AAS Criteria for Congruency

Q.

In a right angled triangle, one acute angle is double the other. Prove that the hypotenuse is double the smallest side. [4 MARKS]

Q.

The image of an object placed at a point A before a plane mirror LM is seen at the point B by an observer at D, as shown in the figure. Prove that the image is as far behind the mirror as the object is in front of the mirror.

Q.

In the figure, if AC is bisector of $\angle BAD$ such that AB = 3 cm and AC = 5 cm, then CD =

1. 4 cm

2. 5 cm

3. 2 cm

4. 3 cm

Q. Question 18
If ABC is a right angled triangle such that AB = AC and bisector of angle C intersects the side AB at D, then prove that AC + AD = BC.

Q.

$\text{In the given figure, ABC is an equilateral triangle;}PQ||AC\text{and AC is produced to R such that CR = BP. Prove that QR bisects PC.}$

Q.

A triangle ABC has $\angle$ B = $\angle$ C.
Prove that :

(i) The perpendiculars from the mid-point of BC to AB and AC are equal.

(ii) The perpendiculars from B and C to the opposite sides are equal.

Q.

$ BE$ and $ CF$ are two equal altitudes of a triangle $ ABC$. Using $ RHS$ congruence rule, prove that the triangle$ ABC$ is isosceles.

Q. In Figure, PS = QR and $\angle SPQ=\angle RQP$.


Prove that PR = QS and $\angle QPR=\angle PQS$.

Q.

$\text{In the given figure, OA = OB and OP = OQ.}\text{Prove that (i) PX = QX, (ii) AX = BX.}$

Q.

$△\mathrm{ABC}$is a triangle in which altitudes $BE$ and$CF$ to sides $AC$and $AB$ are equal (see Fig.) . Show that $\mathrm{\Delta ABE}\cong \mathrm{\Delta ACF}$

Q.

$\text{In the given figure, if x = y and AB = CB then prove that AE = CD.}$

Q.

In the given figure, AD and BC are equal perpendiculars to a line segment AB.

Show that CD bisects AB.

Q.

In two right triangles, one side an acute angle of one are equal to the corresponding side and angle of the other. Prove that the triangles are congruent.

Q.

In the given figure if BE = CF, then which of these following options is correct?

1. $\Delta ABC\cong \Delta ACF$

2. $\Delta ABE\cong \Delta ACF$

3. $\Delta ABE\cong \Delta CAF$

4. $\Delta ABC\cong \Delta AFC$

Q.

$\text{In the given figure, line l is the bisector of an angle}\angle A\text{and B is any point on l.}\text{If BP and BQ are perpendiculars from B to the arms of}\angle A,\text{show that}\left(i\right)\Delta APB\cong \Delta AQB\text{(ii) BP = BQ, i.e., B is equidistant from the armos of}\angle A.$

[3 Marks]

Q.

In triangle ABC, AB = AC; BE $\perp$ AC and CF $\perp$ AB. Prove that :

(i) BE =CF

(ii) AF = AE

Q.

In Fig. it is given that AB = BC and AD = EC. Prove that $\Delta$ABE $\cong$ $\Delta$CBD
[3 MARKS]

Q. Question 9
If ABC is an isosceles triangle in which AC = BC, AD and BE are respectively two altitudes to sides BC and AC, then prove that AE = BD.

Q.

In the given figure, if x = y and AB = CB, then AE is _____.

1. Can't be determined

2. less than CD

3. greater than CD

4. equal to CD

Q.

In triangle ABC, altitudes BE and CF are equal. Prove that the triangle is isosceles.

Q.

Question 4
BE and CF are two equal altitudes of a triangle ABC. Using RHS congruence rule, prove that the triangle ABC is isosceles.

Q. Question 6 (i)
In the given figure, diagonals AC and BD of quadrilateral ABCD intersect at O such that OB = OD. If AB = CD, then show that:
(i) ar(DOC) = ar(AOB)

Q.

Diagonals AC and BD of quadrilateral ABCD intersect at O such that OB = OD.If AB = CD, then show that:
(i) ar ($△$DOC) = ar ($△$AOB) (ii) ar ($△$DCB) = ar ($△$ACB) [4 MARKS]

Q. In the given figure, AC = CE and AB||ED. Using this information, find the value of $x$.

1. 13
2. 15
3. 25
4. 17

Q.

In the parallelogram ABCD, the angles A and C are obtuse. Points X and Y are taken on the diagonal BD such that the angles XAD and YCB are right angles.

Prove that : XA = YC.

Q. BE and CF are two equal altitudes of a triangle ABC. Using RHS congruence rule, prove that the triangle ABC is isosceles.
(3 Marks)

Q.

In the given figure, $ABCD$ is a parallelogram and $PD$ is parallel to $QC.$

S1 : $△APD\cong △BQC$

S2 : ar($△APD)+$ ar($PBCD)=$ ar($△BQC)+$ ar$\left(CDPB\right)$

1. S1 is true but S2 is false

2. S1 is false but S2 is true

3. S1 and S2 are both true and S2 is the explanation for S1

4. S1 and S2 are are both true and S1 is the explanation for S2

Q.

In the following diagram, AP and BQ are equal and parallel to each other.

Prove that :

(i) $\Delta$ AOP $\cong \Delta$ BOQ.

(ii) AB and PQ bisect each other.

Q.

PQRS is a parallelogram. L and M are points on PQ and SR respecitvely such that PL = MR. Show that LM and QS bisect each other.

Q. $ABCD$ is a parallelogram and $AP$ and $CQ$ are perpendiculars from vertices $A$ and $C$ on diagonal $BD$ Show that
i) $△APB\cong △CQD$
ii) $AP=CQ$