Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
History of geometry wikipedia , lookup
Multilateration wikipedia , lookup
Technical drawing wikipedia , lookup
Reuleaux triangle wikipedia , lookup
Perceived visual angle wikipedia , lookup
Rational trigonometry wikipedia , lookup
Euler angles wikipedia , lookup
Trigonometric functions wikipedia , lookup
History of trigonometry wikipedia , lookup
Pythagorean theorem wikipedia , lookup
Proving Triangles Congruent Geometry D – Chapter 4.4 SSS - Postulate If all the sides of one triangle are congruent to all of the sides of a second triangle, then the triangles are congruent. (SSS) Example #1 – SSS – Postulate Use the SSS Postulate to show the two triangles are congruent. Find the length of each side. AC = 5 BC = 7 2 2 AB = 5 7 74 MO = 5 NO = 7 MN = 52 72 74 VABC VMNO Definition – Included Angle J K is the angle between JK and KL. It is called the included angle of sides JK and KL. K L J What is the included angle for sides KL and JL? L K L SAS - Postulate If two sides and the included angle of one triangle are congruent to two sides and the included angle of a second triangle, then the triangles are congruent. (SAS) S L Q P A S A J S S K VJKL VPQR by SAS R Example #2 – SAS – Postulate K L Given: N is the midpoint of LW N is the midpoint of SK Prove: N VLNS VWNK W S N is the midpoint of LW N is the midpoint of SK Given LN NW , SN NK Definition of Midpoint LNS WNK Vertical Angles are congruent VLNS VWNK SAS Postulate Definition – Included Side J JK is the side between J and K. It is called the included side of angles J and K. K L J What is the included side for angles K and L? KL K L ASA - Postulate If two angles and the included side of one triangle are congruent to two angles and the included side of a second triangle, then the triangles are congruent. (ASA) J X Y K L VJKL VZXY by ASA Z H A Example #3 – ASA – Postulate W Given: HA || KS AW WK Prove: VHAW VSKW K S HA || KS, AW WK Given HAW SKW Alt. Int. Angles are congruent HWA SWK Vertical Angles are congruent VHAW VSKW ASA Postulate Identify the Congruent Triangles. Identify the congruent triangles (if any). State the postulate by which the triangles are congruent. A J R B C H I S K M O L P VABC VSTR by SSS VPNO VVUW by SAS N V T U W Note: VJHI is not SSS, SAS, or ASA. A Example #4 – Paragraph Proof Given: VMAT is isosceles with vertex MAT bisected by AH. Prove: MH HT T H M • Sides MA and AT are congruent by the definition of an isosceles triangle. • Angle MAH is congruent to angle TAH by the definition of an angle bisector. • Side AH is congruent to side AH by the reflexive property. • Triangle MAH is congruent to triangle TAH by SAS. • Side MH is congruent to side HT by CPCTC. Example #5 – Column Proof Q P QM MO QM PO, MO has midpoint N Given: QM || PO, Prove: QN PN QM || PO, QM PO QM MO PO MO mQMN 90o mPON 90o QMN PON MO ON VQMN VPON QN PN M N O Given A line to one of two || lines is to the other line. Perpendicular lines intersect at 4 right angles. Substitution, Def of Congruent Angles Definition of Midpoint SAS CPCTC Summary Triangles may be proved congruent by Side – Side – Side (SSS) Postulate Side – Angle – Side (SAS) Postulate, and Angle – Side – Angle (ASA) Postulate. Parts of triangles may be shown to be congruent by Congruent Parts of Congruent Triangles are Congruent (CPCTC).