First, let’s review how to prove a parallelogram. Given: AB CD ABD CDB Prove: ABCD is a Statements 1. AB CD 2. ABD CDB 3. AB || CD 4. ABCD is a B C A D Reasons 1. Given 2. Given 3. If alternate interior angles are , then the lines are || 4. If opp. sides of a quadrilateral are both || and , then the figure is a parallelogram. There are several quadrilaterals that we will need to prove are specific shapes. They include: Rectangle Kite Rhombus Square Isosceles Trapezoid For several of these, we will first need to prove the figure is a parallelogram. Let’s begin. Proving Rectangles When proving rectangles you have 3 options. 1st – If you can prove a quadrilateral has 4 right angles, then the quadrilateral is a rectangle. 2nd – If you can 1st prove the quadrilateral is a parallelogram, then prove at least 1 right angle, then the quadrilateral is a rectangle. 3rd – If you can 1st prove the quadrilateral is a parallelogram, then prove the diagonals are congruent, then the quadrilateral is a rectangle. Proving Kites When proving kites you have 2 options. 1st - If 2 disjoint pairs of consecutive sides are , then the figure is a kite. 2nd - If 1 of the diagonals is the bisector of the other diagonal, then the figure is a kite. (Show that 1 diagonal is to the other AND that the same diagonal bisects the other) Proving Rhombuses When proving rhombuses you have 3 options. 1st - If the diagonals are bisectors of each other, then the figure is a rhombus. 2nd - There must be a step in the proof that says the figure is a PARALLELOGRAM AND If a parallelogram contains a pair of consecutive sides that are , then the figure is a rhombus 3rd - There must be a step in the proof that says the figure is a PARALLELOGRAM AND If either diagonal of a parallelogram bisects 2 ’s of the parallelogram, then the figure is a rhombus. Proving Squares When proving squares you have 1 option. 1st - If a quadrilateral is BOTH a rectangle and a rhombus, then the figure is a square. (So you MUST have a step in the proof that shows the figure is a rectangle and another step that show the figure is a rhombus.) Proving Isosceles Trapezoids When proving Isos. Trapezoids you have 3 options. 1st - If the non-parallel sides of a trapezoid are , then the figure is an isosceles trapezoid. 2nd - If the lower or upper base ’s of a trapezoid are , then the figure is an isosceles trapezoid. 3rd - If the diagonals of a trapezoid are , then the figure is an isosceles trapezoid. In review to prove: Rectangles ( 3 methods) If all 4 ’s are right ’s, then the figure is a rectangle. Prove a parallelogram and 1. 2. If a parallelogram contains at least 1 right rectangle. , or if the diagonals are , then the figure is a Kites (2 methods) If 2 disjoint pairs of consecutive sides are , then the figure is a kite. If 1 of the diagonals is the bisector of the other diagonal, then the figure is a kite. 1. 2. Rhombus (3 methods) If the diagonals are bisectors of each other, then the figure is a rhombus. Prove a parallelogram and 1. 2. If a parallelogram contains a pair of consecutive sides that are , or if either diagonal of a parallelogram bisects 2 ’s of the parallelogram, then the figure is a rhombus. Squares (1 method) 1. If a quadrilateral is BOTH a rectangle and a rhombus, then the figure is a square. Isosceles Trapezoids (3 methods) 1. 2. 3. If the non-parallel sides of a trapezoid are , then the figure is an isosceles trapezoid. If the lower or upper base ’s of a trapezoid are , then the figure is an isosceles trapezoid. If the diagonals of a trapezoid are , then the figure is an isosceles trapezoid. First, let’s review how to prove a parallelogram. Given: GJMO is a OH JK MK is an alt. of ▲ MKJ Prove: OHKM is a rectangle O G Statements M J H K Reasons 1. 2. 3. 4. 5. 6. GJMO is a OM || GK OH GK MK is an alt. of ▲MKJ MK GK OH || MK 1. 2. 3. 4. 5. 6. 7. OHKM is a 7. 8. 9. OHK is a right OHKM is a rectangle 8. 9. Given Opp. Sides of a are || Given Given Def. of altitude If two coplanar lines are to a 3rd line, then they are ||. If both pairs of opp. sides of a quad. are ||, then it is a Def. of If a contains at least one right , it is a rectangle.

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