)

By SSS criteria AQP AND PQB ARE CONGURENT.

we can say ANGLE APC = ANGLE BPC by CPCT

After this to show PQ is the perpendicular bisector of AB.

we will have to prove ACP & PCB congruent by SAS.

AP = PB (GIVEN)

ANGLE APC = BPC ( By CPCT from the above part)

AC = CA (Common)

Hence by SAS proved that APC & PCB are congruent then, by CPCT AC= CB

hence proved that PQ is the bisector of line AB

but to show it perpendicular bisector we are supposed to show ACP and BCP a line and this we do by linear pair.

As it is given that AB is a line therefore according to linear pair axiom the sum of the adjacent angles would be 180°. We can say that angle ACP = PCB by CPCT. therefore let the unknown angle be x then,

x + x = 180°

2x = 180°

x = 90° then

angle ACP = PCB = 90° each

Hence proved PQ is the perpendicular bisector of AB.

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