congruents of triangle
TRANSCRIPT
![Page 1: Congruents of Triangle](https://reader030.vdocuments.us/reader030/viewer/2022032620/55c6c706bb61ebc66e8b4660/html5/thumbnails/1.jpg)
Project in MathCongruent of Triangles
S.Y 2014 – 2015Submitted by:
![Page 2: Congruents of Triangle](https://reader030.vdocuments.us/reader030/viewer/2022032620/55c6c706bb61ebc66e8b4660/html5/thumbnails/2.jpg)
Congruent Triangles
Triangles are congruent when they have exactly the same three sides and exactly the same three angles.
![Page 3: Congruents of Triangle](https://reader030.vdocuments.us/reader030/viewer/2022032620/55c6c706bb61ebc66e8b4660/html5/thumbnails/3.jpg)
What is "Congruent" ... ? It means that one shape can become
another using Turns, Flips and/or Slides:
Rotation Turn!
Reflection Flip!
Translation Slide!
![Page 4: Congruents of Triangle](https://reader030.vdocuments.us/reader030/viewer/2022032620/55c6c706bb61ebc66e8b4660/html5/thumbnails/4.jpg)
Congruent Triangles
When two triangles are congruent they will have exactly the same three sides and exactly the same three angles.
The equal sides and angles may not be in the same position (if there is a turn or a flip), but they are there.
![Page 5: Congruents of Triangle](https://reader030.vdocuments.us/reader030/viewer/2022032620/55c6c706bb61ebc66e8b4660/html5/thumbnails/5.jpg)
Same SidesWhen the sides are the same then the
triangles are congruent. For example:
is congruent to: and
because they all have exactly the same sides.
![Page 6: Congruents of Triangle](https://reader030.vdocuments.us/reader030/viewer/2022032620/55c6c706bb61ebc66e8b4660/html5/thumbnails/6.jpg)
But:
is NOT congruent to:
because the two triangles do not have exactly the same sides.
![Page 7: Congruents of Triangle](https://reader030.vdocuments.us/reader030/viewer/2022032620/55c6c706bb61ebc66e8b4660/html5/thumbnails/7.jpg)
Same Angles Does this also work with angles? Not
always! Two triangles with the same
angles might be congruent:
is congruent to:
only because they are the same size
![Page 8: Congruents of Triangle](https://reader030.vdocuments.us/reader030/viewer/2022032620/55c6c706bb61ebc66e8b4660/html5/thumbnails/8.jpg)
But they might NOT be congruent because of different sizes:
is NOT congruent to:
because, even though all angles match, one is larger than the other.
So just having the same angles is no guarantee they are congruent.
![Page 9: Congruents of Triangle](https://reader030.vdocuments.us/reader030/viewer/2022032620/55c6c706bb61ebc66e8b4660/html5/thumbnails/9.jpg)
How To Find if Triangles are Congruent
Two triangles are congruent if they have: exactly the same three sides and exactly the same three angles. But we don't have to know all three
sides and all three angles ...usually three out of the six is enough.
![Page 10: Congruents of Triangle](https://reader030.vdocuments.us/reader030/viewer/2022032620/55c6c706bb61ebc66e8b4660/html5/thumbnails/10.jpg)
There are five ways to find if two triangles are congruent: SSS, SAS, ASA, AAS and HL.
SAS(side-angle-side) congruence
› Two triangles are congruent if two sides and the included angle of one triangle are equal to the two sides and the included angle of other triangle.
![Page 11: Congruents of Triangle](https://reader030.vdocuments.us/reader030/viewer/2022032620/55c6c706bb61ebc66e8b4660/html5/thumbnails/11.jpg)
SAS CONGRUENCE-A
B C
P
Q R
S(1) AC = PQ
A(2) ∠C = ∠R
S(3) BC = QR
Now If,
Then ∆ABC ≅ ∆PQR (by SAS congruence)
![Page 12: Congruents of Triangle](https://reader030.vdocuments.us/reader030/viewer/2022032620/55c6c706bb61ebc66e8b4660/html5/thumbnails/12.jpg)
ASA(angle-side-angle) congruence
› Two triangles are congruent if two angles and the included side of one triangle are equal to two angles and the included side of other triangle.
![Page 13: Congruents of Triangle](https://reader030.vdocuments.us/reader030/viewer/2022032620/55c6c706bb61ebc66e8b4660/html5/thumbnails/13.jpg)
ASA CONGRUENCE-A
B C
D
E FNow If, A(1) ∠BAC =
∠EDF
S(2) AC = DF
A(3) ∠ACB = ∠DFE
Then ∆ABC ≅ ∆DEF (by ASA congruence)
![Page 14: Congruents of Triangle](https://reader030.vdocuments.us/reader030/viewer/2022032620/55c6c706bb61ebc66e8b4660/html5/thumbnails/14.jpg)
ASA(angle-side-angle) congruence
•Two triangles are congruent if two angles and the included side of one triangle are equal to two angles and the included side of other triangle.
![Page 15: Congruents of Triangle](https://reader030.vdocuments.us/reader030/viewer/2022032620/55c6c706bb61ebc66e8b4660/html5/thumbnails/15.jpg)
AAS CONGRUENCE-A
B C P
Q
R
Now If, A(1) ∠BAC = ∠QPR
A(2) ∠CBA = ∠RQP
S(3) BC = QR
Then ∆ABC ≅ ∆PQR (by AAS congruence)
![Page 16: Congruents of Triangle](https://reader030.vdocuments.us/reader030/viewer/2022032620/55c6c706bb61ebc66e8b4660/html5/thumbnails/16.jpg)
SSS(side-side-side) congruence
•If three sides of one triangle are equal to the three sides of another triangle, then the two triangles are congruent.
![Page 17: Congruents of Triangle](https://reader030.vdocuments.us/reader030/viewer/2022032620/55c6c706bb61ebc66e8b4660/html5/thumbnails/17.jpg)
SSS CONGRUENCE-
Now If, S(1) AB = PQ
S(2) BC = QR
S(3) CA = RP
A
B C
P
Q R
Then ∆ABC ≅ ∆PQR (by SSS congruence)
![Page 18: Congruents of Triangle](https://reader030.vdocuments.us/reader030/viewer/2022032620/55c6c706bb61ebc66e8b4660/html5/thumbnails/18.jpg)
RHS(right angle-hypotenuse-side) congruence
•If in two right-angled triangles the hypotenuse and one side of one triangle are equal to the hypotenuse and one side of the other triangle, then the two triangles are congruent.
![Page 19: Congruents of Triangle](https://reader030.vdocuments.us/reader030/viewer/2022032620/55c6c706bb61ebc66e8b4660/html5/thumbnails/19.jpg)
RHS CONGRUENCE-
Now If, R(1) ∠ABC = ∠DEF = 90°
H(2) AC = DF
S(3) BC = EF
A
B C
D
E F
Then ∆ABC ≅ ∆DEF (by RHS congruence)
![Page 20: Congruents of Triangle](https://reader030.vdocuments.us/reader030/viewer/2022032620/55c6c706bb61ebc66e8b4660/html5/thumbnails/20.jpg)
PROPERTIES OF TRIANGLE
A
B C
A Triangle in which two sides are equal in length is called ISOSCELES TRIANGLE. So, ∆ABC is a isosceles triangle with AB = BC.
![Page 21: Congruents of Triangle](https://reader030.vdocuments.us/reader030/viewer/2022032620/55c6c706bb61ebc66e8b4660/html5/thumbnails/21.jpg)
Angles opposite to equal sides of an isosceles triangle are equal.
B C
A
Here, ∠ABC = ∠ ACB
![Page 22: Congruents of Triangle](https://reader030.vdocuments.us/reader030/viewer/2022032620/55c6c706bb61ebc66e8b4660/html5/thumbnails/22.jpg)
The sides opposite to equal angles of a triangle are equal.
CB
A
Here, AB = AC
![Page 23: Congruents of Triangle](https://reader030.vdocuments.us/reader030/viewer/2022032620/55c6c706bb61ebc66e8b4660/html5/thumbnails/23.jpg)
INEQUALITIES IN A TRIANGLE
Theorem on inequalities in a triangle
If two sides of a triangle are unequal, the angle opposite to the longer side is larger ( or greater)
10
8
9
Here, by comparing we will get that-Angle opposite to the longer side(10) is greater(i.e. 90°)
![Page 24: Congruents of Triangle](https://reader030.vdocuments.us/reader030/viewer/2022032620/55c6c706bb61ebc66e8b4660/html5/thumbnails/24.jpg)
In any triangle, the side opposite to the longer angle is longer.
10
8
9
Here, by comparing we will get that-Side(i.e. 10) opposite to longer angle (90°) is longer.
![Page 25: Congruents of Triangle](https://reader030.vdocuments.us/reader030/viewer/2022032620/55c6c706bb61ebc66e8b4660/html5/thumbnails/25.jpg)
The sum of any two side of a triangle is greater than the third side.
10
8
9
Here by comparing we get-9+8>108+10>910+9>8
So, sum of any two sides is greater than the third side.