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Download 4-6 Congruence in Right Triangles
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Transcript
Review: Solving Systems x+y x 2y+3 12 Find the values of x and y that make the following triangles congruent. Congruent Triangles (CPCTC) Two triangles are congruent triangles if and only if the corresponding parts of those congruent triangles are congruent. • Corresponding sides are congruent • Corresponding angles are congruent Congruence Statement When naming two congruent triangles, order is very important. Third Angle Theorem If two angles of one triangle are congruent to two angles of another triangle, then the third angles are also congruent. Congruence Shortcuts Side-Angle-Side (SAS) Congruence Postulate: If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the two triangles are congruent. Side-Side-Side Congruence Postulate SSS Congruence Postulate: If the three sides of one triangle are congruent to the three sides of another triangle, then the two triangles are congruent. Congruence Shortcuts Angle-Side-Angle (ASA) Congruence 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. Congruence Shortcuts Angle-Angle-Side (AAS) Congruence Theorem: If two angles and a non-included side of one triangle are congruent to the corresponding two angles and the non-included side of another triangle, then the two triangles are congruent. Base Angles Theorem: If two sides of a triangle are congruent, then the angles opposite them are congruent. Converse of the Base Angles Theorem: If two angles of a triangle are congruent, then the sides opposite them are congruent. Equilateral Triangle Theorem A triangle is equilateral if and only if it is equiangular. Practice Practice Practice Congruence in Right Triangles Vocabulary Right Triangles A Hypotenuse Leg B Leg C Hypotenuse-Leg Theorem Hypotenuse-Leg (HL) Congruence Theorem: If the hypotenuse and a leg of a right triangle are congruent to the hypotenuse and leg of another right triangle, then the two triangles are congruent. • To use the HL Theorem, you must show that three conditions are met: • There are two right triangles • The triangles have congruent hypotenuses • There is one pair of congruent legs Using the HL Theorem Using the HL Theorem Statements 1. AD is the bisector of CE , Reasons 1. CD EA 2. 2. Defn. of 3. CBD & EBA are rt. s 3. 4. CB EB 4. 5. 5. HL Thm. Using the HL Theorem Statements Reasons 1. PRS and RPQ are rt s SP QR 1. 2. 2. Defn. of rt s 3. 3. Refl. Prop. of 4. PRS RPQ 4. Which are congruent by HL? E A F J 3in B 5in C 3in 5in H 3in D 5in G Which are congruent by HL? E A F J 3in B 5in C 3in 5in H 3in D 5in G HFJ DEG Prove the triangles are congruent P S Q R 1. QPR and SRP are right, SP RQ Given 2. QPR SRP All Right angles are congruent 3. PR PR Reflexive 4. SRP QPR HL Theorem What else do you need to prove the triangles are congruent? R Is RT XT? or Is XV TV? X V T Prove the two triangles are congruent R 1. T is the midpoint of RV S and U are right angles RS TU 3. RST TUV T S 1. Given U 2. Definition of midpoint 2. RT TV T is the midpoint of RV 3. HL Theorem V
