Self Studies
Self Studies
Self Studies
Self Studies
Selfstudy
Selfstudy

Triangles Test - 7

Result Self Studies

Triangles Test - 7
  • Score

    -

    out of -
  • Rank

    -

    out of -
TIME Taken - -
Self Studies

SHARING IS CARING

If our Website helped you a little, then kindly spread our voice using Social Networks. Spread our word to your readers, friends, teachers, students & all those close ones who deserve to know what you know now.

Self Studies Self Studies
Weekly Quiz Competition
  • Question 1
    1 / -0
    In the given figure, find the measure of ∠PQS if ∠PRQ = 30° and ∠QRS = 50°.


    Solution
    In ΔPQS and ΔPRS,
    PQ = PR (Given)
    QS = RS (Given)
    PS = PS (Common)
    ∴ ΔPQS ≅ ΔPRS (By SSS rule)
    ∴ ∠PQS = ∠PRS (By C.P.C.T.)
    ∠PQS = ∠PRQ + ∠QRS = 30° + 50° = 80°
  • Question 2
    1 / -0
    If ΔABC ≅ ΔXYZ, then which of the following is true?

    (A) ∠ABC = ∠XYZ, ∠BCA = ∠YZX, ∠CAB =∠ZXY
    (B) ∠ABC = ∠YXZ, ∠BCA = ∠XYZ, ∠CAB = ∠YZX
    (C) ∠ABC = ∠YZX, ∠CAB = ∠XYZ, ∠BCA = ∠ZXY
    (D) None of these
    Solution
    Since ΔABC ≅ ΔXYZ, their corresponding parts are equal.
    Therefore, ∠ABC = ∠XYZ, ∠BCA = ∠YZX, ∠CAB = ∠ZXY
  • Question 3
    1 / -0
    In the given figure, if PS = PT and SR = TQ, and PM is the angle bisector of ∠P, then

    Solution
    We have, PS = PT and SR = TQ
    PS + SR = PT + TQ
    So, PR = PQ
    Now, in ΔPRM and ΔPQM,
    PR = PQ
    ∠RPM = ∠QPM (As PM is an angle bisector, it divides the angle in two equal parts.)
    PM = PM (Common)
    So, ΔPRM ≅ ΔPQM (By SAS)
  • Question 4
    1 / -0
    Consider the following figures:



    The two given triangles are
    Solution
    As from the figure we can see that two sides of the triangle are equal, we can say that it is an isosceles triangle.
    Also, both the triangles have sides:
    AB = PQ
    AC = PR
    BC = QR
    Therefore, all the corresponding sides are equal.
    So,
  • Question 5
    1 / -0
    In the given triangles, ΔPQR is congruent to ΔXYZ. Find the sum of the sides QR and XZ if the missing side of the triangle XYZ is 4 cm and ∠ZXY is 40°.

    Solution
    In congruent triangles, corresponding parts are equal.
    ΔPQR is congruent to ΔXYZ.
    So, XY = PQ
    YZ = QR
    Missing side of triangle XYZ:
    YZ = 4 cm
    Since YZ = QR = 4 cm
    XZ = 6cm
    Sum of these sides = 10 cm
  • Question 6
    1 / -0
    Given that PS is the perpendicular bisector of QR, by which congruence rule are the two triangles △PSQ and △PSR congruent?

    Solution
    In triangles ΔPSQ and ΔPSR,
    ∠PSQ = ∠PSR (Each 90°)
    PQ = PR (Given)
    PS = PS (Common)
    So, ΔPSQ ≅ ΔPSR (RHS property)
  • Question 7
    1 / -0
    In the given figure, ΔPQR is an isosceles triangle with PR = QR. If M is the mid-point of PQ, then which of the following options is true?

    Solution
    In triangles PRM and QRM,
    PR = QR (Given) ... (1)
    Since M is the mid-point of PQ, it bisects PQ.
    PM = MQ ... (2)
    Also, RM = RM (Common) ... (3)
    From (1), (2) and (3),
    ΔPRM ≅ ΔQRM (By SSS congruency rule)
  • Question 8
    1 / -0
    In the following figure, ABC is an equilateral triangle. AD and BE are lines which respectively join the mid-points of BC and AC with the opposite vertex.

    Which of the following statements is false?
    Solution
    In and ,
    AE = BD
    AB = AB


    Hence, the triangles are congruent.
    So, AD = BE
    AB is not equal to BE.
  • Question 9
    1 / -0
    In triangle PQR, if ∠P = 70°, ∠Q = 65° and ∠R = 45°, then
    Solution

    In any triangle, the side opposite to the greater angle is longer.
    ∠Q > ∠R
    So, PR is greater than PQ.
  • Question 10
    1 / -0
    In the given figure, ΔPQR is an isosceles triangle with QP = QR. X and Y are points on PR such that XP = XQ and YQ = YR.

    Which of the following options is correct?
    Solution
    ΔPQR is an isosceles triangle with QP = QR.
    Therefore, ∠QPX = ∠QRY (Angles opposite to equal sides of a triangle are equal.) ... (1)
    Also, XP = XQ (Given)
    Therefore, ΔPXQ is an isosceles triangle.
    Therefore, ∠QPX = ∠PQX ... (2)
    Now, YQ = YR (Given)
    Therefore, ΔRYQ is an isosceles triangle.
    ∠RQY = ∠QRY ... (3)
    In fact ∠QPX = ∠PQX = ∠RQY = ∠QRY ... (4) [Using (1), (2) and (3)]

    In ΔPQX and ΔRQY,
    PQ = QR (Given)
    ∠QPX = ∠QRY [Using (4)]
    ∠PQX = ∠RQY [Using (4)]
    Therefore, ΔPQX ≅ ΔRQY (by ASA congruency rule)
  • Question 11
    1 / -0
    In ΔPQR, if the bisector PM of ∠P is perpendicular to side QR, then
    Solution
    In ΔPQM and ΔPMR,
    ∠QPM = ∠RPM (Given)
    PM = PM (Common)
    ∠PMQ = ∠PMR = 90° (Given)
    So, ΔPQM ≅ ΔPRM (By ASA rule)



    So, PQ = PR (C.P.C.T.)
    Or ΔPQR is an isosceles triangle.
  • Question 12
    1 / -0
    DEF is a right-angled triangle at ∠E. If it follows all general rules of geometry, then what will be the measure of the external angle to ∠E?
    Solution
    In triangle DEF, ∠E is 90°.
    The other two angles will therefore be (180 - 90)° = 90°.
    Sum of the angles ∠D and ∠F will be 90°.
    Exterior angle to ∠E will be equal to the sum of interior opposite angles = 90°
    Therefore, the correct answer will be option 2.
  • Question 13
    1 / -0
    In the given figure, ΔMQN, ΔMPO and ΔMNO are isosceles, and ∠QNO is congruent to ∠PON.



    Which of the following rules can be used to prove that ΔMQN is congruent to ΔMPO?
    Solution

    We know that ΔMNO is isosceles, so MN = MO. ... (i)
    ∠MNO = ∠MON (Angles opposite to equal sides)

    Also, ∠QNO = ∠PON, ∠QNM = ∠POM (By C.P.C.T.) ... (ii)
    ΔMQN and ΔPMO are isosceles, so MQ = QN and MP = PO.
    ∠QMN = ∠QNM (Angles opposite to equal sides)
    ∠POM = ∠PMO (Angles opposite to equal sides)
    So, ∠QMN = ∠PMO ... (iii)

    Therefore, from (i), (ii) and (iii) we can say that ΔMQN is congruent to ΔMPO by using ASA congruency rule.
  • Question 14
    1 / -0
    In the following figures, XYZ is an isosceles triangle with XY = XZ. The bisectors of ∠Y and ∠Z intersect each other at P. If PY = 4 cm, then what will be the value of PZ?


    Solution
    It is given that in triangle XYZ, XY = XZ
    ∠XYZ = ∠XZY (Angles opposite to equal sides of a triangle are equal.)
    ∠XYZ = ∠XZY
    ∠PYZ = ∠PZY
    Therefore, PY = PZ (Sides opposite to equal angles of a triangle are also equal.)
    Hence, PZ = PY = 4 cm
  • Question 15
    1 / -0
    Consider the following two triangles ABC and DEF and find the value of ∠DFE if ∠BAC is 35° and AB = DE.
    Solution


    We know that both the triangles ABC and DEF are right-angled triangles, so ∠ABC = ∠DEF.
    Also, AB = DE (Given).
    So, using HL congruency rule, ΔABC ≅ ΔDEF
    We know that in △ABC, ∠ABC = 90° and ∠BAC = 35°.
    Therefore, ∠ACB = 180° - (90° + 35°) = 55°
    Since triangles ABC and DEF are congruent, ∠ACB = ∠DFE = 55°.
  • Question 16
    1 / -0
    In the given figure, ABC is an isosceles triangle, and BE, DC and AF are the medians.



    Which of the following statements is not true regarding the triangle?
    Solution
    We know that the centroid divides the median in the ratio 2 : 1, and also in any triangle, the median divides the triangle into two triangles of equal area. Also, the centroid divides the triangle into six smaller triangles of equal area. Therefore, statement 4 is not true. Hence, option 4 is the correct answer.
  • Question 17
    1 / -0
    What is the name of the congruence rule which proves the congruency of two triangles using the right angle?
    Solution
    If the hypotenuse and one leg of a right triangle are equal to the hypotenuse and one leg of another right triangle, then the two right triangles are congruent. Therefore, HL congruency rule will be used to prove that two right-angled triangles are congruent.
  • Question 18
    1 / -0
    In the given figure, AC = EF. Which of the following triangles is congruent to BDC?

    Solution

    Sides opposite to equal angles are equal.

    AC = BC ……..(i)
    =
    Sides opposite to equal angles are equal.
    DE = EF ……..(ii)
    Also, AC = EF ...........(iii)
    From (i), (ii) and (iii)
    BC = DE
    Now,
    In BCD and DEB
    = (each 90°)
    BC = DE (proved above)
    BD = BD (common)
    BCD DEB by RHS congruency criterion.
  • Question 19
    1 / -0
    PQR is an isosceles triangle such that PQ = PR and perpendiculars from vertices Q and R meet their opposite sides at points M and N, respectively. Which of the following conditions will make ΔNQR ΔMRQ?
    Solution


    PQR is an isosceles triangle.
    =
    Now, in ΔNQR and MRQ,
    (each 90°)
    = (proved above)
    QR = RQ (common side)
    ΔNQR ΔMRQ by AAS congruency criterion.
  • Question 20
    1 / -0
    In the given figure, D is the midpoint of BE, E is the midpoint of CD, AB > BD and AD > CE. Which of the following relations is definitely true?

    Solution
    ED = DB [D is the midpoint of BE]



    And, ED = EC [E is the midpoint of CD]
    ED = DB = EC
    Also, AB > BD
    AB > ED [ BD = ED]
  • Question 21
    1 / -0
    Fill in the blanks:

    I. The sum of any two sides of a triangle is __P__ than the third side.
    II. In triangles PQR and FJK, if PQ = FJ, PR = FK and QR = JK, then ΔPQR ≅ ΔFJK by ___Q___ congruency rule.
    III. According to a property of triangles, the side opposite to the __R__ angle is the shortest side of the triangle.
    Solution
    I. The sum of any two sides of a triangle is greater than the third side.
    II. In triangles PQR and FJK, if PQ = FJ, PR = FK and QR = JK, then ΔPQR ≅ ΔFJK by SSS congruency rule.
    III. According to a property of triangles, the side opposite to the smallest angle is the shortest side of the triangle.
  • Question 22
    1 / -0
    Match the following:

    Column I Column II
    A.
    ΔSRO ≌ ΔOQP    		 1. ASA Congruence Rule
    B.

    ΔABC ≌ ΔXYZ
    		 2. Hypotenuse-Leg Congruence Rule
    C.

    ΔDEF ≌ ΔIJK    		 3. RHS Congruence Rule
    D.
    ΔPQS ≌ ΔPRS
    		 4. SAS Congruence Rule
    Solution
    A. SAS (Side-Angle-Side) Congruence Rule is a rule used to prove whether a given set of triangles is congruent. If two sides and the included angle of one triangle are equal to two sides and included angle of another triangle, then the triangles are congruent.

    B. Hypotenuse-Leg Theorem states that two right triangles are congruent if their hypotenuses are congruent and a corresponding leg is congruent.

    C. ASA Congruence Rule states that two triangles are congruent if any two angles and their included side are equal in both triangles. There are five ways to test that two triangles are congruent.

    D. RHS Congruence Rule of states that if the hypotenuse and one side of a right-angled triangle are equal to the corresponding hypotenuse and one side of another right-angled triangle, then both the right-angled triangles are said to be congruent.
  • Question 23
    1 / -0
    In the given quadrilateral AECB, vertices A and E are extended to meet at point D on side BC such that AD and AE are respectively equal to AB and diagonal AC.



    If ∠BAC = ∠EAD, then which of the following options is true?
    Solution


    It is given that ∠BAC = ∠EAD.
    AB = AD (Given)
    AC = AE (Given)
    We can see that ΔABC is congruent to ΔADE.
    Using this we get that ∠ACB = ∠AED.
  • Question 24
    1 / -0
    Which of the following is the CORRECT statement?
    Solution
    1. If three sides of one triangle are equal to the three sides of another triangle, then the two triangles are congruent by ASA rule. (INCORRECT) This is true by SSS rule.
    2. The reflexive property of congruence shows that all geometric figures are congruent to each other. (INCORRECT) The reflexive property of congruence shows that any geometric figure is congruent to itself.
    3. AAA congruence rule does not work when we have to determine if two triangles are congruent because it can produce similar but not congruent triangles. (CORRECT)
    4. If two sides and the included angle of one triangle are equal to two sides and the included angle of another triangle, then the two triangles are congruent by SSS congruence rule. (INCORRECT) This is true by SAS congruence rule.
  • Question 25
    1 / -0
    O and M are graphically calculated to be the mid-points of equal sides PQ and PR, which are each 12 cm in length. If these lines form two equal adjacent sides of a parallelogram PQSR, then which of the following options is true?
    Solution


    In ΔPQM and ΔPRO,
    PQ = PR (Given)
    ∠P = ∠P (Common)
    PM = PO (Halves of equal sides)
    So, ΔPQM ≅ ΔPRO (By SAS rule)
    Therefore, QM = OR (C.P.C.T.)
Self Studies
User
Question Analysis

  • Correct -

  • Wrong -

  • Skipped -

My Perfomance
  • Score

    -

    out of -
  • Rank

    -

    out of -
Re-Attempt Weekly Quiz Competition
Self Studies Get latest Exam Updates
& Study Material Alerts!
No, Thanks
Self Studies
Click on Allow to receive notifications
Allow Notification
Self Studies
Self Studies Self Studies
To enable notifications follow this 2 steps:
  • First Click on Secure Icon Self Studies
  • Second click on the toggle icon
Allow Notification
Get latest Exam Updates & FREE Study Material Alerts!
Self Studies ×
Open Now