Explain How to Determine if Two Figures Are Congruent

Explain How to Determine if Two Figures Are Congruent.

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

T
he meaning of congruence in Maths is when two figures are similar to each other based on their shape and size.
In that location are basically 4 congruence rules that proves if ii triangles are congruent. But information technology is necessary to find all half-dozen dimensions. Hence, the congruence of triangles can be evaluated by knowing only 3 values out of vi.The corresponding sides and angles of coinciding triangles are equal.
Also, learn almost Congruent Figures here.

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

CPCT is the term, we come up beyond when nosotros acquire almost the coinciding triangle. Let’due south see the status for triangles to be congruent with proof.

Coinciding meaning in Maths

The meaning of congruent in Maths is addressed to those figures and shapes that can be repositioned or flipped to coincide with the other shapes. These shapes tin can exist reflected to coincide with similar shapes.

Two shapes are congruent if they have the aforementioned shape and size. We can also say if 2 shapes are congruent, then the mirror prototype of i shape is the aforementioned equally the other.

Congruent Triangles

A closed polygon made of iii line segments forming 3 angles is known as a Triangle.

Ii triangles are said to exist congruent if their sides have the same length and angles have same measure. Thus, two triangles can exist superimposed side to side and bending to angle.

Congruence Of Triangles

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

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

Sides:
AB=PQ, QR= BC and Air-conditioning=PR;

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

Congruent triangles are triangles having corresponding sides and angles to be equal. Congruence is denoted by the symbol “≅”.From the to a higher place case, we tin write

ABC ≅

PQR.
They accept the same area and the aforementioned perimeter.

For More Information On Introduction To Coinciding Triangles, Watch The Below Video:

CPCT Full Form

CPCT is the term we come up across when we learn near the congruent triangle. CPCT ways “Corresponding Parts of Congruent Triangles”. Every bit nosotros know that the corresponding parts of congruent triangles are equal. While dealing with the concepts related to triangles and solving questions, we frequently make use of the abridgement

cpct

in short words instead of full grade.

CPCT Rules in Maths

The full course of CPCT is Respective parts of Congruent triangles.
After proving triangles congruent, the remaining

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

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

Permit us acquire them all in detail.

SSS (Side-Side-Side)

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 coinciding by SSS rule.

SSS-Congruence Of Triangles

In the above-given figure, AB= PQ, BC = QR and Air conditioning=PR, hence Δ ABC ≅ Δ PQR.

SAS (Side-Angle-Side)

If whatever two sides and the angle included between the sides of one triangle are equivalent to the respective ii sides and the bending between the sides of the second triangle, so the two triangles are said to exist congruent past SAS rule.

SAS-Congruence Of Triangles

In to a higher place given figure, sides AB= PQ, Air-conditioning=PR and bending between Air conditioning and AB equal to bending between PR and PQ i.e. ∠A = ∠P. Hence, Δ ABC ≅ Δ PQR.

ASA (Bending-Side- Angle)

If whatsoever two angles and the side included between the angles of 1 triangle are equivalent to the corresponding two angles and side included between the angles of the 2nd triangle, and so the two triangles are said to be congruent by ASA dominion.

ASA-Congruence Of Triangles

In to a higher place given effigy, ∠ B = ∠ Q, ∠ C = ∠ R and sides between ∠B and ∠C , ∠Q and ∠ R are equal to each other i.eastward. BC= QR. Hence, Δ ABC ≅ Δ PQR.

For More Information On SAS And ASA Congruency Rules, Picket The Beneath Video:

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

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

AAS congruence tin can be proved in easy steps.

Suppose we accept ii triangles ABC and DEF, where,

AAS congruence

∠B = ∠E [Corresponding angles] ∠C = ∠F [Corresponding angles] And
Ac = DF [Next sides]

By angle sum property of triangle, we know that;
∠A + ∠B + ∠C = 180 ………(1)
∠D + ∠E + ∠F = 180 ……….(2)

From equation 1 and ii we tin say;
∠A + ∠B + ∠C = ∠D + ∠E + ∠F
∠A + ∠E + ∠F = ∠D + ∠Due east + ∠F [Since, ∠B = ∠E and ∠C = ∠F] ∠A = ∠D
Hence, in triangle ABC and DEF,
∠A = ∠D
Air-conditioning = 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 ii right triangles are said to exist congruent by RHS rule.

RHS-Congruence Of Triangles

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

RST.

Solved Example

Permit’s Work Out:
Example 1: In the following figure, AB = BC and Ad = CD. Prove that BD bisects AC 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)

At present, consider ∆ABE and ∆CBE,

AB = BC                                                (Given)

∠ABD = ∠CBD                                      (Proved above)

BE = Exist                                                (Common)

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

∠BEA = ∠BEC                                      (CPCTC)

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

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

∠BEA = 180°/2 = xc° = ∠BEC

AE = EC                                                (CPCTC)

Hence, BD is a perpendicular bisector of Ac.

Example 2: In a Δ ABC, if AB = Air-conditioning and ∠ B = seventy°, observe ∠ A.

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

Congruence of triangle Example 2

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

Therefore, ∠ B = ∠ C = seventy°

Sum of angles in a triangle = 180°

∠ A + ∠ B + ∠ C = 180°

∠ A + 70° + 70° = 180°

∠ A = 180° – 140°

∠ A = 40°

Practise Bug

Q.1: PQR is a triangle in which PQ = PR and is whatsoever bespeak on the side PQ. Through S, a line is drawn parallel to QR and intersecting PR at T. Show that PS = PT.
Q.2: If perpendiculars from any betoken within an angle on its arms are congruent. Prove that it lies on the bisector of that angle.

Video Lesson

For more details visit and register with us at BYJU’South. Besides, download BYJU’S-The Learning App to enjoy personalised learning.

Frequently Asked Questions

What are Coinciding Triangles?

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

What is the Full 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 5 principal rules of congruency for triangles:

  • SSS Criterion: Side-Side-Side
  • SAS Benchmark: Side-Angle-Side
  • ASA Criterion: Angle-Side- Angle
  • AAS Criterion: Angle-Angle-Side
  • RHS Criterion: 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 2nd triangle, then the 2 triangles are said to exist congruent by SSS dominion.

What is SAS congruence of triangles?

If any two sides and angle included between the sides of ane triangle are equivalent to the corresponding two sides and the angle between the sides of the second triangle, and so the 2 triangles are said to be congruent past SAS rule.

What is ASA congruency of triangles?

If whatsoever two angles and side included between the angles of 1 triangle are equivalent to the corresponding 2 angles and side included between the angles of the 2nd triangle, then the two triangles are said to be congruent by ASA rule.

What is AAS congruency?

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

What is RHS congruency?

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, so the ii right triangles are said to be congruent by RHS rule.

Explain How to Determine if Two Figures Are Congruent

Source: https://byjus.com/maths/congruence-of-triangles/