Olympiad Mathematics - Congruence of Triangles
Exam Duration: 45 Mins Total Questions : 30
In \(\triangle\)ABC, AB = AC and AD is perpendicular to BC. State the property by which \(\triangle\)ADB \(\cong \) \(\triangle\)ADC.
- (a)
S.A.S. property
- (b)
S.S.S.property
- (c)
R.H.S.property
- (d)
A.S.A. property
Two students drew a line segment each. What is the condition for them to be congruent?
- (a)
They should be drawn with a scale.
- (b)
They should be drawn on the same sheet of paper.
- (c)
They should have different lengths
- (d)
They should have the same length.
In the given figure, if AD = BCand ADIIBC, which of the following is true?
- (a)
AB = AD
- (b)
AB = DC
- (c)
BC=CD
- (d)
AC=AD
In\(\triangle\) PQR and \(\triangle\) XYZ,\(\angle\) P = 50°, XY = PQ, and XZ = PR.By which property are\(\triangle\) XYZ and \(\triangle\)PQR congruent?
- (a)
S.S.S,property
- (b)
S.A.S.property
- (c)
A.S.A property
- (d)
R.H.S.property
For the triangles \(\triangle\)ART and \(\triangle\)PEN given, if S.A.S. criterion should be used given \(\angle\)T = \(\angle\)N, what are the respective measures of PN and RT?
- (a)
TR and PE
- (b)
AR and PE
- (c)
AT and EN
- (d)
AR and PN
In two triangles, the three angles of one triangle are correspondingly equal to three angles of another triangle. Which of the following is a correct statement?
- (a)
One triangle is an enlarged copy of other.
- (b)
The two triangles are necessarily congruent
- (c)
The two triangles are congruent by AAA. congruency criterion
- (d)
All of the above
Which part of \(\triangle\)ABC corresponds to \(\overline {RP}\) ?
- (a)
\(\overline{AB}\)
- (b)
\(\overline{AC}\)
- (c)
\(\overline{CA}\)
- (d)
\(\overline{BC}\)
\(\triangle\)ABC is congruent to\(\triangle\)XYZ. Find the measures of \(\angle\) x and \(\angle\)y respectively.
- (a)
80°,60°
- (b)
60°,40°
- (c)
80°,40°
- (d)
60°,80°
\(\triangle\)ABC\(\cong \) \(\triangle\)FDE What is the measure of \(\angle\)F?
- (a)
70°
- (b)
50°
- (c)
130°
- (d)
60°
Observe the triangles given in the figure
State the condition under which \(\triangle\)ABC \( \cong \) \(\triangle\)PQR
- (a)
A.S.A.
- (b)
S.S.S.
- (c)
SAS.
- (d)
R.H.S
By which congruency property, are the two triangles PQS and PRSgiven in the following figure congruent?
- (a)
S.S.S.property
- (b)
SAS. property
- (c)
A.SA property
- (d)
R.H.S.property
For the congruence of \(\triangle\)ABC and \(\triangle\)PQR, which one of the following sets of conditions is not sufficient?
- (a)
\(\angle\) ABC = \(\angle\)PQR.a = p, c = r
- (b)
\(\angle\)CAB= \(\angle\) RPQ,\(\angle\)ABC= \(\angle\) PQR,c = r
- (c)
b = q, \(\angle\) CAB =\(\angle\) RPQ,a = p
- (d)
a = p,c = r, \(\angle\)ABC = \(\angle\)PQR
Which of the following is important in congruence of triangles?
- (a)
Naming the angles of the triangles using capital letters
- (b)
Measures of angles in degrees
- (c)
The order of letters of the triangles
- (d)
Exact length of the sides of the triangles
In the figure given, which of the following statements is true?
- (a)
\(\triangle\)QPR\(\cong \)\(\triangle\)SPR
- (b)
\(\triangle\)PSR \(\cong \) \(\triangle\)RQP
- (c)
\(\triangle\)PRS\(\cong \) \(\triangle\)QPR
- (d)
\(\triangle\)QRP \(\cong \) \(\triangle\)PSR
Are triangles in the given figure congruent by R.H.S. condition?
- (a)
Yes
- (b)
Insufficient data
- (c)
No
- (d)
Either (B) or (C)
Which of the following statements is INCORRECT?
- (a)
Two triangles having same area are congruent.
- (b)
If two sides and one angle of a triangle are equal to the corresponding two sides and the angle of another triangle, then the two triangles are congruent.
- (c)
If the hypotenuse of one right angled triangle is equal to the hypotenuse of another right angled triangle, then the triangles are congruent.
- (d)
All of these
In \(\Delta\)ABC, AB = AC and AD is perpendicular bisector of BC. The property by which \(\Delta\)ADB is not congruent to \(\Delta\)ADC is _______________
- (a)
SAS property
- (b)
SSS property
- (c)
RHS property
- (d)
AAA property
In the given figure, PA \(\bot \)AB, QB \(\bot \) AB and \(\Delta \)OAP \(\cong \) \(\Delta \)OBQ, then
- (a)
PA = OB
- (b)
AP = QB
- (c)
OP = BQ
- (d)
OA = OQ
In the given figure, triangles ABC and DCB are right angled at A and D respectively and AC = DB, then \(\Delta\)ABC \(\cong \) \(\Delta\)DCB by ____________ criterion.
- (a)
AAA
- (b)
SAS
- (c)
ASS
- (d)
None of these
In the given figure, ABC is an isosceles triangle in which AB = AC. If E and F be the midpoints of AC and AB respectively, then BE is equal to ____________
- (a)
CF
- (b)
AB
- (c)
CE
- (d)
BF
\(\Delta\) PQR \(\cong \) \(\Delta\)XYZ by ______________ congruency rule.
- (a)
SSS
- (b)
AAA
- (c)
SAS
- (d)
ASA
Which congruence criterion can be used to state that to \(\Delta\)XOY \(\cong \) \(\Delta\)POQ?
- (a)
ASA
- (b)
SAS
- (c)
SSS
- (d)
RHS
Study the figure and information given below carefully and answer the following questions.
CF and AE are equal perpendiculars on BD and BF = FE = ED.
\(\angle\)BAE = ______________
- (a)
\(\angle\)BCD
- (b)
\(\angle\)CBA
- (c)
\(\angle\)ADC
- (d)
\(\angle\)DCF
In two triangles PQR and LMN, PQ = QR, \(\angle\)P = \(\angle\)M and QR = LN, then which of the following statements is CORRECT?
- (a)
Triangles are congruent only
- (b)
Triangles are isosceles only.
- (c)
Triangles are both congruent and isosceles.
- (d)
None of these
Ananya is designing the window shown in the figure. She wants to make \(\Delta\)PRQ congruent to PRS.She designs the window so that PR \(\bot \) QS. Which of the following conditions will make the two triangles congruent?
- (a)
QR = RS
- (b)
PQ = PS
- (c)
Both (A) and (B)
- (d)
None of these
Akira gave a problem to her sister Kiara. However, Kiara got stuck. Help Kiara identify whether the triangles are congruent and choose the correct option.
- (a)
Yes, \(\Delta\)ABC \(\cong \) \(\Delta\)DCE
- (b)
No, they are not congruent
- (c)
Yes, \(\Delta\)DCE \(\cong \)\(\Delta\)CAB
- (d)
Yes, \(\Delta\)DEC \(\cong \) \(\Delta\)CAB
Three students Pia, Sia and Tia wrote a statement on a blackboard.
Pia wrote, "All rectangles are congruent".
Sia wrote, "All equilateral triangles are congruent".
Tia wrote, "All right angled triangles are congruent".
Who wrote the INCORRECT statement?
- (a)
Pia
- (b)
Sia
- (c)
Tia
- (d)
All of them
Tiara wants to know the width of the given river. While doing so, she stands on the edge of the river and look straight across to a point on the other edge without changing the inclination of the neck and head. She turns side ways until the vision is in line with a point on the side of the stream.
From the above description, find the value of QR.
- (a)
25 units
- (b)
12 units
- (c)
15 units
- (d)
Can't be determined
Match the figures in Column-I with their corresponding congruence criterion given in Column-II.
- (a)
(i) \(\rightarrow\) (b), (ii) \(\rightarrow\) (d), (iii) \(\rightarrow\) (a), (iv) \(\rightarrow\) (c)
- (b)
(i) \(\rightarrow\) (c), (ii) \(\rightarrow\) (a), (iii) \(\rightarrow\) (b), (iv) \(\rightarrow\) (d)
- (c)
(i) \(\rightarrow\) (b), (ii) \(\rightarrow\) (c), (iii) \(\rightarrow\) (a), (iv) \(\rightarrow\) (d)
- (d)
(i) \(\rightarrow\) (a), (ii) \(\rightarrow\) (c), (iii) \(\rightarrow\) (b), (iv) \(\rightarrow\) (d)
Which of the following statements is CORRECT?
Statement-1: Two triangles are said to be congruent if two sides and an angle of one triangle are respectively equal to the two sides and an angle of the other.
Statement-2: Two triangles are congruent if two sides and the included angle of the one triangle equal to the corresponding two sides and included angle of the other.
- (a)
Only Statement-1
- (b)
Only Statement-2
- (c)
Both Statement-1 and Statement-2
- (d)
Neither Statement-1 nor Statement-2