ASA Criteria for Congruency

Q.

If two angles and the included side of a triangle are equal to two angles and the included side of another triangle, the two triangles are congruent by _____ congruence condition.

1. ASA

2. AAA

3. SAS

4. RHS

Q.

In a parallelogram ABCD, E and F are the mid-points of sides AB and CD respectively (see the given figure). Show that the line segments AF and EC trisect the diagonal BD.

Q.

Prove that the centre of the circum circle of a right angled triangle is the midpoint of its hypotenuse.

Q. In triangles ABC and PQR, ∠C = ∠R, BC = QR and ∠B = ∠Q, then by which congruency criteria are these two triangles congruent?

1. A.S.A. criterion
2. S.A.S. criterion
3. A.A.A. criterion
4. S.S.S. criterion

Q. 40. Triangle PQR is a right triangle which is right angled at Q. QX perpendicular to PR, XY perpendicular to RQ and XZ perpendicular to PQ are drawn. Prove that ZX2=PZ x ZQ

Q. Question 7
P and Q are the mid – points of the opposite sides AB and CD of a parallelogram ABCD. AQ intersects DP at S and BQ intersects CP at R. Show that PRQS is a parallelogram.

Q.

In ∆ABC and ∆DEF, AB = DF and ∠A = ∠D. The two triangles will be congruent by SAS axiom if:

1. BC = DE

2. AC = EF

3. BC = EF

4. AC = DE

Q.

In the figure given below, $PQR$is an isosceles triangle in which $PQ=PR$, $PQ$and $PR$are produced to $S$and $T$respectively such that $QS=RT$. Show that $RS=QT$

Q.

$\text{ABCD is a quadrilateral such that diagonal AC bisects that angles}\angle Aand\angle C.\text{Prove that AB = AD and CB = CD.}$

Q.

ABCD is a quadrilateral in which AD = BC and ∠DAB = ∠CBA (See the given figure). Prove that

(i) ΔABD ≅ ΔBAC

(ii) BD = AC

(iii) ∠ABD = ∠BAC.

Q.

BD and CE are bisectors of $\angle Band\angle C\text{of an isosceles}\Delta ABCwithAB=AC.$ Prove that BD = CE.

Q.

In the adjoining figure, the side AC of ∆ABC is produced to E such that CE = 1/2 AC. If D is the midpoint of BC and ED produced meets AB at F, and CP, DQ are drawn parallel to BA, then FD is

1. FD = 1/3 FE

2. FD = 2/3 FE

3. FD = 1/2 FE

4. FD = 1/4 FE

Q.

If AP bisects angle BAC and M is any point on AP, prove that the perpendiculars drawn from M to AB and AC are equal.

Q.

$In\Delta ABCand\Delta DEF,\text{it is given that}\angle B=\angle Eand\angle C=\angle F.\text{In order that}\Delta ABC\cong \Delta DEF,\text{we must have}\left(a\right)AB=DF\left(b\right)AC=DE\left(c\right)BC=EF\left(d\right)\angle A=\angle D$

Q.

For what value of $x$ will $△ABC$ be congruent to $△XYZ$ ?

1. $x={50}^{\circ }$
2. $x={40}^{\circ }$
3. $x={60}^{\circ }$
4. $x={30}^{\circ }$

Q.

Prove that the four triangles got by joining the midpoints of a triangle are all congruent and that they are similar to the original triangle.

Q. In triangles ABC and PQR, if ∠A = ∠R, ∠B = ∠P and AB = RP, then which one of the following congruence conditions applies:

(a) SAS

(b) ASA

(c) SSS

(d) RHS

Q. In the adjacent figure, $△$ ABC, D is the midpoint of BC. DE$\perp$ AB, DF$\perp$ AC and DE = DF. Show that $△$ BED $\cong △$ CFD

Q.

In triangles $\mathrm{ABC}$ and $\mathrm{PQR}$, $\angle \mathrm{A}=\angle \mathrm{Q}$ and $\angle \mathrm{B}=\angle \mathrm{R}$.

Which side of $△\mathrm{PQR}$ should be equal to side $\mathrm{BC}$ of $△\mathrm{ABC}$ so that the two triangles are congruent.

Q.

$△ABC$ is an isosceles triangle in which $AC=BC$. $AD$ and$BE$ are respectively two altitudes to sides $BC$ and $AC$. Prove that $AE=BD$.

Q.

In the adjoining figure, explain how one can find the breadth of the river without crossing it.

Q.

In the given figure, two parallel lines l and m are intersected by two parallel lines p and q.

$\text{Show that}\Delta ABC\cong \Delta CDA.$

Q. ABCD is a quadrilateral such that AB=AD and CB=CD.Prove that AC is the perpendicular bisector of BD.

Q. If CE is parallel to DB in the given figure, then the value of ‘x’ is ${30}^{\circ }$ .

1. True
2. False

Q. In a triangle ABC, if AD is the bisector of ∠BAC, which meets BC at point D such that AD ⊥ BC, then which of the following is always true?

1. BC = AC
2. BD = AD
3. ΔABD ≅ ΔCAD
4. ∠ABD = ∠ACD

Q.

$\text{In the given figure, BE and CF are two equal altitudes of}\Delta ABC.\text{Show that (i)}\Delta ABE\cong \Delta ACF,\left(ii\right)AB=AC.$

Q.

In triangles ABC and PQR, if $\angle A=\angle R.$

$\angle B=\angle PandAB=RP,\text{then which one of the following congruence condition applies:}$

1. ASA

2. SSS

3. SAS

4. RHS

Q.

If in $△ABC$, $AB=10cm$, $\angle CAB$ = ${30}^{\circ }$ and $\angle CBA$ = ${60}^{\circ }$ and in $△DEF$, $FE=10cm$, $\angle DFE={30}^{\circ }$, $\angle DEF={60}^{\circ }$; then, select the correct statement.

1. ΔBAC $\cong$ ΔFDE

2. ΔBCA $\cong$ ΔEFD

3. ΔABC $\cong$ ΔFDE

4. ΔABC $\cong$ ΔFED

Q.

From the given diagram, in which ABCD is a parallelogram, ABL is a line segment and E is mid point of BC.

Prove that :

(i) $\Delta$ DCE $\cong \Delta$ LBE

(ii) AB = BL.

(iii) AL = 2DC

Q. Prove that two triangles are congruent if any two sides and the
included angle of one triangle is equal to any two sides and the included angle of other triangle.