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Congruence of Triangles Test - 8

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Congruence of Triangles Test - 8
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Weekly Quiz Competition
  • Question 1
    1 / -0
    Which of the following statements is false?
    Solution
    There are five ways to find congruence of two triangles:
    1. SSS (Side - Side - Side)
    2. SAS (Side - Angle - Side)
    3. ASA (Angle - Side - Angle)
    4. AAS (Angle - Angle - Side)
    5. RHS (Right Angle - Hypotenuse - Side)
    There is no AAA congruence rule.

  • Question 2
    1 / -0
    If BC = QR, then by which of the following rules of congruence is ABC congruent to PQR?

    Solution
    Both triangles can be made congruent by ASA rule.
    ∠ABC = ∠PQR (Given)
    BC = QR (Given)

    In ABC,
    ∠ACB = 180° - ∠CAB - ∠CBA
    ∠ACB = 180° - 30° - 80°
    ∠ACB = 70°
    ∠ACB = ∠PRQ = 70°

    Therefore, ABC is congruent to PQR by ASA rule.

  • Question 3
    1 / -0
    In the given figure, OQ = OS, QP = SR and QPO SRO, then which of the following is true?

    Solution
    By RHS criterion, both triangles are congruent and PO = OR.
    Hence, option 2 is correct.
  • Question 4
    1 / -0
    In the given figure, triangles PQR and SRQ are right angled at Q and R, respectively; and PR = SQ.


    By which of the following criteria are these triangles congruent?
    Solution
    None of the given criteria is correct because these triangles are congruent by RHS (Right angle - Hypotenuse - Side) criterion.
    QR = RQ (Common side)
    PR = SQ (Given)
    ∠PQR = ∠SRQ = 90°
  • Question 5
    1 / -0
    In the given figure, O is the midpoint of PY, and PQ = YX.

    Which of the following relations is true?
    Solution
    PQ = YX (Given)
    O is the midpoint of PY. (Given)
    Therefore, PO = YO (As O is the midpoint)
    Now, ∠ POQ = ∠YOX (Angles opposite to equal sides)
    PQO YXO (By Angle-Angle-Side criterion)
    Hence, OQ = OX (By CPCT)


  • Question 6
    1 / -0
    In the given figure, AD is a perpendicular drawn on side BC. If you have to prove that ABD is congruent to ACD by SAS criterion, then which of the following will be required?

    Solution
    In ADB and ADC,
    AD is common side.
    ∠ADB = ∠ADC = 90° (Given)
    Now, if BD = CD, then ADB ADC by SAS (Side-Angle-Side) criterion



  • Question 7
    1 / -0
    Which congruence criterion can be used to make the given triangles congruent?

    Solution
    ∠OXY = ∠OZY (Right angle)
    XY = ZY (Given)
    OY = YO (Common side)
    Therefore, XOY ZOY by RHS criterion.
  • Question 8
    1 / -0
    If , then the measure of ∠QPR is

    Solution
    In triangle ABC,
    ∠BAC + ∠ABC + ∠BCA = 180° (Angle sum property)
    Now, ∠BAC = 180° - ∠ABC + ∠BCA = 180° - 50° - 65° = 180° - 115° = 65°
    So, ∠QPR = ∠BAC = 65° (By CPCT)

  • Question 9
    1 / -0
    If ΔABC ΔKML, then ∠A and ML are respectively equal to
    Solution
    Since ΔABC ΔKML, their corresponding parts are equal.
    ∴ ∠A = ∠K and BC = ML
  • Question 10
    1 / -0
    If PQ = PS and QR = SR, then which of the following is true?

    Solution
    In triangles PSR and PQR, three pairs of equal sides are as given below.
    PQ = PS (Given)
    QR = SR (Given)
    and PR = PR (Common in both)
    So, by SSS criterion.
    ∠PSR = ∠PQR and ∠PRQ = ∠PRS. (By CPCT)
  • Question 11
    1 / -0
    In the given figure, if POQ SOR, then which of the following relations is false?
    Solution

    In the figure, the two triangles are congruent.
    So, PO = SO, RO = QO and RS = QP
    But QO SO

  • Question 12
    1 / -0
    △ABC ≅ △DEF by __________ congruence rule.

    Solution


    ∠B = ∠E
    AB = DE and BC = EF.
    Hence, both the triangles are congruent by SAS criterion.
  • Question 13
    1 / -0
    Which of the following congruence criteria can be used to state that △BCA ≅ △ECD?
    Solution

    In the figure,
    BC = CD (Given)
    AC = CE (Given)
    ∠ACB = ∠DCE (Vertically opposite angles)
    So, the triangles are congruent by SAS criterion.
  • Question 14
    1 / -0
    Directions: Study the figure and information given below carefully to answer the following question.



    In the given figure, F and G are the midpoints of AE and DE respectively. AF = GE and BE = CE.

    △BEF is congruent to
    Solution
    F is the midpoint of AE and G is the midpoint of DE.
    AF = GE
    ∴ AF = FE = GE = DG
    Also,
    ∠BEF = ∠CEG (Vertically opposite angles)
    By SAS criterion,
    △BEF is congruent to △CEG.

  • Question 15
    1 / -0
    Directions: Study the figure and information given below carefully to answer the following question.



    In the given figure, F and G are the midpoints of AE and DE respectively. AF = GE and BE = CE.
    ∠ABE is equal to
    Solution
    F is the midpoint of AE and G is the midpoint of DE.
    AF = GE (Given)
    ∴ AF = FE = GE = DG
    ∴ AE = DE

    In ABE and DCE
    ∠BEA = ∠CED (Vertically opposite angles)
    BE = CE (Given)
    So, ABE DCE (By SAS criterion)
    ∴ ∠ABE = ∠DCE (By CPCT)
  • Question 16
    1 / -0
    Directions: Find the odd one out.
    Solution


    (D) is the odd one out because these triangles are not congruent.
  • Question 17
    1 / -0
    If for PQR and TUV, the correspondence RPQ VTU gives a congruence, then which of the following is not true?
    Solution
    If RPQ VTU gives a congruence, then
    RP = VT, PQ = TU, RQ = VU, ∠R = ∠V, ∠P = ∠T and ∠Q = ∠U.
    Hence, PQ = VU is not correct.
  • Question 18
    1 / -0
    Which of the following is correct?

    Solution
    In △ABC and △DBC,
    AB = DB
    AC = DC
    BC is the common side.
    Hence,
    △ABC ≅△DBC (By SSS)
  • Question 19
    1 / -0
    If in two triangles PQR and DEF, PQ = QR, QR = EF and ∠D = ∠P, then which of the following statements is correct?

    A. Both these triangles are congruent as well as equilateral.
    B. Both these triangles are congruent.
    C. Both these triangles are equilateral.
    D. None of these
    Solution
    In PQR and DEF,
    PQ = QR, QR = EF and ∠D = ∠P
    Hence, we cannot say that these triangles are congruent or equilateral as nothing about the sides DE and DF is mentioned.

  • Question 20
    1 / -0
    In the figure, AD = BC and BD = AC. Which of the following is true?

    Solution
    AD = BC
    BD = AC
    AB is the common side.
    So,
    △ABD ≅△BAC (By SSS)
    Also,
    ∠DAB = ∠CBA (By CPCT)
  • Question 21
    1 / -0
    Rohan wants to design a window as shown in figure. He decides to make △LNM congruent to △LNO. He designed the window such that LN ⊥ MN. Which of the following conditions make(s) the two triangles congruent?

    Solution
    Both the conditions in option (1) and option (2) will make △LNM congruent to △LNO by SSS criterion.

  • Question 22
    1 / -0
    Two security guards are standing in a building area in such a way that both are always at equal distance from the front and back exit gates of the area.



    Which of the following statements is/are true?
    Solution

    The given problem can be understood with the help of the above diagram.
    Now,
    PS = PQ (Given)
    SR = QR (Given)
    PR is common in triangle A and triangle B.
    Therefore, △PSR ≌ △PQR (SSS criterion)
    ∠1 = ∠2 (By CPCT)
    So, none of the statements is correct.
    Hence, option 4 is correct.
  • Question 23
    1 / -0
    Ashish asked Tanmay to find congruent triangles in the following figure, if any. Help Tanmay identify whether the triangles are congruent, and mark the correct option.

    Solution
    Given,
    AB = DE
    BC = EF
    Now,
    DC = CF = AF
    And
    DC + CF = DF
    AF + CF = AC

    Therefore,
    DF = AC
    This means △ACB ≅ △DFE (By SSS criterion).
    Hence, option 3 is correct.
  • Question 24
    1 / -0
    Directions: Given below are three statements about congruence written by three friends. Read the statements carefully and answer the question that follows.

    Tanmay: All circles are congruent.
    Simran: All acute angled triangles are congruent.
    Rohan: All scalene triangles are congruent.

    Who wrote the incorrect statement?
    Solution
    Only circles of the same radius are congruent.
    For triangles, size of the sides and measurement of angles are needed to check congruence.
    Not all scalene triangles are congruent.
    So, all of them wrote incorrect statements.
  • Question 25
    1 / -0
    There are two right triangles and both these triangles are congruent. If PS = CB and PQ = CA, then find the length of AB.

    Solution

    Using Pythagoras theorem in triangle PQS,







    QS = 8 cm

    Since both triangles are congruent, thus
    QS = AB (By CPCT)
    AB = 8 cm
  • Question 26
    1 / -0
    Directions: Match the figures in column I with congruence criterion used in column II.

    Column I Column II
    (i) a. SAS Criterion
    (ii) b. SSS Criterion
    (iii) c. RHS Criterion
    (iv) d. AAS Criterion
    Solution
    Figures in (i) are congruent by AAS criterion, so it matches with d.
    Figures in (ii) are congruent by RHS criterion, so it matches with c.
    Figures in (iii) are congruent by SSS criterion, so it matches with b.
    Figures in (iv) are congruent by SAS criterion, so it matches with a.
  • Question 27
    1 / -0
    Directions: Read the given statements carefully and choose the correct option.

    Statement 1: Two triangles are congruent triangles, if they havethree congruent sides and three congruent angles.
    Statement 2: If two sides and the included angle of one triangle are equal to two sides and the included angle of another triangle, then the triangles are congruent.
    Solution
    Statement 1 is about SSS congruence criterion, so it is correct.
    Statement 2 is about SAS congruence criterion, so it is also correct.
    Hence, both statements 1 and 2 are correct.
  • Question 28
    1 / -0
    Which of the following statements is correct?

    (A) SSA is a criterion of congruence.
    (B) If two triangles are equilateral triangles, then they are congruent.
    (C) All rectangles are congruent.
    (D) None of the above statements is correct.
    Solution
    None of the statements is correct.
    (A) SSA is not a criterion of congruence.
    (B) If two triangles are equilateral triangles, then they may or may not be congruent.
    (C) All rectangles are not congruent.
  • Question 29
    1 / -0
    Which of the following pairs of triangles is congruent by RHS criterion? (Figures are not drawn to scale)

    Solution
    In figure (i) and in figure (ii):
    Both triangles have a right angle.
    Size of hypotenuse is same = 10 cm.
    Size of the other two sides is also same.
    Therefore, triangles in figure (i) and figure (ii) are congruent by RHS criterion.
  • Question 30
    1 / -0
    Directions: State 'T' for true and 'F' for false for the following statements.

    I. If three sides of one triangle are equal to three sides of another triangle, the triangles are congruent.
    II. If two angles and the included side of one triangle are equal to the corresponding angles and side of another triangle, the triangles are congruent.
    III. If three angles of one triangle are equal to three angles of another triangle, the triangles are congruent.
    IV. If the hypotenuse and one leg of one right-angled triangle are equal to the corresponding hypotenuse and leg of another right-angled triangle, the two triangles are congruent.
    Solution
    I. If three sides of one triangle are equal to three sides of another triangle, the triangles are congruent. (True) This criterion is called SSS criterion.
    II. If two angles and the included side of one triangle are equal to the corresponding angles and side of another triangle, the triangles are congruent. (True) It is ASA criterion of congruence.
    III. If three angles of one triangle are equal to three angles of another triangle, the triangles are congruent. (False) As can be seen in the following figure



    IV. If the hypotenuse and one leg of one right-angled triangle are equal to the corresponding hypotenuse and leg of another right-angled triangle, then the two triangles are congruent. (True) It is called RHL (Right angle - Hypotenuse - Leg) or RHS (Right angle - Hypotenuse - Side).
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