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Congruent triangles theorem
This document discusses triangle congruence theorems. It defines the five main congruence theorems: ASA, SAS, SSS, AAS, and introduces four additional theorems for right triangles only: HL, HA, LL, LA. Examples are provided to demonstrate applying each theorem to determine if two triangles are congruent based on given side or angle information.
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Introduction to triangle congruence with definitions of congruent triangles.
Postulates and theorems defining congruence: ASA, AAS, SAS, SSS for all triangles, and HL for right triangles.
Examples demonstrating how to determine if triangles are congruent using theorems.
Summarization of congruence postulates and theorems for all triangles and specifically for right triangles.
Ending slide marking the conclusion of the presentation.
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Congruent triangles theorem
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- 2. Congruent triangles havethree congruent sides and and three congruent angles. However, triangles can be proved congruent without showing 3 pairs of congruent sides and angles. Congruent Triangles
- 3. The Triangle Congruence Postulates&Theorems LAHALLHL FOR RIGHT TRIANGLES ONLY AASASASASSSS FOR ALL TRIANGLES
- 4. Theorem If two anglesin one triangle are congruent to two angles in another triangle, the third angles must also be congruent. Think about it… they have to add up to 180°.
- 5. A closer look... Iftwo triangles have two pairs of angles congruent, then their third pair of angles is congruent. But do the two triangles have to be congruent? 85° 30° 85° 30°
- 6. Example 30° 30° Why aren’t thesetriangles congruent? What do we call these triangles?
- 7. So, how dowe prove that two triangles really are congruent?
- 8. ASA (Angle, Side,Angle) If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, . . . then the 2 triangles are CONGRUENT! F E D A C B
- 9. AAS (Angle, Angle,Side) Special case of ASA If two angles and a non- included side of one triangle are congruent to two angles and the corresponding non- included side of another triangle, . . . then the 2 triangles are CONGRUENT! F E D A C B
- 10. SAS (Side, Angle,Side) If in two triangles, two sides and the included angle of one are congruent to two sides and the included angle of the other, . . . then the 2 triangles are CONGRUENT! F E D A C B
- 11. SSS (Side, Side,Side) In two triangles, if 3 sides of one are congruent to three sides of the other, . . . F E D A C B then the 2 triangles are CONGRUENT!
- 12. HL (Hypotenuse, Leg) Ifboth hypotenuses and a pair of legs of two RIGHT triangles are congruent, . . . A C B F E D then the 2 triangles are CONGRUENT!
- 13. HA (Hypotenuse, Angle) Ifboth hypotenuses and a pair of acute angles of two RIGHT triangles are congruent, . . . then the 2 triangles are CONGRUENT! F E D A C B
- 14. LA (Leg, Angle) Ifboth hypotenuses and a pair of acute angles of two RIGHT triangles are congruent, . . . then the 2 triangles are CONGRUENT! A C B F E D
- 15. LL (Leg, Leg) Ifboth pair of legs of two RIGHT triangles are congruent, . . . then the 2 triangles are CONGRUENT! A C B F E D
- 16. Example 1 Given themarkings on the diagram, is the pair of triangles congruent by one of the congruency theorems in this lesson? F E D A C B
- 17. Example 2 Given themarkings on the diagram, is the pair of triangles congruent by one of the congruency theorems in this lesson? A C B F E D
- 18. Example 3 Given themarkings on the diagram, is the pair of triangles congruent by one of the congruency theorems in this lesson? D A C B
- 19. Example 4 Why arethe two triangles congruent? What are the corresponding vertices? A B C D E F SAS A D C E B F
- 20. Example 5 Why arethe two triangles congruent? What are the corresponding vertices? A B C D SSS A C ADB CDB ABD CBD
- 21. Example 6 Given: B C D A CDAB ADBC CDAB DABC CAAC Arethe triangles congruent? S S S Why?
- 22. Example 7 Given: QRPS R H SRSSR Are the Triangles Congruent? QSR PRS = 90 Q RS P T mQSR = mPRS = 90 PSQR Why?
- 23. Summary: ASA - Pairsof congruent sides contained between two congruent angles SAS - Pairs of congruent angles contained between two congruent sides SSS - Three pairs of congruent sides AAS – Pairs of congruent angles and the side not contained between them.
- 24. Summary --- for RightTriangles Only: HL – Pair of sides including the Hypotenuse and one Leg HA – Pair of hypotenuses and one acute angle LL – Both pair of legs LA – One pair of legs and one pair of acute angles
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