RHS Criteria for Congruency

Q.

In an equilateral $\Delta ABC,AD\perp BC$. prove that $A{D}^2=3B{D}^2$

Q. Question 6
If $\angle$A and $\angle$B are acute angles such that cos A = cos B, then show that $\angle$A = $\angle$B.

Q. in triangle ABC, AD is median on BC AB= 10 cm AC = 12 cm and BC = 16 cm then length of AD ?

Q. In triangle ABC angle C = 5 angle A - 5 degree = 3 angle B + 15 degree. find the three angles of triangle ABC.

Q. In triangle ABC, bisectors of angle B and angle C meet each other at point P. Line AP when extended, cuts the side BC at Q. Then AP/PQ

Q. ABC is a triangle, in which angle-B=2angle-C. D is a point on side BC such that AD bisects angle-BAC and AB=CD. Prove that angle-BAC=72°.

Q.

Given below are measurements of some parts of two triangles. By drawing their rough figure, examine whether the two triangles are congruent or not by ASA congruence rule. In case of congruence, write it in symbolic form.

$∆DEF$
$\angle D=60°,\angle F=80°,DF=5cm$

$∆PQR$
$\angle Q=60°,\angle R=80°,QR=5cm$

Q. Question 3
A tangent PQ at a point P of a circle of radius 5 cm meets a line through the centre O at a point Q so that OQ = 12 cm. Length PQ is :
(A) 12 cm
(B) 13 cm
(C) 8.5 cm
(D) $\sqrt{119}cm$

Q. If in a triangle ABC, AB>BC>AC which angle is largest angle?

Q. 46. In a right angled triangle , one acute angle is double the other. Prove that the hypotenuse is double the smallest side.

Q. In a triangle ABC, if angle a + angle b=65 degree and angle b + angle c=140 degree then the measure of angle c is

Q. If the bisector of an angle of a triangle bisects the opposite side, then the triangle is
(a) scalene
(b) equilateral
(c) isosceles
(d) right-angled

Q. The length of a hypotenuse of a right angle triangle exceeds the length of its base by $2$ cm and exceeds twice the length of the altitude by $1$ cm. Find the length of each side of the triangle (in cm).

1. $6,8,10$
2. $7,24,25$
3. $8,15,17$
4. $7,40,41$

Q. In a ∆ABC, if AB = 8 cm, AC = 10 cm and BC = 14 cm, then the length of median AD drawn from point ‘A’ on BC is equal to

Q. If D and E are points on sides AB and AC respectively of triangle ABC such that BD = to CE. Prove that triangle ABC is isosceles.

Q. in triangle abc, M is the circumcentre, AC=BC& angle AMB=130°, then the measure of angle CAB

Q. 18. In an equilateral triangle ABC if D is a midpoint on side BC such that BD then AD;AB=

Q.

In the following figure, check whether triangles are congruent by ASA congruence condition ? If congruent, write the congruence of two triangles in symbolic form:

Q. In the given fugure, AD is the bisector of ∠BAC. If AB = 10 cm, AC = 14 cm and BC = 6 cm. Find BD and DC.
[3 Marks]

Q. Two angles of a triangle are equal and the third angles is greater than each one if them by 18°. Find the angles

Q. In the figure given below, $\angle BCD={60}^{\circ }$,
then the measure of $\angle$BDA is.

Q. A tangent PQ at a point P of a circle of radius 5 cm meets a line through the centre O at a point Q so that OQ = 12 cm. Find the length of PQ.

Q. A tangent PQ at a point P of a circle of radius 5 cm meets a line through the centre O at a point Q so that OQ = 12 cm. Find the length of PQ.

Q. 35. If angles a, b, c of a triangle ABC are in A.P.then B=

Q. In a ΔABC, D and E are points on AC and BC, respectively such that DE || AB. If AD = $2x$, DC = $x+1$, BE = $2x-1$, and EC = $x-1$, then find the value of $x$.

Q.

In the given pairs of triangles, measures of some parts are given. By applying RHS congruence rule, state whether the triangles are congruent or not. In case of congruent triangles, write the result in symbolic form.

Q. Question 3
A tangent PQ at a point P of a circle of radius 5 cm meets a line through the centre O at a point Q so that OQ = 12 cm. Length PQ is :
(A) 12 cm
(B) 13 cm
(C) 8.5 cm
(D) $\sqrt{119}cm$

Q.

In $∆ABC\text{and}∆PQR$, it is given that $AB=AC,\angle C=\angle P\text{and}\angle B=\angle Q$. Then, the two triangles are:

1. isosceles but not congruent

2. neither congruent nor isosceles

3. isosceles and congruent

4. congruent but not isosceles

Q. In a triangle ABX, if ADis the bisector of angle BAC which meets BC at point D such that BD=CD, then which of the following is not always true

Q. For finding AB and BC wit the help of information given in the figure, complete following activity .
$AB=BC............$
$\therefore \angle BAC=...........$
$\therefore AB=BC=........\times AC$
$=........\times \sqrt{8}$
$=........\times 2\times \sqrt{2}$
$=.......$