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Transcript
Congruent Figures Congruent figures have the same size and shape. When two figures are congruent, we can move one so that it fits exactly on the other one. Three ways to make such a move – a slide, a flip, and a turn. Congruent polygons have congruent corresponding parts – their matching sides and angles. Matching vertices are corresponding vertices. When we name congruent polygons, we have to list corresponding vertices in the same order. Two triangles are congruent when they have three pairs of congruent corresponding sides and three pairs of congruent corresponding angles. Theorem: If two angles of one triangle are congruent to two angles of another triangle, then the third angles are congruent. Proof:. A D B C E Given: Angles A and D and B and E and congruent Prove that angles C and F are congruent F Steps Reasons Given A D; B E mA mD;mB mE Def. of Cong mA mB mC 180 TAST mD mE mF 180 TAST Transitive/ mA mB mC mD mE mF Substitution mD mE mC mD mE mF Substitution Steps mC mF C F Reasons SPE Def. of Cong Triangle Congruence by SSS and SAS We do not need to know that all six corresponding parts are congruent in order to conclude that two triangles are congruent. It is enough to know that corresponding sides are congruent. Side-Side-Side (SSS) Postulate If the three sides of one triangle are congruent to the three sides of another triangle, then the two triangles are congruent. The word included is used to refer to angles and the sides of a triangle. Side-Angle-Side (SAS) Postulate If two sides and an included angle of one triangle are congruent to two sides and an included angle of another triangle, then the two triangles are congruent. Given: A AE and BD bisect each other B C Prove: ACB ECD D E Given: AB CM, AB DB D and M is the midpoint C of AB , DB CM B M Prove: AMC MBD A Side-Side-Side (ASA) Postulate If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the two triangles are congruent. Side-Side-Side (AAS) Theorem If two angles and a nonincluded side of one triangle are congruent to two angles and a nonincluded side of another triangle, then the two triangles are congruent. Side-Side-Side (AAS) Theorem A D E B C F Steps A D; C F AB DE B E one triangles are angles of triangle, then are also ABC DEF Reasons Given Given If two angles of congruent to two another the third angles congruent ASA Post. Given: N P, MO QO M N O Prove: MON QOP P Q Given: AE BD, AE BD, E D E A B Prove: AEB BDC D C Assignment: Given: FG JH, F H F J Prove: FGJ HJG G H Given: 1 2 , DH bisects BDF D B 1 2 H Prove: BDH FDH F Using Congruent Triangles: CPCTC With SSS, SAS, ASA, and AAS, we know how to use three parts of triangles to show that triangles are congruent. Once we have congruent triangles, we can make conclusions about their parts because by definition corresponding parts of congruent triangles are congruent. This is abbreviated as CPCTC We can use congruent triangles and CPCTC to measure distances, such as the distance across a river and similar problems indirectly. Isosceles and Equilateral Triangles Isosceles triangles are common in the real world. We can find them in structures such as bridges and buildings. The congruent sides of an isosceles triangle are its legs.. The third side is the base. The two congruent sides from the vertex angle. The other two angles are the base angles. Isosceles Triangle Theorem If two sides of a triangle are congruent, then the angles opposite those sides are congruent. Given: Isosceles triangle ABC with legs B AB and BC. Prove that angles A and C are congruent A D C Steps Isosceles triangle ABC with legs AB and BC. AB BC Construct ray BD, the angle bisector of angle ABC ABD CBD BD BD ABD CBD A C Reasons Given Def of isosceles triangle Construction Def. Of angle bisector Reflexive Prop. SAS Post. CPCTC Steps Isosceles triangle ABC with legs AB and BC. AB BC Construct ray BD, the angle bisector of angle ABC ABD CBD BD BD ABD CBD A C Reasons Given Def of isosceles triangle Construction Def. Of angle bisector Reflexive Prop. SAS Post. CPCTC Isosceles Triangle Theorem If two angles of a triangle are congruent, then the sides opposite the angles are congruent Given: Triangle ABC with congruent B angles A and C Prove that sides AB and BC are congruent. A D C Steps Triangle ABC with congruent angles A and C Construct ray BD, the angle bisector of angle ABC ABD CBD BD BD ABD CBD AB BC Reasons Given Construction Def. Of angle bisector Reflexive Prop. AAS Post. CPCTC Steps Triangle ABC with congruent angles A and C Construct ray BD, the angle bisector of angle ABC ABD CBD BD BD ABD CBD AB BC Reasons Given Construction Def. Of angle bisector Reflexive Prop. AAS Post. CPCTC Theorem: The bisector of the vertex angle of an isosceles triangle is the perpendicular bisector of the base. Given: Isosceles triangle ABC with vertex B angle B Prove that line BD is the perpendicular bisector of segment AC. A D C Steps Isosceles triangle ABC with legs AB and BC. AB BC Construct ray BD, the angle bisector of angle ABC ABD CBD BD BD ABD CBD ADB CDB; AD CD Reasons Given Def of isosceles triangle Construction Def. Of angle bisector Reflexive Prop. SAS Post. CPCTC Steps Reasons ADB and CDB are If two angles are both right angles congruent supplementary then each of them is a right angle Def of a BD is the perpendicular perpendicular bisector of AC bisector Steps Isosceles triangle ABC with legs AB and BC. AB BC Construct ray BD, the angle bisector of angle ABC ABD CBD BD BD ABD CBD ADB CDB; AD CD Reasons Given Def of isosceles triangle Construction Def. Of angle bisector Reflexive Prop. SAS Post. CPCTC Steps Reasons ADB and CDB are If two angles are both right angles congruent supplementary then each of them is a right angle Def of a BD is the perpendicular perpendicular bisector of AC bisector A corollary is a statement that follows immediately from a theorem. Corollary: If a triangle is equilateral, then the triangle is equiangular. A B C Steps Equilateral triangle ABC Reasons Given AB BC AC Def of an equilateral triangle C A B Isosceles Triangle Theorem Def. Of an equiangular triangle Triangle ABC is equiangular Steps Equilateral triangle ABC Reasons Given AB BC AC Def of an equilateral triangle C A B Isosceles Triangle Theorem Def. Of an equiangular triangle Triangle ABC is equiangular Corollary: If a triangle is equiangular, then the triangle is equilateral Steps Equiangular triangle ABC Reasons Given C A B Def. Of an equiangular triangle AB BC AC Triangle ABC is equiangular Converse of Isosceles Triangle Theorem Def of an equilateral triangle Steps Equingular triangle ABC Reasons Given C A B Def. Of an equiangular triangle AB BC AC Triangle ABC is equiangular Converse of Isosceles Triangle Theorem Def of an equilateral triangle Congruence in Right Triangles In a right triangle, the side opposite the right angle is the longest side and is called the hypotenuse. The other two legs are called legs. Right triangles provide a special case for which there is an SSA congruence rule. It occurs when hypotenuse are congruent and one pair of legs are congruent. A C B Hypotenuse-Leg (HL) Theorem If the hypotenuse and a leg of one right triangle are congruent to the hypotenuse and a leg of another right triangle, then the triangles are congruent. A D C B F E D A E C F B Steps Right triangles ABC and DEF with right angles C ad F Reasons Given ACB DFE All right angles are congruent Given Triangle BAF/BDF is Def of an isosceles an isosceles triangle triangle Isosceles Triangle B E Theorem AAS Theorem ACB DFE AC DF; AB DE Steps Right triangles ABC and DEF with right angles C ad F Reasons Given ACB DFE All right angles are congruent Given Triangle BAF/BDF is Def of an isosceles an isosceles triangle triangle Isosceles Triangle B E Theorem AAS Theorem ACB DFE AC DF; AB DE Using Corresponding Parts of Congruent Triangles Some triangle relationships are difficult to see because the triangles overlap. Overlapping triangles may have a common side or angle. We can simplify this by separating and redrawing the triangles In overlapping triangles, a common side or angle is congruent to itself by the reflexive property of congruence. A B E C Given: ACD BDC Prove: CE DE D In overlapping triangles, a common side or angle is congruent to itself by the reflexive property of congruence.
