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Section 4.1

Section 4.1. Congruent Polygons. Polygons. Examples of Polygons . Examples of Non-Polygons. Non-Polygons. Polygons. Naming a Polygon. Octagon CDEFGHAB H A G B F C

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Section 4.1

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  1. Section 4.1 Congruent Polygons

  2. Polygons Examples of Polygons Examples of Non-Polygons Non-Polygons Polygons

  3. Naming a Polygon Octagon CDEFGHAB H A G B F C E D When naming polygons, the rule is to go around the figure, either clockwise or counterclockwise, and list the vertices in order. It does not matter which vertex you list first.

  4. Corresponding Sides and Angles • If two polygons have the same number of sides, it is possible to set up a correspondence between them by pairing their parts. In rhombi RSTU and MNOP, the corresponding angles and sides would be as follows. Pair – Angles R and M, S and N, T and O, and U and P R M U S P N T O

  5. Polygon Congruence Postulate Two Polygons are congruent if and only if there is a correspondence between their sides and angles such that: Each pair of corresponding angles is congruent. Each pair of corresponding sides is congruent. Otherwise, same shape, same size.

  6. Congruence Statement • ABCDE ≅ KRMNG, Angle A corresponds to K, B to R, C to M, D to N, and E to G. A N E B M G D C R K

  7. Section 4.2 Triangle Congruence

  8. SSS (Side-Side-Side) Postulate • If the sides of one triangle are congruent to the sides of another triangle, then the two triangles are congruent.△VEG ≅△TFH G H 12m 12m 19m V 19m T 20m 20m E F

  9. SAS (Side-Angle-Side) Postulate • If two sides and their angle in one triangle are congruent to two sides and their angle in another triangle, then the two triangles are congruent. △ABC ≅△FDE A F 37˚ 37˚ C E B D

  10. ASA (Angle-Side-Angle) Postulate • If two angles and the side between the two angles in one triangle are congruent to two angles and the side between the two angles in another triangle, then the two triangles are congruent. △MNO ≅△TSR M S R 56° 39° O 39° 56° N T

  11. Section 4.3 Analyzing Triangle Congruence

  12. AAS (Angle-Angle-Side) Postulate • If two angles and a non-included side of one triangle are congruent to two angles and a non-included side of another triangle, then the two triangles are congruent. △ABC ≅△EFD A F 101°35° D C 35° 101° B E

  13. Combinations That Do Not Work Unless Under Special Circumstances • AAA (Angle-Angle-Angle) • SSA (Side-Side-Angle)

  14. When Dealing With Right Triangles • HL (Hypotenuse-Leg) Congruence Theorem • If the hypotenuse and a leg of a right triangle are congruent to the hypotenuse and a leg of another right triangle, then the two triangles are congruent. • Other Possibilities • LL (Leg-Leg) Congruence Theorem • LA (Leg-Angle) Congruence Theorem • HA (Hypotenuse-Angle) Congruence Theorem

  15. Section 4.4 Using Triangle Congruence

  16. CPCTC in Flowchart Proofs • CPCTC – Corresponding Parts of Congruent Triangles are Congruent. • CPCTC is used after a triangle congruence postulate (SSS, ASA, SAS, or AAS) has been establish to prove two triangles are congruent. • Flowchart Proofs are used to explain and understand why two or more triangles are congruent.

  17. Given: AC≅BD, CX≅DX, and <C≅<DProve: X is the midpoint of AB. C B X A D

  18. Isosceles Triangle • An isosceles triangle is a triangle with two congruent sides. • The two congruent sides are known as the legs of the triangle. • The remaining side is known as the base. • The angle opposite the base is the vertex angle. • The angles whose vertices are the endpoints of the base are base angles.

  19. Theorems Involving Isosceles Triangles • Isosceles Triangle Theorem – If two sides of a triangle are congruent, then the angles opposite those sides are congruent. • Converse of the Isosceles Triangle Theorem – If two angles of a triangle are congruent, then the sides opposite those angles are congruent. • The bisector of the vertex angle of an isosceles triangle is the perpendicular bisector of the base.

  20. Reminders For an Equilateral Triangle • The measure of each angle of an equilateral triangle is 60°. • The measures of the sides are congruent.

  21. Section 4.5 Proving Quadrilateral Properties

  22. Quadrilateral Hierarchy

  23. Properties of Parallelograms • A diagonal divides the parallelogram into two congruent triangles. • Opposite sides of a parallelogram are ≅. • Opposite angles of a parallelogram are ≅. • Consecutive angles are supplementary. • The two diagonals bisect each other. • Rectangles, rhombi, and squares are parallelograms.

  24. Properties of Parallelograms Properties of Rectangles Properties of Rhombi The diagonals are perpendicular. Four ≅ sides. The diagonals bisect a pair of opposite angles. • The diagonals are ≅. • Four 90° angles.

  25. Properties of Squares • All properties of a parallelogram • All properties of a rectangle • All properties of a rhombus

  26. Section 4.6 Conditions for Special Quadirlaterals

  27. The Conditions That Determine a Figure • If you are given a quadrilateral, check the given information against the definitions of the special figures. • Check for properties that match or you can prove to a special figure. • It is better to be more specific than general. Remember a square is always a parallelogram but a parallelogram is not always a square.

  28. The Triangle Midsegment Theorem • A midsegment of a triangle is a parallel to a side of the triangle and has a measure equal to half of the measure of that side. • CD is the midsegment of △AGH. G H C D A

  29. Review of Other Properties • Substitution Property • Transitive Property • Reflexive Property • Symmetric Property • Transitive Property • SSS (Side-Side-Side) • ASA (Angle-Side-Angle) • SAS (Side-Angle-Side) • AAS (Angle-Angle-Side) • CPCTC • Definition of Midpoint • Alt. Interior Angles • Cons. Interior Angles • Corresponding Angles • Alt. Exterior Angles • Vertical Angles Theorem

  30. Section 4.7 Compass and Straightedge Constructions

  31. Compass and Straightedge Constructions • A segment congruent to a given segment pg. 261 • A triangle congruent to a given triangle pg. 262 • Angle bisector pg. 263 • An angle congruent to a given angle pg. 265 • The perpendicular bisector of a given segment and the midpoint of a given segment pg. 266 • A line through a point perpendicular to a given line pg. 267 • A line through a point parallel to a given line pg. 268

  32. Section 4.8 Constructing Transformations

  33. Transformations • Rotation (Turn) – every point of the preimage is rotated by a given angle about a point. • Reflection (Flip) – every point of the preimage may be connected to its image point by a segment that (a) is perpendicular to the line or plane that is the mirror of the reflection and (b) has its midpoint on the mirror of the reflection.

  34. Transformations • Translation (Slide) – every point of the preimage moves in the same direction by the same amount to form the image. • Dilation – a figures size is increased or decreased by a scale factor, but its shape stays the same.

  35. Other Theorems Betweenness Postulate Triangle Inequality Theorem The sum of the lengths of any two sides of a triangle is greater than the length of the third side. AB + BC > AC A AB + AC > BC AC + BC > AB B C • Converse of the Segment Addition Postulate • Given three points P, Q, and R, if PQ + QR = PR, then P, Q, and R are collinear and Q is between P and R.

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