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Transcript

Geometry Triangle Congruence Theorems Congruent Triangles Congruent triangles have three congruent sides and and three congruent angles. However, triangles can be proved congruent without showing 3 pairs of congruent sides and angles. The Triangle Congruence Postulates &Theorems FOR ALL TRIANGLES SSS SAS ASA AAS FOR RIGHT TRIANGLES ONLY HL LL HA LA Theorem If two angles in 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°. A closer look... If two 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° 85° 30° 30° Example 30° 30° Why aren’t these triangles congruent? What do we call these triangles? So, how do we prove that two triangles really are congruent? ASA (Angle, Side, Angle) A C D If two angles and the included side of one triangle are congruent to two angles and the B included side of another triangle, . . . F E then the 2 triangles are CONGRUENT! AAS (Angle, Angle, Side) Special case of ASA If two angles and a nonincluded side of one triangle are congruent to two angles and the corresponding non- C included side of another triangle, . . . then the 2 triangles are CONGRUENT! A D B F E SAS (Side, Angle, Side) A C D If in two triangles, two sides and the included angle of one are congruent to two sides B and the included angle of the other, . . . F E then the 2 triangles are CONGRUENT! SSS (Side, Side, Side) A C B D F E In two triangles, if 3 sides of one are congruent to three sides of the other, . . . then the 2 triangles are CONGRUENT! HL (Hypotenuse, Leg) If both hypotenuses and a pair of legs of two RIGHT triangles are congruent, . . . A C B D then the 2 triangles are CONGRUENT! F E HA (Hypotenuse, Angle) If both hypotenuses and a pair of acute angles of two RIGHT triangles are congruent, . . . then the 2 triangles are CONGRUENT! A C B D F E LA (Leg, Angle) If both hypotenuses and a pair of acute angles of two RIGHT triangles are C congruent, . . . A B D then the 2 triangles are CONGRUENT! F E LL (Leg, Leg) A If both pair of legs of two RIGHT triangles are congruent, . . . then the 2 triangles are CONGRUENT! C B D F E Example 1 A C B D E F Given the markings on the diagram, is the pair of triangles congruent by one of the congruency theorems in this lesson? Example 2 A C B D E F Given the markings on the diagram, is the pair of triangles congruent by one of the congruency theorems in this lesson? Example 3 A C B D Given the markings on the diagram, is the pair of triangles congruent by one of the congruency theorems in this lesson? D Example 4 Why are the two triangles congruent? SAS F E A What are the corresponding vertices? A D C E C B B F Example 5 A Why are the two triangles congruent? SSS D B What are the corresponding A C vertices? ADB CDB ABD CBD C Example 6 Given: AB CD BC AD B A Are the triangles congruent? Why? AB CD S BC DA S AC CA S C D Example 7 Q P Given: PS QR T mQSR = mPRS = 90° R S Are the Triangles Congruent? Why? QSR PRS = 90° QR PS SR RS H L Summary: ASA - Pairs of congruent sides contained between two congruent angles AAS – Pairs of congruent angles and the side not contained between them. SAS - Pairs of congruent angles contained between two congruent sides SSS - Three pairs of congruent sides Summary --for Right Triangles 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 THE END!!!