SAS Criteria for Congruency

Q.

In adjoining figure, $PS\perp RQ$ and $QT\perp PR$. If $RQ=6,PS=6$ and $PR=12$, then Find $QT$.

Q. Question 14
The perpendicular from A on side BC of a $\Delta ABC$ intersects BC at D such that DB = 3CD .Prove that $2A{B}^2=2A{C}^2+B{C}^2$.

Q. In an isosceles triangle the circumcentre, the orthocentre, the incentre and the centroid are:

1. Collinear
2. Coincide
3. None of above
4. Do not coincide

Q.

Question 13
In the given figure, l||m and line segments AB, CD and EF are concurrent at point P. Prove that
$\frac{AE}{BF}=\frac{AC}{BD}=\frac{CE}{FD}$.

Q. For the proof of BPT theorem, it is said that since a given triangle is obtuse therefore, it ha its altitude external from the triangle. Why is this stated? Which theorem is this?

Q. $ABC$ is an isosceles triangle in which $AB=AC$. $P$ is any point in the interior of $\Delta ABC$ such that $\angle ABP=\angle ACP$. Prove that
a) $BP=CP$
b) $AP$ bisects $\angle BAC$

Q. The incircle of an isosceles triangle ABC, in which AB = AC, touches the sides BC, CA and AB at D, E and F respectively, then BD = DC.If this statement is true , enter 1 otherwise enter 0

Q. In the given figure, ΔABC ~ ΔCDE and BCD is a straight line. If ∠B = 90°, AC = 12 cm and 5CE = 2AC + 1, then the length of AE is

दिये गये चित्र में, ΔABC ~ ΔCDE तथा BCD एक सरल रेखा है। यदि ∠B = 90°, AC = 12 cm तथा 5CE = 2AC + 1, तब AE की लंबाई है

1. $12\sqrt{5}cm$
2. $2\sqrt{15}cm$
3. $10\sqrt{2}cm$
4. 13 cm

Q.

In the given figure, If $PQ\parallel RS$, prove that $\Delta POQ\sim \Delta SOR$.

Q. $ABCD$ is quadrilateral in which $AD=BC$ and $\angle DAB=\angle CBA$. Prove that
(i) $\Delta ABD=\Delta BAC$
(ii) $BD=AC$
(iii) $\angle ABD=\angle BAC$

Q. Chords $AB$ and $CD$ of a circle meet inside the circle at $P$. If $PA=4$ cm, $AB=7$ cm and $PD=6$ cm, then length of $CD$ is

1. $8$ cm
2. $6$ cm
3. $2$ cm
4. None of these

Q. If two corresponding sides and the angle between them of a triangle are equal to another triangle. Then the angles are :

1. similar but not congruent
2. congruent but not similar
3. neither congruent nor similar
4. congruent and similar.

Q.

In $△$ PQR, right angled at Q, PR + QR = 25 cm and PQ = 5 cm. Determine the value of cos R.

1. $\frac{12}{13}$

2. $\frac{15}{12}$

3. $\frac{5}{12}$

4. $\frac{5}{13}$

Q. Let an acute-angled triangle $ABC$ be inscribed in a circle whose centre is the origin. Let $B=(3,4)$ and $C=(-4,3)$. Then $\angle BAC$ is

1. $\frac{\pi }{5}$
2. $\frac{\pi }{4}$
3. $\frac{\pi }{2}$
4. $\frac{\pi }{3}$

Q. In figure if $AD=6cm,DB=9cm,AE=8cm$ and $EC=12cm$ and $\angle ADE={48}^o$. Find $\angle ABC$.

Q. If $△ABC\cong △DEF$ by SSS congruence, then, which of the following options is correct?

Q. A triangle ABC in which DE||BC, and intersects AB in D and AC in E
Hence,
$\frac{AD}{DB}=\frac{AE}{EC}$

1. True
2. False

Q.

In the given figure, If $PQ\parallel RS$, prove that $\Delta POQ\sim \Delta SOR$.
[2 MARKS]

Q. Using the information, given in each of the following figures, find the value of $a$.

1. $9$
2. $7$
3. $10$
4. $8$

Q. $\cos A$ $=\frac{12}{m}$, where m is,

Q. In a tringle, the lengths of the two larger sides are $10$ and
$9$ respectively. If the angles are in $A.P.,$ then the
length of the third side can be

1. $5+\sqrt{6}$
2. $5-\sqrt{6}$
3. $3\sqrt{3}$
4. $5$
5. None of these

Q. Find the value $x$ in the given figure (in degrees)

Q. Prove that the four triangles formed by joining in pairs, the mid-points of three sides of a triangle, are congruent to each other.

Q. Triangle ABC is similar to triangle PQR. If AD and PM are altitudes of the two triangles, Hence, $\frac{AB}{PQ}=\frac{AD}{PM}$.
If the above statement is true then mention answer as 1, else mention 0 if false

Q. $ABCD$ is a quadrilateral in which $AD=BC$ and $\angle DAB=\angle CBA$. Prove that
$\left(i\right)△ABD\cong △BAC$
$\left(ii\right)BD=AC$
$\left(iii\right)\angle ABD=\angle BAC.$

Q.

In SAS congruence condition two sides of one triangle and their ______________ angle are equal to two corresponding sides and their _____________ angle of other triangle.

Q.

In the given figure AD is the bisector of $\angle$A and AB = AC. Then, by which criterion $△$ACD and $△$ABD are congruent?

Q. $\Delta ABC$ and $\Delta PQR$ are similar triangles such that $\angle A={32}^0$ and $\angle R={65}^{\circ }$, then the value of $\angle B$ is

1. ${83}^{\circ }$
2. ${32}^{\circ }$
3. ${65}^{\circ }$
4. ${97}^{\circ }$

Q.

The side BC of a $\Delta$ ABC is bisected at D; O is any point in AD. BO and CO produced meet AC and AB in E and F respectively and AD is produced to X so that O is the mid-point of OX. Prove that AO : AX = AF : AB and show that FE $\parallel$ BC.

Q. A regular octagon is to be formed by cutting equal isosceles right triangles from the corners of a square. If the square has sides of one unit, the leg of each of the triangles has length:

1. $\frac{2-\sqrt{2}}{3}$
2. $\frac{2+\sqrt{2}}{3}$
3. $\frac{2-\sqrt{2}}{2}$
4. $\frac{1+\sqrt{2}}{2}$
5. $\frac{1+\sqrt{2}}{3}$