**Example 1:** In given figures, lengths of the sides of the triangles are indicated. By applying SSS congruent rule, state which pairs of triangle are congruent. In case of congruent triangles, write the results in symbolic form:

**Answer:** (i) ΔABC ≈ ΔPQR

(ii) ΔDEF ≈ ΔNML

(iii) Not congruent

(iv) ΔADB ≈ ΔADC

**Example 2:** In the given figure, AB = AC and D is the mid-point of BC

(a) State the three pairs of equal parts in ΔADB and ΔADC.

**Answer:** AD = AD, DB = DC and AB = AC

(b) Is ΔADB ≈ΔADC? Give reasons.

**Answer:** By SSS criterion, triangles are congruent.

(c) Is angle B = angle C? Why?

**Answer:** Since, triangle are congruent, AC = AB and DC = DB hence, angle B = angle C

**Example 3:** In the given figure, AC = BD and AD = BC. Which of the following statements is meaningfully written? (i) ΔABC ≈ ΔADB (ii) ΔABC ≈ ΔBAD

**Answer:** (ii) ΔABC ≈ ΔBAD

**Example 4:** Which angle is included between the sides DE and EF of ΔDEF?

**Answer:** ∠E

**Example 5:** By applying SAS congruence rule, you want to establish that, ΔPQR ≈ ΔFED. It is given that PQ = FE and RP = DF. What additional information is needed to establish the congruence?

**Answer:** Measurement of QR and ED or measurement of angles P and F.

**Example 6:** In the given figures, measures of some parts of the triangles are indicated. By applying SAS congruent rule, state the pairs of congruent triangles, if any, in each case. In case of congruent triangles, write them in symbolic form.

**Answer:** (i) ΔABC ≈ ΔDEF

(ii) ΔCAB ≈ ΔPRQ

(iii) ΔDEF ≈ ΔPRQ

(iv) ΔQPR ≈ ΔSRP

**Example 7:** In the given figure, AB and CD bisect each other at O.

(a) State the three pairs of equal parts in two triangles AOC and BOD.

**Answer:** OC = OD, OA = OB and ∠COA = ∠DOB

(b) Which of the following statements are true?

(i) ΔAOC ≈ ΔDOB

(ii) ΔAOC ≈ ΔBOD

**Answer:** (ii) ΔAOC ≈ ΔBOD

**Example 8:** What is the side included between the angles M and N of ΔMNP?

**Answer:** MN

**Example 9:** You want to establish ΔDEF ≈ ΔMNP, using the ASA congruent rule. You are given that angle D = angle M and angle F = angle P. What information is needed to establish the congruence?

**Answer:** Measurements of sides DF and MP

**Example 10:** In the given figure, measures of some part are indicated. By applying ASA congruence rule, state which pairs of triangles are congruent. In case of congruence, write the result in symbolic form.

**Answer:** (i) ΔABC ≈ ΔFED

(ii) Not congruent

(iii) ΔPQR ≈ ΔMNL

(iv) ΔDAB ≈ ΔCBA

**Example 11:** Given below are measurements of some parts of two triangles. Examine whether the two triangles are congruent or not, by ASA congruent rule. In case of congruence, write it in symbolic form.

S. No. | ΔDEF | ΔPQR |
---|---|---|

(a) | D = 60°, F = 80°, DF = 5 cm | Q = 60, R = 80°, QR = 5 cm |

(b) | D = 60°, F = 80°, DF = 6 cm | Q = 60°, R = 80°, QP = 6 cm |

(c) | E = 80°, F = 30°, EF = 5 cm | P = 80°, R = 30°, PQ = 5 cm |

**Answer:** a → Yes, b → No, c → No

**Example 12:** In the given figure, ray AX bisects angle DAB as well as angle DCB.

(a) State the three pairs of equal parts in triangles BAC and DAC.

(b) Is ΔBAC ≈ ΔDAC? Give reasons.

**Answer:** By ASA criterion, triangles are congruent

(c) Is AB = AD? Justify your answer.

**Answer:** Since triangles are congruent hence, AB = AD because both sides originate from equal angles in two triangles.

(d) Is CD = CB? Give reasons.

**Answer:** Since triangles are congruent hence, CD = CB because both sides originate from equal angles in two triangles.

**Example 13:** In the given figure, measures of some parts of triangles are given. By applying RHS congruence rule, state which pairs of triangles are congruent? In case of congruent triangles, write the results in symbolic form.

**Answer:** (i) Not congruent

(ii) ΔACB ≈ ΔBDA

(iii) ΔABC ≈ ΔADC

(iv) ΔPSQ ≈ ΔPSR

**Example 14:** It is to be established by RHS congruence rule that ΔABC ≈ ΔRPQ. What additional information is needed, if it is given that angle B = angle P = 90° and AB = RP?

**Answer:** Measurement of hypotenuse of two triangles.

**Example 15:** In the given figure, BD and CE are altitudes of ΔCBD and ΔBCE such that BD = CE.

(a) State the three pairs of equal parts in ΔCBD and ΔBCE.

**Answer:** BD = CE, BC = CB and angle D = angle E

(b) Is ΔCBD ≈ΔBCE? Why or why not?

**Answer:** By RHS criterion, triangles are congruent.

(c) Is ∠DCB = ∠EBC? Why or why not?

**Answer:** They are equal because they are included between equal sides of the given congruent triangles.

**Example 16:** ABC is an isosceles triangle with AB = AC and AD is one of its altitudes.

(a) State the three pairs of equal parts in ΔADB and ΔADC.

**Answer:** AD = AD, AB = AC and DB = DC

(b) Is ΔADB ≈ ΔADC? Why or why not?

**Answer:** By SSS criterion, triangles are congruent.

(c) Is angle B = angle C? Why or why not?

**Answer:** Yes, because they are included between equal sides of given congruent triangles.

(d) Is BD = CD? Why or why not?

**Answer:** Yes, because altitude of an isosceles triangle bisects its base when the base is not one of the two equal sides.

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