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THEOREMS OF GEOMETRY Angles 1. Two adjacent angles are complementary when the sum of their measures is 90̊. 2. Two adjacent angles are supplementary when the sum of their measures is 180̊. 3. Vertically opposite angles are congruent. 4. Corresponding angles formed by parallel lines and a transversal are congruent. 5. Alternate interior angles formed by parallel lines and a transversal are congruent. 6. Alternate exterior angles formed by parallel lines and a transversal are congruent. 7. If the corresponding angles are congruent, or the alternate interior angles are congruent, or the alternate exterior angles are congruent, then the two lines intersected by a transversal are parallel. Triangles 1. The sum of the measures of the interior angles of a triangle is 180º. 2. An isosceles triangle has two congruent sides. 3. In an isosceles triangle, the angles opposite the congruent sides are congruent. 4. The axis of symmetry of an isosceles triangle is a median, a perpendicular bisector, an angle bisector & an altitude of the triangle. 5. An equilateral triangle has three congruent sides. 6. In an equilateral triangle, each angle measures 60º. 7. All three axes of symmetry of an equilateral triangle are the medians, perpendicular bisectors, angle bisectors & altitudes of the triangle. 8. In a right triangle, the side opposite a 30º angle is one-half the length of the hypotenuse. 9. In a right triangle, the square of the hypotenuse is equal to the sum of the squares on the other two sides. (Pythagoras Theorem) 10. In any right triangle, the acute angles are complementary. 11. In any isosceles right triangle, each of the acute angles measures 45º. Quadrilaterals 1. The sum of the measures of the interior angles of a quadrilateral is 360º. Squares 1. A square has four congruent sides. 2. In a square, each angle measures 90º. 3. The diagonals of a square bisect each other perpendicularly. Rhombuses 1. A rhombus has four congruent sides. 2. The diagonals of a rhombus are perpendicular to each other. Rectangles 1. The opposite sides of a rectangle are congruent. 2. In a rectangle, each angle measures 90º. 3. The diagonals of a rectangle bisect each other. Parallelograms Trapezoids 1. The opposite sides of a parallelogram are congruent. 2. The opposite angles of a parallelogram are congruent. 3. The consecutive angles of a parallelogram are supplementary. 4. The diagonals of a parallelogram bisect each other. 1. The bases are parallel 2. An isosceles trapezoid has one pair of congruent sides. 3. A right trapezoid has 2 right angles. Circles Kites 1. A kite has two pairs of congruent sides. 2. The diagonals of a kite are perpendicular to each other. 1. In a circle, the measure of the radius is half the measure of the diameter. 2. All the diameters of a circle are congruent. 3. All the radii of a circle are congruent. ISOMETRIC TRIANGLES 1. Theorem of Congruence SAS: Two triangles with corresponding congruent angle contained between two congruent corresponding sides are isometric. 2. Theorem of Congruence ASA: Two triangles with corresponding congruent side contained between two congruent corresponding angles are isometric. 3. Theorem of Congruence SSS: Two triangles with corresponding congruent sides are isometric. 4. When 2 triangles are isometric, their corresponding elements are congruent. SIMILAR TRIANGLES 1. Theorem of Similarity AA: Two triangles with two corresponding congruent angles are similar. 2. Theorem of Similarity SAS: Two triangles with corresponding congruent angle contained between two proportional corresponding sides are similar. 3. Theorem of Similarity SSS: Two triangles with corresponding proportional sides are similar. 4. When 2 triangles are similar, their corresponding sides are proportional. Parallel Line to a Triangle’s Side: Any line parallel to a triangle’s side determines two similar triangles. Segment Joining the Midpoints of Two Sides in a Triangle: Any segment joining the midpoints of two sides in a triangle is parallel to the third side and is half the measure of this third side. Thales’ Theorem: Two transversal lines intersected by parallel lines are separated into corresponding segments of proportional length. METRIC RELATIONS IN A RIGHT TRIANGLE Each side of the right angle is the geometric mean between its projection onto the hypotenuse and the hypotenuse. 2 2 a =mxc b =nxc The altitude is the geometric mean of the projections of the sides onto the hypotenuse. 2 h =mxn The product of the sides of the right angle is equal to the product of the hypotenuse and the altitude. a b h m n c axb=hxc