Geometry 1 Why Study Chapter 3? Knowledge of triangles is a key application for: Support beams Theater Kaleidoscopes Painting Car stereos Rug design Tile floors Gates

Geometry 2 Baseball field Sign making Architecture Sailboats Fire and lifeguard towers Sports Airplanes Bicycles Surveying

Geometry 3 Canyon Snowboarding Advertising (logos) Kites Chess Stenciling

Geometry 4 Section 3.1 Congruent Figures Definitions  Congruent: having the same size and shape.  Congruent triangles: all pairs of corresponding parts are congruent.  Corresponding parts –If triangles ABC and DEF are congruent, then what parts must match up? Refer to page 111 in text. A CB D FE

Geometry 5 Problem Is ? Explain your answer. Refer to page 112 in the text. A CB D FE

Geometry 6 Definitions Plane: a two-dimensional figure usually represented by a shape that looks like a wall or floor even though the plane extends without end Polygon: a closed plane figure with the following properties: 1) It is formed by three or more line segments called sides. 2) Each side intersects exactly two sides, one at each endpoint, so that no two sides with a common endpoint are collinear.

Geometry 7 A Triangle is a Polygon with Three Sides Congruent Polygons all pairs of corresponding parts are congruent

Geometry 8 Section 3.1 Congruent Figures Reflection  when a figure has a mirror line  Refer to page 112 in text.

Geometry 9 Section 3.1 Congruent Figures Other Types of Correspondences  Slide: where a copy of the figure has been shifted by some set amount  Refer to page 113 in text. AB C D E PQ R S T

Geometry 10 Section 3.1 Congruent Figures Other Types of Correspondences  Rotational: when the figure has been rotated around a common point  Refer to page 113 in text. R T A Y B

Geometry 11 Section 3.1 Congruent Figures These correspondences can be combined!  Try a reflection and a rotation on the figure below.

Geometry 12 Section 3.1 Congruent Figures Reflexive property  any segment or angle is congruent to itself. Since the triangles overlap, this angle is reflexive to triangles!

Geometry 13 Section Ways to Prove Triangles Congruent Introduction  Included sides and angles  To be included means to be flanked by or trapped between –Thus, the points of a line segment are included between its endpoints. –In a triangle, sides can be included by angles and angles can be included by sides. M RZ List the inclusions in triangle MRZ. Refer to page 115 in text.

Geometry 14 Section Ways to Prove Triangles Congruent Side Side Side (SSS) Postulate  SSS: If three sides of one triangle are congruent to three sides of another triangle then the triangles are congruent A CB D FE

Geometry 15 Section Ways to Prove Triangles Congruent Given: Prove:

Geometry 16 Section Ways to Prove Triangles Congruent

Geometry 17 Section Ways to Prove Triangles Congruent

Geometry 18 Section Ways to Prove Triangles Congruent Angle Side Angle (ASA) Postulate  ASA: If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the triangles are congruent. A CB D FE

Geometry 19 Section Ways to Prove Triangles Congruent

Geometry 20 Section Ways to Prove Triangles Congruent

Geometry 21 Section Ways to Prove Triangles Congruent

Geometry 22 Section Ways to Prove Triangles Congruent

Geometry 23 Section Ways to Prove Triangles Congruent

Geometry 24 Section Ways to Prove Triangles Congruent Side Angle Side (SAS) Postulate  SAS: 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. A CB D FE

Geometry 25 Section Ways to Prove Triangles Congruent

Geometry 26 Section Ways to Prove Triangles Congruent Given: Prove:  APQ   BPQ StatementsReasons

Geometry 27 Section Ways to Prove Triangles Congruent Given: Prove:  XWY   XZY StatementsReasons

Geometry 28 Section 3.3 Circles and CPCTC If you have proven via SSS can you state that  5 is congruent to  7? WHY? CPCTC Principle:  Corresponding Parts of Congruent Triangles are Congruent You MUST prove the triangles congruent FIRST!!!

Geometry 29 Section 3.3 Circles and CPCTC Circle: The set of all points in a plane that are a given distance from a given point in the plane. O. P R.. Segment OP is a radius Segments OP and OR are radii Remember your formulas?  Area of a circle  Circumference of a circle Theorem: All radii of a circle are congruent.

Geometry 30 Section 3.4 Beyond CPCTC Median: A line segment drawn from any vertex of the triangle to the midpoint of the opposite side. How many medians does a triangle have? (3) A median divides into two congruent segments, or bisects the side to which it is drawn.

Geometry 31 Section 3.4 Beyond CPCTC Altitude: A line segment drawn from any vertex of the triangle to the opposite side, extended if necessary, and perpendicular to that side. How many altitudes does a triangle have? (3) An altitude of a triangle forms right (90º) angles with one of the sides.

Geometry 32 Section 3.4 Beyond CPCTC Auxiliary Lines: A line introduced into a diagram for the purpose of clarifying a proof. Postulate: Two points determine a line (or ray or segment). A C BD R S T U

Geometry 33 Solving Proofs: Remember the Steps 1. Draw the diagram. 2. Carefully read the problem and mark the diagram. 3. Place a question mark (?) in the area that you need to prove. 4. Create a flow diagram. 5. Use one given at a time and draw as much information as possible from that given. (Disregard information that is not needed.) 6. Sequentially list the statements and reasons. 7. The last statement should be the prove statement. 8. Check the proof. It should follow a logical order.

Geometry 34 Section 3.4 Practice Proof Given: is an altitude Prove: is a median D E C F

Geometry 35 Given: Prove:. O G H J

Geometry 36 Given: is a median ST = x + 40 SW = 2x + 30 WV = 5x – 6 Find: SW, WV, and ST S T V W

Geometry 37 Test Tomorrow Study lessons 3.1 to 3.4 and class notes Define the following: 1. Reflexive Property 2. SSS, SAS, and ASA Postulates 3. CPCTC, radius, radii, and diameter 4. Formulas for the area and circumference of a circle 5. Median, altitude, auxiliary lines Study PowerPoint slides 1-36

Geometry 38 Section 3.5 Overlapping Triangles

Geometry 39 Section 3.6 Types of Triangles Scalene Triangle: a triangle in which no two sides are congruent.

Geometry 40 Section 3.6 Types of Triangles Isosceles Triangle: a triangle in which at least two sides are congruent.  Congruent sides are called legs  Non-congruent side is called the base  Angles included between leg and base are called base angles Leg Base Angle Vertex Base

Geometry 41 Section 3.6 Types of Triangles Equilateral Triangle: a triangle in which all sides are congruent.

Geometry 42 Section 3.6 Types of Triangles Equiangular Triangle: a triangle in which all angles are congruent.

Geometry 43 Section 3.7 Angle-Side Theorems Theorem: If two sides of a triangle are congruent, then the angles opposite the sides are congruent. Theorem: If two angles of a triangle are congruent, the sides opposite the angles are congruent. Ways to Prove that a Triangle is Isosceles: 1.If at least two sides of a triangle are congruent 2.If at least two angles of a triangle are congruent

Geometry 44 Section 3.7 Angle-Side Theorems Theorem: If two sides of a triangle are not congruent, then the angles opposite them are not congruent, and the larger angle is opposite the longer side. Theorem: If two angles of a triangle are not congruent, then the sides opposite them are not congruent, and the longer side is opposite the larger angle.

Geometry 45 Section 3.8 The HL Postulate HL Postulate: If there exists a correspondence between the vertices of two right triangles such that the hypotenuse and a leg of one triangle are congruent to the corresponding parts of the other triangle, the two right triangles are congruent.