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.
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.
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.
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.
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.
In the figure
AB ≅ PQ Sides (S)
BC ≅ QR Sides (S)
CA ≅ RP Sides (S)
Therefore, ΔABC ≅ ΔPQR
When the lines are parallel in a triangle, then they intersect each other which divides the sides of a triangle proportionally.
Verification:
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.
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
Solution:
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=45^{o}\(\angle\)D=40^{o}
\(\angle\)B=100^{o}\(\angle\)E=70^{o}
\(\angle\)C=35^{o}\(\angle\)F=70^{o}
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=52^{o} \(\angle\)=88^{o} \(\angle\)C=40^{o}
\(\angle\)P=52^{o} \(\angle\Q=88^{o} \(\angle\)R=40^{o}
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:
Here, \(\angle\) A=40^{o}\(\angle\) B=70^{o}\(\angle\) C=70^{o}
\(\angle\) P=41^{o}\(\angle\) Q=62^{o}\(\angle\) R=77^{o}
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:
Here, \(\angle\)A=38^{o} \(\angle\)B=90^{o} \(\angle\)C=52^{o}
\(\angle\)P=38^{o} \(\angle\)Q=90^{o} \(\angle\)r=52^{o}
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.
Which of the following triangles are always similar?
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.
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.
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?
The given triangle ABC and triangle DEF are pair of Congruent triangles. Find the value of x.
By which postulate, the given two triangles ABC and DEF are congruent? Which is the corresponding sides of AC?
In the given figure, if triangle PQR is congruent to triangle XYZ, find the value of two unknown angles and the value of y.
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rajeeb khaska
how to find angle
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How to solve congruent triangles and draw
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