Triangles CONGRUENCY AND SIMILARITY
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Transcript of Triangles CONGRUENCY AND SIMILARITY
LAVESH DHADIWAL &DHIRAJ PATIDARCLASS – 10TH
SUBJECT – MATHMATECISSUMMATIVE ASSESMENT – IIFORMATIVE ASSESMENT – IIIACTIVITY – V
PRESENTING BY
.
TRIANGLES
CONTENTS•INTRODUCTION •CONGRUENT TRIANGLE• SIMILAR TRIANGLE • CONGRUENCY RULES • SIMILARITY RULES
INTRODUCTIONTRIANGLE IS A CLOSED FIGURE BOUNDED BY THREE LINE SEGMENTS .
IT HAS THREE VERTEX ,THREE ARMS (SIDES) AND THREE ANGLES .
A
CB
VERTEX
ANGLE
ARM(SIDE)
TRIANGLE
LINE SEGMENTS AB,BC,CA
SIDES AB,BC,CA
VERTEX A,B,C
ANGLE ⦟ABC , BCA , ⦟BAC⦟
CONCLUSION
TWO TRIANGLES ARE SAID TO BE CONGRUENT, WHEN ALL SIDES AND ALL ANGLES OF ONE ARE EQUAL TO CORRESPONDING SIDES
AND ANGLES OF THE OTHER.
CONGRUENCY OF TRIANGLES
TRIANGLES ‘ABC’ AND ‘PQR’ ARE SAID TO BE CONGRUENT IF SIDES AB,BC AND CA OF TRIANGLE ‘ABC’ ARE EQUAL TO SIDES PQ,QR
AND PR OF TRIANGLE PQR AND A⦟ , B ⦟ AND C ⦟ OF TRIANGLE ABC ARE EQUAL TO P⦟ , Q⦟ , R ⦟ OF TRIANGLE PQR RESPECTIVELY .
A
CB
P
RQ
IF △ ABC ≌ △PQRTHEN• AB=PQ• BC=QR • AC=PR • ⦟A= P ⦟• ⦟B= Q ⦟• ⦟C= R ⦟
CONCLUSION
SIMILARITY OF TRIANGLESTWO TRIANGLES ARE SAID TO BE SIMILAR, IF THEIR
CORRESPONDING ANGLES ARE EQUAL AND CORRESPONDING SIDES ARE PROPORTIONAL.
B
A
C
P
RQ
TRIANGLES ‘ABC’ AND ‘PQR’ ARE SAID TO BE SIMILAR IF A⦟ , B ⦟AND C ⦟ OF TRIANGLE ABC ARE EQUAL TO CORRESPONDING P⦟ , Q⦟ , R ⦟ OF TRIANGLE PQR AND SIDES AB, BC, AC OF TRIANGLE ABC ARE PROPORTIONAL TO CORRESPONDING SIDES PQ, QR AND PR OF TRIANGLE PQR .
IF △ABC∼ △PQR , THEN• ⦟A= P ⦟• ⦟B = Q ⦟• ⦟C= R⦟
• AB/PQ= BC/QR=AC/PR
CRITERIA OF CONGRUENCY OF TRIANGLES
• Side Angle Side (SAS) congruency rule. • Angle Side Angle (ASA) congruency rule. • Side Side Side (SSS) congruency rule. • Angle Angle Side (AAS) congruency rule. • Right Angle-Hypotenuse-Side (RHS) congruency rule. ≅
(SIGN OF CONGRUENCY)
Side Angle Side (SAS) congruence criterion
RQ
PA
CB
\\| |
△ ABC ≅ △ PQR SINCE , IN △ ABC and △ PQR AB = PQ (sides) BC = QR (sides) ⦟B = Q ⦟(angles)
Two triangles are congruent if two sides and included angle of one triangle are equal to corresponding two sides
and the included angle of the other triangle.
By SAS congruence criterion
Included angles refers to equal angle included between the sides.
A
CB RQ
P
\| |
\\\ \\
Side Side Side (SSS) congruency criterion
Two triangles are congruent if three sides of one triangle are equal to corresponding sides of the other triangle.
SINCE , IN △ ABC and △ PQR AB = PQ BC = QR AC = PRBy SSS congruency criterion△ ABC ≅ △ PQR
A
CB
P
RQ
| |
Angle Angle Side (AAS) congruency rule
Two triangles are congruent if two angles and any one side of one triangle are equal to two angles and corresponding
side of the other triangle.
SINCE , IN △ ABC and △ PQR ⦟B = Q⦟ AC = PR ⦟C = R⦟△ ABC ≅ △ PQR By AAS congruence criterion
A
CB
P
RQ| |
Angle Side Angle (ASA) congruency rule
Two triangles are congruent if two angles and included side of one triangle are equal to two angles and the
included side of the other triangle.
Included side refers to equal side included
between the angles.
SINCE , IN △ ABC and △ PQR ⦟B = Q⦟ BC = QR ⦟C = R⦟△ ABC ≅ △ PQR By ASA congruence criterion
P
RQB
A
C
\ \| |
Right Angle-Hypotenuse-Side (RHS) congruency rule
Two right angled triangles are congruent if one side and the hypotenuse of one triangle are equal to one side and the
hypotenuse of the other triangle.SINCE , IN △ ABC and △ PQR ⦟B = Q ( right angle)⦟ BC = QR (one side or base) AC = PR (hypotenuse)By RHS congruency criterion△ ABC ≅ △ PQR
CRITERIA OF SIMILARITY OF TRIANGLES
• Angle Angle Angle (AAA) similarity rule. • Side Side Side (SSS) similarity rule. • Angle Angle (AA) similarity rule. ∼(SIGN OF SIMILARITY)
A
CB
P
RQ
Angle Angle (AA) similarity rule
Two triangles are similar if two angles of one triangle are equal to corresponding angles of the other triangle.
SINCE, IN △ ABC and △ PQR ⦟A = P⦟ ⦟B = Q ⦟By AA similarity criterion△ ABC ∼ △ PQR
Angle Angle Angle (AAA) similarity rule
Two triangles are similar if al l angles of one triangle are equal to corresponding angles of the other triangle.
SINCE , IN △ ABC and △ PQR ⦟A = P⦟ ⦟B = Q⦟ ⦟C = R⦟By AAA similarity criterion△ ABC ∼ △ PQR
A
CB
P
RQ
Side Side Side (SSS) similarity criterionA
CB RQ
P
Two triangles are similar if corresponding sides of both triangle are equal in the same rati o.
SINCE , IN △ ABC and △ PQR AB/PQ BC/QR AC/PRAND ALSO, AB/PQ=BC/QR=AC/PR By SSS similarity criterion△ ABC ∼ △ PQR
TRIANGLESCONGRUENCYCONGRUENCY
RULESSIMILARITY SIMILARITY
RULES SAS ASA, SSS, AAS RHS AA
AAA, SSSAND THAT’S
ALL AND
WHAT YOU HAVE LEARNT ?
ROUND TABLE CONVERSATION
.