Choose the Correct options
0 of 49 Questions completed
Questions:
You have already completed the quiz before. Hence you can not start it again.
Quiz is loading…
You must sign in or sign up to start the quiz.
You must first complete the following:
0 of 49 Questions answered correctly
Your time:
Time has elapsed
You have reached 0 of 0 point(s), (0)
Earned Point(s): 0 of 0, (0)
0 Essay(s) Pending (Possible Point(s): 0)
Average score 

Your score 

If two angles and a side of one triangle are equal to two angles and a side of another triangle, then the two triangles must be congruent. Is the statement true ? Why?
Upload your answer to this question.
This response will be awarded full points automatically, but it can be reviewed and adjusted after submission.
The statement is true because the triangles will be congruent either by ASA rule or AAS rule. This is because two angles and one side are sufficient to construct two congruent triangles.
In the two triangles ABC and DEF, AB = DE, and AC = EF. Name two angles from the triangles that must be equal so that the two triangles are congruent. Give reason for your answer.
Upload your answer to this question.
This response will be awarded full points automatically, but it can be reviewed and adjusted after submission.
Given
The two angle are
(SAS)
(ASS)
(SSA)
Prove that the sum of three altitudes of a triangle is less than the sum of the three sides of the triangle.
Upload your answer to this question.
This response will be awarded full points automatically, but it can be reviewed and adjusted after submission.
In ,
is the longest side
In ,
is the longest side
In
is the longest side
So adding all the three we get that perimeter of a triangle is greater than sum of its three altitudes
AD is a median of the triangle ABC. Is it true that AB + BC + CA > 2 AD. Give reason for your answer.
Upload your answer to this question.
This response will be awarded full points automatically, but it can be reviewed and adjusted after submission.
is a median,
(Proved)
M is a point on side BC of a triangle ABC such that AM is the bisector of \(\angle \)BAC . Is it true to say that AB + BC + CA > 2 AM ? Give reason for your answer.
Upload your answer to this question.
This response will be awarded full points automatically, but it can be reviewed and adjusted after submission.
In
(Sum of two sides is greater than third side)
Adding equation (1) and (2)
Arrange the sides of \(\triangle \)ABC in ascending order of lengths.
Upload your answer to this question.
This response will be awarded full points automatically, but it can be reviewed and adjusted after submission.
\(\)\angle A+\angle B=110^o\ â€¦(1)\\ \\ \angle B+2C=130^o\ â€¦.(2)\\ \\ \angle C\angle A=20^o\\ \\ \angle C=\angle A+20^o\\ \\ \angle C=180110=70^o\\ \\ \angle A=7020=50\\ \\ \angle B=180(70+50)=60^o\\ \\ \angle A<\angle B<\angle C\\ \\ BC<AC
Is it possible to construct a triangle with lengths of its sides as 8 cm, 9 cm, and 2 cm,? Give reason for your answer.
Upload your answer to this question.
This response will be awarded full points automatically, but it can be reviewed and adjusted after submission.
Yes it is possible as sum of two sides is greater than the third side
In an equilateral triangle \(ABC\), if \(AD \) is a median, then prove that \(\angle ADC=90^o \).
Upload your answer to this question.
This response will be awarded full points automatically, but it can be reviewed and adjusted after submission.
In and
(given)
But (linear pair)
and
In the figure, PR = QR, \(\angle \)PRA = \(\angle \)QRB and \(\angle \)BPR = \(\angle \)AQR.Prove that BP = QA.
Upload your answer to this question.
This response will be awarded full points automatically, but it can be reviewed and adjusted after submission.
In and
(given)
(given)
(addition of same angles with equal angle will also be equal)
(AAS)
(cpct)
In \(\triangle \)ABC, AD is perpendicular bisector of BC. Show that \(\triangle \)ABC is an isosceles triangle in which AB = AC.
Upload your answer to this question.
This response will be awarded full points automatically, but it can be reviewed and adjusted after submission.
In and
(common)
(each )
(given)
(SAS)
(cpct)
Therefore ABC is isosceles triangle in which
Prove that in an isosceles triangle, angles opposite to equal sides are equal.
Upload your answer to this question.
This response will be awarded full points automatically, but it can be reviewed and adjusted after submission.
In and
(given)
(each )
(common)
(RHS)
(cpct)
In the figure. \(\angle ABD=\angle ACE \) and \(AB=AC \). Prove that \(\triangle ABD\cong\triangle ACE \)
Upload your answer to this question.
This response will be awarded full points automatically, but it can be reviewed and adjusted after submission.
In and
(given)
(common)
(given)
(AAS)
In \(\triangle \)ABC, AB = AC. D is a point inside \(\triangle \)ABD such that BD = DC. Prove that \(\angle \)ABD = \(\angle \)ACD.
Upload your answer to this question.
This response will be awarded full points automatically, but it can be reviewed and adjusted after submission.
In and
(given)
(common)
(SSS)
(cpct)
In the figure X and Y are two points on equal sides AB and AC of a \(\triangle \)ABC such that AX = AY. Prove that XC = YB.
Upload your answer to this question.
This response will be awarded full points automatically, but it can be reviewed and adjusted after submission.
In and
(given)
(given)
(common)
(SAS)
(cpct)
In the figure, ABCD is a square and P is the midpoint of AD, BP and CP are joined. Prove that \(\angle \)PCB = \(\angle \)PBC
Upload your answer to this question.
This response will be awarded full points automatically, but it can be reviewed and adjusted after submission.
In and
(given)
(each )
( is a square)
(cpct)
(Proved)
In the figure, the diagonal AC of quadrilateral ABCD bisects \(\angle \)BAD and \(\angle \)BCD . Prove that BC = CD.
Upload your answer to this question.
This response will be awarded full points automatically, but it can be reviewed and adjusted after submission.
In and
(given)
(given)
(common)
(AAS)
(cpct)
In the figure, AB > AC, BO and CO are the bisectors of \(\angle \)B and \(\angle \)C respectively. Show that OB > OC.
Upload your answer to this question.
This response will be awarded full points automatically, but it can be reviewed and adjusted after submission.
and are bisectors
is oppsite to and is opposite to
So (Proved)
In the figure, PQ = PR and \(\angle \)Q = \(\angle \)R, Prove that PQS \(\cong\ \triangle \)PRT
Upload your answer to this question.
This response will be awarded full points automatically, but it can be reviewed and adjusted after submission.
In and
(given)
(given)
(common)
(AAS)
(Proved)
In the figure. AB \(\bot \) AC, DE \(\bot \) DF such that BA = DE and BF = EC. Show that \( \triangle\)ABC \(\cong\ \triangle \)DEF
Upload your answer to this question.
This response will be awarded full points automatically, but it can be reviewed and adjusted after submission.
In and
(given)
(each )
(RHS)
In the figure, D is any point on the side BC of a triangle ABC. Prove that AB + BC + CA > 2AD.
Upload your answer to this question.
This response will be awarded full points automatically, but it can be reviewed and adjusted after submission.
In
(Sum of two sides is greater than third side) â€¦(1)
In
â€¦(2)
Adding equation (1) and (2)
(Proved)
In the figure, \(AB = AC\) and \(\angle1=\angle2 \). Prove that \(\angle PBC=\angle PCB \).
Upload your answer to this question.
This response will be awarded full points automatically, but it can be reviewed and adjusted after submission.
In and
(given)
(given)
(common)
(SAS)
(cpct)
(Angle opposite to equal sides are equal)
In the figure, two lines AB and CD interect each other at O such that BCDA and BC=DA. Show that O is the midpoint of both the line segments AB and CD.
Upload your answer to this question.
This response will be awarded full points automatically, but it can be reviewed and adjusted after submission.
In and
(given)
(alternate angle)
(alternate angle)
(ASA)
(cpct)
(cpct)
In the figure, AD is the bisector of \(\angle \)BAC . Prove that AB > BD.
Upload your answer to this question.
This response will be awarded full points automatically, but it can be reviewed and adjusted after submission.
Since is the bisector of
(exterior angle is greater than each of the opposite interior angle)
In the figure, l  m and M is the midpoint of a line segment AB. Show that M is also the midpoint of any line segment CD having its end points on l and m respectively.
Upload your answer to this question.
This response will be awarded full points automatically, but it can be reviewed and adjusted after submission.
In and
(given)
(V.O.A)
(alt. )
(AAS)
(cpct)
In the figure, D is any point on the base BC produced of an isosceles triangle \(\triangle \)ABC . Prove that AD > AB.
Upload your answer to this question.
This response will be awarded full points automatically, but it can be reviewed and adjusted after submission.
( ABC is an isosceles )
In
ext.
In right triangle ABC, \(\angle C=90^o \), M is midpoint of hypotenuse AB, C is joined to M and produced to a point D such that DM = CM. Point D is joined to point B. Show that
(i) \(\triangle AMC\cong\triangle BMD \)
(ii) \(\triangle DBC\cong\triangle ACB \)
Upload your answer to this question.
This response will be awarded full points automatically, but it can be reviewed and adjusted after submission.
In and
(given)
(VOA)
(alt )
(AAS)
(ii) In and
(common)
(common)
(common)
(AAS)
In the figure. PR > PQ and PS bisects \(\angle \)PQR . Prove that \(\angle \)PSR > \( \angle\)PSQ .
Upload your answer to this question.
This response will be awarded full points automatically, but it can be reviewed and adjusted after submission.
In
In
Adding to both sides
In the figure, the perpendicular AD, BE and CF drawn from the vertices A,B and C respectively of \(\triangle \)ABC are equal. Prove that the triangle is an equilateral triangle.
Upload your answer to this question.
This response will be awarded full points automatically, but it can be reviewed and adjusted after submission.
In and
(given)
(common)
(ASA)
In and
(Common)
(ASA)
(cpct)
(Proved)
Prove that medians of an equilateral triangle are equal.
Upload your answer to this question.
This response will be awarded full points automatically, but it can be reviewed and adjusted after submission.
are the medians of
and
( each)
(SAS)
Similarly
So
In the figure, \(\triangle \)LMN is an isosceles triangle with LM = LN and LP bisects \(\angle \)NLQ. Prove that LP  MN.
Upload your answer to this question.
This response will be awarded full points automatically, but it can be reviewed and adjusted after submission.
In
From (1) and (2)
But they form corresponding angle
So,
In the figure. D is a point on side BC of \(\triangle \)ABC such that AD = AC. Show that AB > AD.
Upload your answer to this question.
This response will be awarded full points automatically, but it can be reviewed and adjusted after submission.
(exterior angle property)
(Proved)
O is any point in the interior of \(\triangle \)ABC. Show that OB + OC < AB + AC
Upload your answer to this question.
This response will be awarded full points automatically, but it can be reviewed and adjusted after submission.
In
In
â€¦(1)
In
â€¦(2)
In
….(3)
….(4)
In
…(5)
Adding (4) and (5)
In a right angled triangle, one acute angle is double the other. Prove that the hypotenuse is double the smallest side.
Upload your answer to this question.
This response will be awarded full points automatically, but it can be reviewed and adjusted after submission.
In and
(by construction)
(Common)
(SAS)
(cpct),
ABC is a right angled triangle with AB = AC. Bisector of \(\angle \)A meets BC at D. Prove that BC = 2AD.
Upload your answer to this question.
This response will be awarded full points automatically, but it can be reviewed and adjusted after submission.
So â€¦.(1)
Similarly
….(2)
Adding equation (1) and (2)
Show that in a quadrilateral ABCD, AB + BC + CD + DA < 2 (BD + AC).
Upload your answer to this question.
This response will be awarded full points automatically, but it can be reviewed and adjusted after submission.
In â€¦(1)
In â€¦.(2)
In â€¦.(3)
In â€¦.(4)
In the figure, A is a point equidistant from two lines \(l_1 \) and \(l_2 \) intersecting at a point P, show that AP bisects the angle between \(l_1 \) and \(l_2 \).
Upload your answer to this question.
This response will be awarded full points automatically, but it can be reviewed and adjusted after submission.
In and
(given)
(Common)
(each )
(RHS)
(cpct)
(Proved)
In a triangle PQR, PR > PQ and PS is the bisector of \(\angle \)QPR. Prove that \(\angle \)PSR > \(\angle \)PSQ.
Upload your answer to this question.
This response will be awarded full points automatically, but it can be reviewed and adjusted after submission.
Let
In
â€¦(1)
In ,
….(2)
In the figure, \(AC = AE, AB = AD \) and \(\angle BAD=\angle EAC \). Show that \(BC=DE \).
Upload your answer to this question.
This response will be awarded full points automatically, but it can be reviewed and adjusted after submission.
In and
(given)
(given)
(SAS)
(cpct)
In the figure, show that 2(AC + BD) > (AB + BC + CD + DA)
Upload your answer to this question.
This response will be awarded full points automatically, but it can be reviewed and adjusted after submission.
In
â€¦.(1)
(Sum of two sides is greater than the third side)
In
â€¦(2)
In ,
….(3)
In
….(4)
\(O \) is a point in the interior of \(\triangle PQR \). Prove that \(OP + OQ + OR >\dfrac{1}{2}(PQ + QR + PR) \)
Upload your answer to this question.
This response will be awarded full points automatically, but it can be reviewed and adjusted after submission.
In ,
â€¦(1)
In
â€¦.(2)
In
â€¦.(3)
Adding 3 equation we get
Show that perimeter of a triangle is greater than the sum of its medians.
Upload your answer to this question.
This response will be awarded full points automatically, but it can be reviewed and adjusted after submission.
We know that sum of any two sides of a triangle is greater than twice the median
Adding we get
AB and CD are respectively the smallest and the longest sides of a quadrilateral ABCD as shown in the figure. Prove that \(\angle A>\angle C \) and \(\angle B>\angle D \)
Upload your answer to this question.
This response will be awarded full points automatically, but it can be reviewed and adjusted after submission.
In ,
â€¦(1)
In ,
is the largest side
â€¦.(2)
Adding (1) and (2)
Similarly by joining we can prove that
In the figure, if two isosceles triangles have a common base, prove that the line segment joining their vertices bisects the common base at right angles.
Upload your answer to this question.
This response will be awarded full points automatically, but it can be reviewed and adjusted after submission.
In and
(sides of one isosceles )
(sides of one isosceles )
(common)
(cpct)
(cpct)
But (Linear pair)
In the figure, \(AC=BC,\ \angle DCA=\angle ECB \) and \(\angle DCB=\angle EAC \). Prove that (i) \(\triangle DBC\cong\triangle EAC \) ; (ii) \(DC=EC \) and \(BD=AE \).
Upload your answer to this question.
This response will be awarded full points automatically, but it can be reviewed and adjusted after submission.
In and
(given)
(given)
(AAS)
(cpct)
(cpct)
In the figure, BAPQ, CA RS and BP=RC. Prove that (i) BS = PQ ; (ii) RS = CQ.
Upload your answer to this question.
This response will be awarded full points automatically, but it can be reviewed and adjusted after submission.
Similarly
In and
(ASA)
(cpct)
(cpct)
In a right triangle ABC, right angled at C, M is the mid point of hypotenuse AB. C is joined to M and produced to a point D such that DM = CM. Point D is joined to point B. Show that :
(i) \(\triangle AMC\cong\triangle BMD \)
(ii) \(\angle DBC=90^o \)
(iii) \(\triangle DBC\cong\triangle ACB \)
Upload your answer to this question.
This response will be awarded full points automatically, but it can be reviewed and adjusted after submission.
In and
(given)
(VOA)
(SAS)
(cpct)
(cpct)
But they form linear pair
So,
In and
(Common)
(Proved above)
(Proved above)
(RHS)
In the figure, \(\angle \)QPR = \(\angle \)PQR and M and N are respectively points on sides QR and PR of \(\triangle \)PQR , such that QM = PN. Prove that OP = OQ, where O is the point of intersection of PM and QN.
Upload your answer to this question.
This response will be awarded full points automatically, but it can be reviewed and adjusted after submission.
In and
(given)
(given)
(SAS)
(cptc)
In and
(given)
(VOA)
(Proved above)
(AAS)
(cpct)
In the figure, PQ and RS are perpendicular to QS,QA = BS and PB = AR. Prove that \(\angle \)QPB = \( \angle\)SRA
Upload your answer to this question.
This response will be awarded full points automatically, but it can be reviewed and adjusted after submission.
In and
(given)
(given)
(each )
(RHS)
(cpct)
In the figure, if AD is the bisector of \(\angle \)BAC , then prove that :
(i) AB > BD
(ii) AC > CD
Upload your answer to this question.
This response will be awarded full points automatically, but it can be reviewed and adjusted after submission.
( is bisector of )
(exterior angle is greater than each interior opposite angle
(Side opposite to greater angle is larger)
(Side opposite to greater angle is greater)