Note on Congurency and Similarities

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Congruent Triangles

The triangles having same size and shape are called congruent triangles. Two triangles are congruent when the three sides and three angles of one triangle have the measurements as three sides and three angles of another triangle. The symbol for congruent is ≅.

In the following figure, ΔABC and ΔPQR are congruent. We denote this as ΔABC ≅ ΔPQR.

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Postulate and Theorems for Congruent Triangles

Postulate (SAS)

If two sides and the angle between them in one triangle are congruent to the corresponding parts in another triangle, then the triangles are congruent.

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In the given figure,

AB ≅ PQ Sides (S)

∠B ≅ ∠Q Angle (A)

BC ≅ QR Side (S)

Therefore, ΔABC ≅ ΔPQR

Theorem (ASA)

A unique triangle is formed by two angles and the included side.

Therefore, if two angles and the included side of one triangle are congruent to two angles and the included side of the another triangle, then the triangles are congruent.

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In the figure,

∠B ≅ ∠E Angle (A)

BC ≅ EF Side (S)

∠C ≅ ∠F Angle (A)

Therefore, ΔABC ≅ ΔDEF

Theorem ( AAS)

A unique triangle is formed by two angles and non-included side. Therefore, if two angles and the side opposite to one of them in a triangle are congruent to the corresponding parts in another triangle, then the triangles are congruent.

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In the figure,

∠A ≅ ∠X Angle (A)

∠C ≅ ∠Z Angle (A)

BC ≅ YZ Side (S)

Therefore, ΔABC ≅ ΔXYZ

Theorem (SSS)

A unique triangle is formed by specifying three sides of a triangle, where the longest side (if there is one) is less than the sum of the two shorter sides.

Therefore, if their sides of a triangle are congruent to three sides of another triangle, then the triangles are congruent.

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In the figure

AB ≅ PQ Sides (S)

BC ≅ QR Sides (S)

CA ≅ RP Sides (S)

Therefore, ΔABC ≅ ΔPQR

Similar Triangles

Methods of providing triangles similar

  1. If the corresponding sides of a triangle are proportion to another triangle then the triangles are similar.
    Example
    .
    If∠A ≅ ∠D and ∠B ≅ ∠E, Then ΔABC ∼ ΔDEF

  2. If the corresponding angle of a triangle is congruent to another triangle then, the triangles are similar.
    Example
    .
    If \(\frac{AB}{DE}\) = \(\frac{BC}{EF}\) = \(\frac{AC}{DF}\), then ΔABC ∼ ΔDEF

  3. Conversing first and the second method we can prove triangle similar as their sides being proportional and angles congruent.
    Example
    .
    ∠A ≅ ∠D and \(\frac{AB}{DE}\) = \(\frac{AC}{DF}\) then ΔABC ∼ ΔDEF

In case of overlapping triangles

When the lines are parallel in a triangle, then they intersect each other which divides the sides of a triangle proportionally.

Verification:

N

In ΔPQR and ΔSPT

Statements Reasons
ST⁄⁄QR Given
\(\angle\)PST \(\cong\) \(\angle\)QSR Corresponding angles
ΔPQR \(\cong\) ΔSPT Common Angle P
\(\frac{PS}{SQ}\) = \(\frac{PT}{TR}\) ST⁄⁄QR

 

Example

Given the following triangles, find the length of x.

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Solution:

The triangles are similar by AA rule.So, the ratio of lengths are equal.

\(\frac{6}{3}\) = \(\frac{10}{x}\)

or, 6x = 30

or, x = \(\frac{30}{6}\)

\(\therefore\) x = 5 cm

  • There is three easy way to prove similarity. If two pairs of corresponding  angles in a pair of triangles are congruent, then the triangles are similar.
  • When the three angle pairs are all equal, the  three pairs of the side must  be proportion.
  • When triangles are congruent and one triangle is placed on the top of other sides and angles that are in the same position are called corresponding parts.
  • Congruent and similar shapes can make calculations and design work easier.
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Very Short Questions

Solution:

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Here, AB=1.6 cm DE=1.9 cm

BC=1.3 cm EF=1.4 cm

CA=1.9 cm DF=1.9 cm

\(\angle\)A=45o\(\angle\)D=40o

\(\angle\)B=100o\(\angle\)E=70o

\(\angle\)C=35o\(\angle\)F=70o

The angles arms of the triangle is not equal so it is not congurent.

Solution:

Here given, congurent triangle is PQ and LM, QR and MN, PR and LN. So that, congurent angles is \(\angle\)P and(\angle\)L,(\angle\)R and(\angle\)N,(\angle\)Q and(\angle\)M.

Solution:

Here given, congurent triangle is XY and AB, YZ and BC, XZ and AC. So that, congurent angles is \(\angle\)X and(\angle\)A,(\angle\)Y and(\angle\)B,(\angle\)Z and(\angle\)C.

Solution:

Here, \(\angle\)A=52o \(\angle\)=88o \(\angle\)C=40o

\(\angle\)P=52o \(\angle\Q=88o \(\angle\)R=40o

AB=1 cm BC=1.2 cm CA=2 cm

PQ=1.3 cm QR=1.6 cm PR=2.6 cm

\(\angle\)A= \(\angle\)P, \(\angle\)B= \(\angle\)Q and \(\angle\)C= \(\angle\)R

\(\frac{AB}{PQ}\)=\(\frac{1.0}{1.3}\),\(\frac{BC}{QR}\),\(\frac{1.0}{1.3}\),=\(\frac{CA}{RP}\)=\(\frac{1.0}{1.3}\)

Hence, given triangle is similar triangle.

Solution:

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Here, \(\angle\) A=40o\(\angle\) B=70o\(\angle\) C=70o

\(\angle\) P=41o\(\angle\) Q=62o\(\angle\) R=77o

AB=1.5 cm BC=1 cm CA=1.5 cm

PQ=1.1 cm QR=1 cm RP=1.6 cm

Here,\(\angle\)A \(\neq \)\(\angle\)P,\(\angle\)B\(\neq \)\(\angle\)Q and \(\angle\)C\(\neq \) \(\angle\)R

Hence, given triangle is not similar triangle.

Solution:

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Here, \(\angle\)A=38o \(\angle\)B=90o \(\angle\)C=52o

\(\angle\)P=38o \(\angle\)Q=90o \(\angle\)r=52o

AB=1.6 cm BC=1.2 cm

CA=2 cm PQ=0.8 cm

QR=0.6 cm PR=1 cm

\(\angle\)A= \(\angle\)P, \(\angle\)B= \(\angle\)Q and \(\angle\)C= \(\angle\)R

and \(\frac{AB}{PQ}\)=\(\frac{1.6}{0.8}\)=2, \(\frac{BC}{QR}\)=\(\frac{1.2}{0.6}\)=2, \(\frac{CA}{RP}\)=\(\frac{2}{1}\)=2

Hence, given triangle is similar triangle.

From the given figure,

∠STU ≅ ∠SVW and TU ≅ VW

Here, ∠TSU and ∠VSW are vertical angles. Since vertical angles are congruent,

∠TSU ≅ ∠VSW.

Finally, put the three congruency statements in order. ∠STU is between ∠TSU and TU, and ∠SVW is between ∠VSW and VW in the diagram.

∠TSU ≅ ∠VSW (Angle)

∠STU ≅ ∠SVW (Angle)

TU ≅ VW (Side)

Hence, the given triangles are congurent as it forms AAS theorem.

From the given figure,

BC ≅ BH and ∠BCF≅∠BHG.

Here, ∠CBF and ∠GBH are vertical angles. Since vertical angles are congruent,

∠CBF ≅ ∠GBH.

Finally, put the three congruency statements in order. BC is between ∠BCF and ∠CBF, and BH is between ∠BHG and ∠GBH in the diagram.

∠BCF ≅ ∠BHG Angle

BC ≅ BH Side

∠CBF ≅ ∠GBH Angle

Hence, the congruent sides and angles form ASA. The triangles are congruent by the ASA Theorem.

From the figure,

∠XWY≅∠YWZ and ∠WXY≅∠WZY.

Here, the triangles share WY. By the reflexive property of congruence, WY ≅ WY.

Finally, put the three congruency statements in order. ∠WXY is between ∠XWY and WY, and ∠WZY is between ∠YWZ and WY in the diagram.

∠XWY ≅ ∠YWZ Angle

∠WXY ≅ ∠WZY Angle

WY ≅ WY Side

Hence, the congruent sides and angles form AAS. The triangles are congruent by the AAS Theorem.

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  • Which of the following triangles are always similar?

    Isosceles triangles
    Right triangles
    Similar triangles
    Equilateral triangles
  • The sides of a triangle are 5cm, 6cm and 10cm.Find the length of the longest side of a similar triangle whose shortest side is 15cm.

     30cm
     35cm
     45cm
     60cm
  • Calculate the height of a building that casts a shadow of 6.5 meters if at the same time and in the same place a pole of 4.5m in height produces a shadow of 0.9m.

     35m
     33m
     32.5m
     37m
  • The legs of a right angled triangle measure 24cm and 10cm.What is the length of the legs of a similar triangle to this one whose hypotenuse is 52cm?

     22cm, 45cm
     22cm, 45cm
     20cm, 48cm
     25cm, 50cm
  • The given triangle ABC and triangle DEF are pair of Congruent triangles. Find the value of x.

    .

     5cm
     4cm
     6cm
     7cm
  • By which postulate, the given two triangles ABC and DEF are congruent? Which is the corresponding sides of AC?

    .`
    `

    ASA Theorem, EF
    AAS Theorem, EF
    SSS Theorem, EF
    SAS postulate, EF.
  • In the given figure, if triangle PQR is congruent to triangle XYZ, find the value of two unknown angles and the value of y.

    .

    (angle)Q=60o, (angle)X=45o, Y=7cm
    (angle)Q=45o, (angle)X=60o, Y= 6cm
    (angle)Q=30o, (angle)X=45o, Y=8cm
    (angle)Q= 60o, (angle)X= 55o, Y= 5cm
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rajeeb khaska

how to find angle


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Aakash khanal

How to solve congruent triangles and draw


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Zeref dragneel

Nope......


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