Class: 7GGROUP 2
project maths
CONGRUENCE OF TRIANGLES
Date :2/9/2022
Index
conclusion
10.
challenge question
9.
amazing facts
8.
historic facts
7.
real life EXAMPLES
6.
application
5.
classification [rhs]
classification [asa]
3.
clasSification [sss & sas]
2.
DEFINITION
1.
CONTENT NAME
SLIDE NO.
INTRODUCTION
11.
01. introduction
Congruent means ‘exactly equal’ in terms of shape and sizeIf any geometrical figures are superimposed on each other , they are termed as congruent figuresThis property applies to triangles , quadrilaterals and so on . apart from figures , line segments and angles are also termed as congruent if they are of equal measures .
SOME CONGRUENT SHAPES
02. DEFINITION OF CONGRUENCE OF TRIANGLES
DEFINITION:
Two triangles are said to be congruent :
- If their corresponding sides are equal in length
- If their corresponding angles are of the same measure
- The symbol of congruence is
- Two congruent triangles can be equal in area but 2 triangles which are equal in area may not be congruent .
ABC
FDE
AB = FD BC = DE AC = FE
∠A = ∠F
∠B = ∠D
∠C = ∠E
03. classificaton
SAS THEOREM
SAS means two sides and one included angle
SSS THEOREM
If two sides and the included angle of one triangle are congruent to two sides and the included angles of another triangle, then the two triangles are congruent by SAS.
SSS means Side-Side-Side
SSS rule states that if all three sides of one triangle are equal to all three corresponding sides of another triangle, this makes the two triangles congruent.
ABC
PQR
AB = PQ (given)BC = QR(given) ∠B = ∠Q(given)
ABC
EDF
AB = DE (6cm)BC = FD(7cm) AC = FE (8cm)
04.
ASA THEOREM
ASA Congruence rule stands for Angle-Side-Angle. Under this rule, two triangles are said to be congruent if any two angles and the side included between them of one triangle are equal to the corresponding angles and the included side of the other triangle.
Lorem ipsum dolor sit amer
ABC
DEF
∠B = ∠E BC = EF ∠C = ∠F
05.
RHS THEOREM
RHS stands for Right angle - Hypotenuse - Side
Lorem ipsum dolor sit amer
Under RHS rule, we consider only the hypotenuse and one corresponding side of the given two right triangles to prove the congruency of triangles.
RHS Theorem states that, if the hypotenuse and side of one right-angled triangle are equal to the hypotenuse and the corresponding side of another right-angled triangle, the two triangles are congruent.
AC = PR (given) ∠B = ∠Q (90°)BC = QR (given)
PQR
ABC
06. The APpLICATION
PROVE THAT THE BISECTOR OF THE VERTICAL ANGLE OF AN ISOSCELES TRIANGLE BISECTS THE BASE AT RIGHT ANGLES.
PROVE: ∠ADB= ∠ADC= 90° & BD =DC PROOF: In ADB & ADC, we have : AB=AC (given) ∠BAD=∠CAD (given) AD=AD (common) ∴ ADB ≅ ADC (SAS property ) so, BD = DC ∠ADB = ∠ADC ∠ADB + ∠ADC =180°(linear pair property) ∠ABD =∠ADC=90° Hence AD bisects BC at right angles.
A pair of bookends with triangles in their design would typically make the triangles congruent, so they present the symmetrical characteristic of such items.
07. IN REAL-LIFE
We can measure the height of the building using triangles.
Many pairs of triangles designed into a building, for example as roof ends, would be congruent, so the roof beam and the top edges of the walls are horizontal.
We can use triangles to find the areas of a polygon by breaking them into smaller triangles.
08. The HISTORIC FACTS
- The SSS criteria was created by Sal Khan
- Pascal discovered congruence of triangles.
- The congruency of triangles' theorem was introduced in the 4th century
Pascal
09. The amazing FACTS
- A Greek mathematician Pappus (early 4th century) discovered that triangles can also be congruent even in another orientation.
- In plain geometry, SSA does not imply congruence.
- Congruent triangles cannot be expanded or contracted and still be congruent.
Pappus of Alexandria
10. The Challenge
In the given figure ΔBAC =ΔQRP by SAS criterion of congruence. Find the value of x and y.
11. conclusion
Now , we have come to the conclusion of this presentation . So far , we have discussed about , Congruence of triangles , the important theorems and its real life examples.
Thank you for your attention
THE TEAM
6]parvathi [application] [slides] 7]mridula (REAL-LIFE) 8]kushal (HISTORIC FACTS) 9]mannat (AMAZING FACTS) 10]lavanya (CHALLENGe question)
1]pradyot[introduction] 2]N.gautAm [definition] 3]himani [sss & sas] 4]misbah [asa] 5]hasini [rhs]
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Transcript
Class: 7GGROUP 2
project maths
CONGRUENCE OF TRIANGLES
Date :2/9/2022
Index
conclusion
10.
challenge question
9.
amazing facts
8.
historic facts
7.
real life EXAMPLES
6.
application
5.
classification [rhs]
classification [asa]
3.
clasSification [sss & sas]
2.
DEFINITION
1.
CONTENT NAME
SLIDE NO.
INTRODUCTION
11.
01. introduction
Congruent means ‘exactly equal’ in terms of shape and sizeIf any geometrical figures are superimposed on each other , they are termed as congruent figuresThis property applies to triangles , quadrilaterals and so on . apart from figures , line segments and angles are also termed as congruent if they are of equal measures .
SOME CONGRUENT SHAPES
02. DEFINITION OF CONGRUENCE OF TRIANGLES
DEFINITION:
Two triangles are said to be congruent :
ABC
FDE
AB = FD BC = DE AC = FE
∠A = ∠F ∠B = ∠D ∠C = ∠E
03. classificaton
SAS THEOREM
SAS means two sides and one included angle
SSS THEOREM
If two sides and the included angle of one triangle are congruent to two sides and the included angles of another triangle, then the two triangles are congruent by SAS.
SSS means Side-Side-Side
SSS rule states that if all three sides of one triangle are equal to all three corresponding sides of another triangle, this makes the two triangles congruent.
ABC
PQR
AB = PQ (given)BC = QR(given) ∠B = ∠Q(given)
ABC
EDF
AB = DE (6cm)BC = FD(7cm) AC = FE (8cm)
04.
ASA THEOREM
ASA Congruence rule stands for Angle-Side-Angle. Under this rule, two triangles are said to be congruent if any two angles and the side included between them of one triangle are equal to the corresponding angles and the included side of the other triangle.
Lorem ipsum dolor sit amer
ABC
DEF
∠B = ∠E BC = EF ∠C = ∠F
05.
RHS THEOREM
RHS stands for Right angle - Hypotenuse - Side
Lorem ipsum dolor sit amer
Under RHS rule, we consider only the hypotenuse and one corresponding side of the given two right triangles to prove the congruency of triangles.
RHS Theorem states that, if the hypotenuse and side of one right-angled triangle are equal to the hypotenuse and the corresponding side of another right-angled triangle, the two triangles are congruent.
AC = PR (given) ∠B = ∠Q (90°)BC = QR (given)
PQR
ABC
06. The APpLICATION
PROVE THAT THE BISECTOR OF THE VERTICAL ANGLE OF AN ISOSCELES TRIANGLE BISECTS THE BASE AT RIGHT ANGLES.
PROVE: ∠ADB= ∠ADC= 90° & BD =DC PROOF: In ADB & ADC, we have : AB=AC (given) ∠BAD=∠CAD (given) AD=AD (common) ∴ ADB ≅ ADC (SAS property ) so, BD = DC ∠ADB = ∠ADC ∠ADB + ∠ADC =180°(linear pair property) ∠ABD =∠ADC=90° Hence AD bisects BC at right angles.
A pair of bookends with triangles in their design would typically make the triangles congruent, so they present the symmetrical characteristic of such items.
07. IN REAL-LIFE
We can measure the height of the building using triangles.
Many pairs of triangles designed into a building, for example as roof ends, would be congruent, so the roof beam and the top edges of the walls are horizontal.
We can use triangles to find the areas of a polygon by breaking them into smaller triangles.
08. The HISTORIC FACTS
Pascal
09. The amazing FACTS
Pappus of Alexandria
10. The Challenge
In the given figure ΔBAC =ΔQRP by SAS criterion of congruence. Find the value of x and y.
11. conclusion
Now , we have come to the conclusion of this presentation . So far , we have discussed about , Congruence of triangles , the important theorems and its real life examples.
Thank you for your attention
THE TEAM
6]parvathi [application] [slides] 7]mridula (REAL-LIFE) 8]kushal (HISTORIC FACTS) 9]mannat (AMAZING FACTS) 10]lavanya (CHALLENGe question)
1]pradyot[introduction] 2]N.gautAm [definition] 3]himani [sss & sas] 4]misbah [asa] 5]hasini [rhs]