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A Unit 4 E Congruent??????? B D M.Sigley, Baker MS Unit 4 M C R N P O 1 2 Unit 4 Unit 4 M.Sigley, Baker MS M.Sigley, Baker MS 3 M.Sigley, Baker MS 4 Unit 4 POSTULATE Polygon Congruence Postulate SSS Two polygons are congruent if and only if: If the corresponding sides of one triangle are congruent to the corresponding sides of another triangle then the triangles are congruent. Each pair of corresponding sides are congruent Each pair of corresponding angles are congruent 2 M.Sigley, Baker MS Unit 4 1 M.Sigley, Baker MS Unit 4 POSTULATE POSTULATE SAS ASA If two sides and their included angle of a triangle are congruent to two sides and their included angle of another triangle, then the triangles are congruent. If two angles and the included side in one triangle are congruent to two angles and the included side in another triangle, then the two triangles are congruent. M.Sigley, Baker MS POSTULATE Unit 4 4 M.Sigley, Baker MS 3 Unit 4 M.Sigley, Baker MS Unit 4 5 Unit 4 Unit 4 6 M.Sigley, Baker MS A CPCTC C M.Sigley, Baker MS 7 M.Sigley, Baker MS B 8 Unit 4 THEOREM THEOREM Unit 4 HL AAS If the hypotenuse and a leg of a right triangle are congruent to the hypotenuse and a leg on another right triangle, then the two triangles are congruent. 6 M.Sigley, Baker MS PROPERTIES Unit 4 If two angles and the non-included side of one triangle are congruent to two angles and the nonincluded side of another triangle, then the triangles are congruent. 5 M.Sigley, Baker MS POSTULATE Unit 4 Isosceles Triangle Corresponding parts of congruent triangles are congruent. AB = AC (legs) BC is base Angle C and Angle B are base angles. Angle A is vertex angle. M.Sigley, Baker MS 8 M.Sigley, Baker MS 7 Unit 4 Unit 4 1 1 2 2 3 9 M.Sigley, Baker MS 10 M.Sigley, Baker MS Unit 4 Unit 4 M.Sigley, Baker MS 3 12 11 M.Sigley, Baker MS 12 Unit 4 COROLLARY THEOREM Unit 4 Isosceles Triangle Theorem Base angles of an isosceles triangle are congruent. (angle 2 and 3) The measure of each angle of an equilateral triangle is 60°. 10 M.Sigley, Baker MS Unit 4 If two angles of a triangle are congruent then the sides opposite the angles are congruent. THEOREM COROLLARY Unit 4 The bisector of the vertex angle of an isosceles triangle is the perpendicular bisector of the base. A diagonal of a parallelogram divides the parallelogram into two congruent triangles. M.Sigley, Baker MS 9 M.Sigley, Baker MS 12 M.Sigley, Baker MS 11 Unit 4 M.Sigley, Baker MS Unit 4 13 14 M.Sigley, Baker MS Unit 4 Unit 4 2 1 M.Sigley, Baker MS 15 M.Sigley, Baker MS 16 Unit 4 THEOREM THEOREMS Unit 4 Opposite sides of a parallelogram are congruent. If two pairs of opposite sides of a Opposite angles of a parallelogram are congruent. quadrilateral are congruent, then the quadrilateral is a parallelogram. If one pair of opposite sides of a quadrilateral is parallel and congruent, then the quadrilateral is a parallelogram. 14 M.Sigley, Baker MS Unit 4 Unit 4 THEOREM Consecutive angles of a parallelogram are supplementary. 13 M.Sigley, Baker MS THEOREM The diagonals of a parallelogram bisect each other. If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram. M.Sigley, Baker MS 16 M.Sigley, Baker MS 15 Unit 4 Unit 4 Rhombus is? M.Sigley, Baker MS Rectangle is? 17 18 Unit 4 Unit 4 M.Sigley, Baker MS M.Sigley, Baker MS 19 M.Sigley, Baker MS 20 Unit 4 THEOREM A rhombus is a parallelogram. A rectangle is a parallelogram. If one pair of adjacent sides of a If one angle of a parallelogram is a right parallelogram is congruent, then the parallelogram is a rhombus. angle, then the parallelogram is a rectangle. 18 M.Sigley, Baker MS Unit 4 THEOREM 17 M.Sigley, Baker MS THEOREM Unit 4 The diagonals of a rhombus are perpendicular. The diagonals of a rectangle are congruent. If the diagonals of a parallelogram bisect Housebuilder Theorem: If the the angles of the parallelogram then the parallelogram is a rhombus. diagonals of a parallelogram are congruent, then the parallelogram is a rectangle. M.Sigley, Baker MS THEOREM Unit 4 If the diagonals of a parallelogram are perpendicular, then the parallelogram is a rhombus. 20 M.Sigley, Baker MS 19 Unit 4 M.Sigley, Baker MS Unit 4 21 22 Unit 4 Unit 4 M.Sigley, Baker MS M.Sigley, Baker MS 23 M.Sigley, Baker MS 24 Unit 4 THEOREMS A square is a rectangle. THEOREM Unit 4 The diagonals of a kite are perpendicular. A square is a rhombus. 22 M.Sigley, Baker MS Unit 4 Diagonals of a Square Triangle Midsegment Theorem Congruent A midsegment of a triangle is parallel to a side of the triangle and has a measure equal to half the measure of that side. M.Sigley, Baker MS THEOREM Unit 4 THEOREM 21 M.Sigley, Baker MS Perpendicular bisectors of each other 24 M.Sigley, Baker MS 23 Unit 4 Unit 4 A C M.Sigley, Baker MS B 25 26 Unit 4 Unit 4 M.Sigley, Baker MS M.Sigley, Baker MS 27 M.Sigley, Baker MS 28 Unit 4 THEOREM THEOREM Unit 4 Triangle Inequality Theorem Euler Path The sum of the lengths of any two sides of a triangle is greater than the length of the third side. A graph contains an Euler path if and only if there are at most two odd vertices. AB + BC > AC BC + AC > AB M.Sigley, Baker MS 26 AC + AB > BC 25 Unit 3 Unit 3 M.Sigley, Baker MS M.Sigley, Baker MS 28 M.Sigley, Baker MS 27