Question

1) If two triangles have two sides and the included angle of the one equal to the corresponding sides and the incident angle of the other then the triangles are congruent.

2) The angles opposite to two equal sides of a traiangle are equal.

3) If two angles of a traiangle are equal then the sides opposite to them are also equal.

4) If the three sides of one traiangle are equal to the corresponding three sides of another traiangle, then prove that two traiangles are congruent.

5) In the given figure(attachment) OA = OB and OC = OD

Show that :-
(i) △AOC ≅ △BOD
(ii) AC || BD

ＮＯ ＳＰＡＭＭＩＮＧ ❌
ＱＵＡＬＩＴＹ ＲＥＱＵＩＲＥＤ ✔️

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Answer 1

Step-by-step explanation:

ANSWER:-

1) If two triangles have two sides and the included angle of the one equal to the corresponding sides and the incident angle of the other then the triangles are congruent $.$ [ SAS Congruence]

2) The angles opposite to two equal sides of a triangle are equal. [Inequalities in a Triangle]

3) If two angles of a triangle are equal then the sides opposite to them are also equal. [ASA Congruence ]

4) Let ABC and DEF be the 2 triangles. So,

$\longrightarrow$ AB = DE [given]

$\longrightarrow$BC=EF [given]

$\longrightarrow$AC=DF [given]

By SSS Congruence, [side-side-side]

Triangle ABC is congruent to(≅ ) Triangle DEF.

5)

i)△AOC ≅ △BOD (To prove)

So in ABC and BOD,

$\longrightarrow$OA = OB {given}

$\longrightarrow$OC = OD{given}

$\longrightarrow$Angle AOC = Angle DOB [Vertically Opposite Angles]

By SAS Congruence, [side-angle-side]

△AOC ≅ △BOD

ii)AC || BD (To prove)

According to what is proved above,

We can say that

$\longrightarrow$Angle ACO = Angle BDO [CPCT]

$\longrightarrow$Angle OAC = Angle DBO, [CPCT]

As these angles are equal and in alternate interior angles, so we can prove that AC || BD.

$\longrightarrow$CPCT stands for Congruent Parts of Congruent Triangle.

Hence, Proved.

Answer 2

$\huge\fcolorbox{black}{aqua}{Solution☣:-}$

ANSWER:-

1) If two triangles have two sides and the included angle of the one equal to the corresponding sides and the incident angle of the other then the triangles are congruent .. [ SAS Congruence]

2) The angles opposite to two equal sides of a triangle are equal. [Inequalities in a Triangle]

3) If two angles of a triangle are equal then the sides opposite to them are also equal. [ASA Congruence ]

4) Let ABC and DEF be the 2 triangles. So,

⟶ AB = DE [given]

⟶ BC=EF [given]

⟶ AC=DF [given]

By SSS Congruence, [side-side-side]

Triangle ABC is congruent to(≅ ) Triangle DEF.

5)

i)△AOC ≅ △BOD (To prove)

So in ABC and BOD,

⟶ OA = OB {given}

⟶ OC = OD{given}

⟶ Angle AOC = Angle DOB [Vertically Opposite Angles]

By SAS Congruence, [side-angle-side]

△AOC ≅ △BOD

ii)AC || BD (To prove)

According to what is proved above,

We can say that

⟶ Angle ACO = Angle BDO [CPCT]

⟶ Angle OAC = Angle DBO, [CPCT]

As these angles are equal and in alternate interior angles, so we can prove that AC || BD.

⟶ CPCT stands for Congruent Parts of Congruent Triangle.

Hence, Proved.☢