Are the Triangles Congruent Why or Why Not

Are the Triangles Congruent Why or Why Not

Congruence of triangles: 2 triangles are said to exist congruent if all three corresponding sides are equal and all the three corresponding angles are equal in measure. These triangles can be slides, rotated, flipped and turned to be looked identical. If repositioned, they coincide with each other. The symbol of congruence is’ ≅’.

T
he meaning of congruence in Maths is when ii figures are similar to each other based on their shape and size.
At that place are basically 4 congruence rules that proves if two triangles are congruent. But it is necessary to observe all six dimensions. Hence, the congruence of triangles can be evaluated by knowing just three values out of 6.The corresponding sides and angles of congruent triangles are equal.
Also, learn about Congruent Figures hither.

Congruence is the term used to ascertain an object and its mirror image. 2 objects or shapes are said to be congruent if they superimpose on each other. Their shape and dimensions are the same. In the case of geometric figures, line segments with the same length are congruent and angle with the same measure are coinciding.

CPCT is the term, nosotros come across when nosotros larn about the congruent triangle. Let’s see the condition for triangles to be congruent with proof.

Congruent pregnant in Maths

The significant of coinciding in Maths is addressed to those figures and shapes that can be repositioned or flipped to coincide with the other shapes. These shapes can exist reflected to coincide with like shapes.

Two shapes are coinciding if they have the aforementioned shape and size. We can also say if two shapes are congruent, and so the mirror image of one shape is the aforementioned as the other.

Congruent Triangles

A closed polygon made of three line segments forming iii angles is known as a Triangle.

Two triangles are said to be congruent if their sides take the same length and angles have same measure out. Thus, two triangles can be superimposed side to side and angle to angle.

In the above figure, Δ ABC and Δ PQR are congruent triangles. This means,

Vertices:
A and P, B and Q, and C and R are the aforementioned.

Sides:
AB=PQ, QR= BC and Ac=PR;

Angles:
∠A = ∠P, ∠B = ∠Q, and ∠C = ∠R.

Congruent triangles are triangles having corresponding sides and angles to exist equal. Congruence is denoted past the symbol “≅”.From the above instance, we can write

ABC ≅

PQR.
They have the aforementioned expanse and the aforementioned perimeter.

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For More than Information On Introduction To Congruent Triangles, Sentinel The Below Video:

CPCT Total Course

CPCT is the term we come across when we larn about the congruent triangle. CPCT means “Corresponding Parts of Congruent Triangles”. As we know that the corresponding parts of coinciding triangles are equal. While dealing with the concepts related to triangles and solving questions, nosotros oft make use of the abbreviation

cpct

in short words instead of full class.

CPCT Rules in Maths

The full form of CPCT is Corresponding parts of Congruent triangles.
After proving triangles congruent, the remaining

dimension
can be predicted without really measuring the sides and angles of a triangle. Different rules of congruency are as follows.

  • SSS (Side-Side-Side)
  • SAS (Side-Angle-Side)
  • ASA (Angle-Side-Angle)
  • AAS (Bending-Angle-Side)
  • RHS (Right angle-Hypotenuse-Side)

Permit usa learn them all in detail.

SSS (Side-Side-Side)

If all the three sides of i triangle are equivalent to the respective three sides of the second triangle, then the two triangles are said to be congruent past SSS dominion.

SSS-Congruence Of Triangles

In the above-given effigy, AB= PQ, BC = QR and AC=PR, hence Δ ABC ≅ Δ PQR.

SAS (Side-Angle-Side)

If any two sides and the bending included betwixt the sides of 1 triangle are equivalent to the corresponding ii sides and the angle betwixt the sides of the second triangle, and then the ii triangles are said to be congruent by SAS dominion.

SAS-Congruence Of Triangles

In above given figure, sides AB= PQ, Ac=PR and angle between Ac and AB equal to angle between PR and PQ i.due east. ∠A = ∠P. Hence, Δ ABC ≅ Δ PQR.

ASA (Angle-Side- Angle)

If any two angles and the side included between the angles of i triangle are equivalent to the corresponding ii angles and side included between the angles of the second triangle, then the two triangles are said to exist congruent by ASA rule.

ASA-Congruence Of Triangles

In above given figure, ∠ B = ∠ Q, ∠ C = ∠ R and sides between ∠B and ∠C , ∠Q and ∠ R are equal to each other i.e. BC= QR. Hence, Δ ABC ≅ Δ PQR.

For More than Information On SAS And ASA Congruency Rules, Lookout The Below Video:

AAS (Angle-Angle-Side) [Application of ASA]

AAS stands for Bending-Bending-Side. When two angles and a not-included side of a triangle are equal to the corresponding angles and sides of some other triangle, then the triangles are said to be congruent.

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AAS congruence can be proved in piece of cake steps.

Suppose we accept two triangles ABC and DEF, where,

AAS congruence

∠B = ∠Eastward [Corresponding angles] ∠C = ∠F [Corresponding angles] And
Ac = DF [Adjacent sides]

By angle sum property of triangle, we know that;
∠A + ∠B + ∠C = 180 ………(1)
∠D + ∠E + ∠F = 180 ……….(ii)

From equation ane and ii nosotros tin can say;
∠A + ∠B + ∠C = ∠D + ∠E + ∠F
∠A + ∠E + ∠F = ∠D + ∠E + ∠F [Since, ∠B = ∠East and ∠C = ∠F] ∠A = ∠D
Hence, in triangle ABC and DEF,
∠A = ∠D
AC = DF
∠C = ∠F
Hence, by ASA congruency,
Δ ABC ≅ Δ DEF

RHS (Right angle- Hypotenuse-Side)

If the hypotenuse and a side of a right- angled triangle is equivalent to the hypotenuse and a side of the second right- angled triangle, then the two right triangles are said to be congruent past RHS rule.

RHS-Congruence Of Triangles

In above effigy, hypotenuse XZ = RT and side YZ=ST, hence
XYZ ≅

RST.

Solved Instance

Let’south Work Out:
Case ane: In the following effigy, AB = BC and Advertizement = CD. Show that BD bisects Air conditioning at right angles.

Congruence of Triangle ExampleSolution: We are required to prove ∠BEA = ∠BEC = 90° and AE = EC.Consider ∆ABD and ∆CBD,AB = BC                                                (Given)AD = CD          (Given)BD = BD                                                (Common)Therefore, ∆ABD ≅ ∆CBD                      (Past SSS congruency)∠ABD = ∠CBD                                      (CPCTC)

Now, consider ∆ABE and ∆CBE,

AB = BC                                                (Given)

∠ABD = ∠CBD                                      (Proved above)

BE = Be                                                (Common)

Therefore, ∆ABE≅ ∆CBE                       (By SAS congruency)

∠BEA = ∠BEC                                      (CPCTC)

And ∠BEA +∠BEC = 180°                      (Linear pair)

2∠BEA = 180°                                       (∠BEA = ∠BEC)

∠BEA = 180°/ii = 90° = ∠BEC

AE = EC                                                (CPCTC)

Hence, BD is a perpendicular bisector of AC.

Example ii: In a Δ ABC, if AB = Ac and ∠ B = 70°, find ∠ A.

Solution: Given: In a Δ ABC, AB = AC and ∠B = 70°

Congruence of triangle Example 2

∠ B = ∠ C [Angles opposite to equal sides of a triangle are equal]

Therefore, ∠ B = ∠ C = 70°

Sum of angles in a triangle = 180°

∠ A + ∠ B + ∠ C = 180°

∠ A + 70° + 70° = 180°

∠ A = 180° – 140°

∠ A = xl°

Practice Bug

Q.i: PQR is a triangle in which PQ = PR and is whatsoever point on the side PQ. Through South, a line is drawn parallel to QR and intersecting PR at T. Bear witness that PS = PT.
Q.ii: If perpendiculars from any point within an angle on its arms are coinciding. Testify that information technology lies on the bisector of that bending.

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Video Lesson

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Frequently Asked Questions

What are Congruent Triangles?

Two triangles are said to exist congruent if the three sides and the three angles of both the angles are equal in any orientation.

What is the Total Grade of CPCT?

CPCT stands for Respective parts of Congruent triangles. CPCT theorem states that if two or more than triangles which are congruent to each other are taken then the corresponding angles and the sides of the triangles are also congruent to each other.

What are the Rules of Congruency?

There are five main rules of congruency for triangles:

  • SSS Criterion: Side-Side-Side
  • SAS Criterion: Side-Angle-Side
  • ASA Criterion: Angle-Side- Angle
  • AAS Criterion: Angle-Bending-Side
  • RHS Benchmark: Right angle- Hypotenuse-Side

What is SSS congruency of triangles?

If all the three sides of one triangle are equivalent to the corresponding three sides of the second triangle, then the two triangles are said to be congruent by SSS rule.

What is SAS congruence of triangles?

If any ii sides and angle included between the sides of one triangle are equivalent to the corresponding two sides and the angle between the sides of the 2nd triangle, then the two triangles are said to exist coinciding by SAS rule.

What is ASA congruency of triangles?

If any two angles and side included between the angles of one triangle are equivalent to the corresponding two angles and side included between the angles of the second triangle, then the 2 triangles are said to exist congruent by ASA rule.

What is AAS congruency?

When two angles and a non-included side of any two triangles are equal then they are said to be coinciding.

What is RHS congruency?

If the hypotenuse and a side of a correct- angled triangle is equivalent to the hypotenuse and a side of the second right- angled triangle, so the 2 right triangles are said to be coinciding by RHS rule.

Are the Triangles Congruent Why or Why Not

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