Advanced Geometry

First you must prove or be given that the figure is a parallelogram, then A PARALLELOGRAM is a rectangle if… 1. It contains at least one right angle, 2. Or if the diagonals are congruent

A QUADRILATERAL is a rectangle if… All FOUR angles are right angles Two isn’t enough! In this case, you DO NOT have to prove parallelogram first. Notice that we are going from QUAD to RECTANGLE here!

A QUADRILATERAL is a kite if… 1. It has two pairs of consecutive congruent sides, or 2. If one of the diagonals is the perpendicular bisector of the other

A PARALLELOGRAM is a rhombus if… 1. Parallelogram with a pair of consecutive congruent sides ~ or ~ 2. Parallelogram where either diagonal bisects two angles

A QUADRILATERAL is a rhombus if… The diagonals are perpendicular bisectors of each other. of each other

A QUADRILATERAL is a square if… It is both a rectangle and a rhombus Both are Parallelograms, so: Opposite sides parallel Opposite angles congruent Consecutive angles supplementary From Rhombus : ALL sides congruent Diagonals are perpendicular bisectors of each other Diagonals bisect opposite angles From Rectangle : 4 Right Angles Diagonals are congruent

A QUADRILATERAL is a square if… It is both a rectangle and a rhombus Both are Parallelograms, so: Opposite sides parallel Opposite angles congruent Consecutive angles supplementary From Rhombus : ALL sides congruent Diagonals are perpendicular bisectors of each other Diagonals bisect opposite angles From Rectangle : 4 Right Angles Diagonals are congruent Each diagonal creates two 45 ⁰ -45 ⁰ -90 ⁰ ISOSCELES, RIGHT TRIANGLES!

A TRAPEZOID is isosceles if… The nonparallel sides are congruent, or The lower or upper base angles are congruent, 0r The diagonals are congruent X √ Trapezoid : X X X Isosceles Trapezoid: √√√ √√√

Sample Problem 1: What is the most descriptive name for quadrilateral ABCD with vertices A(-3, -7) B(-9, 1) C(3, 9) D(9, 1) Step 1: Graph ABCD Step 2: Find the slope of each side. A(-3, -7) (-9, 1)B C(3, 9) (9, 1) D – (-9) (-7) -9- (-3) (1) -3 – 9 Slope: AB  -8/6 = -4/3 BC  8/12 = 2/3 CD  8/-6 = -4/3 AD  -8/-12 = 2/

Sample Problem 1: What is the most descriptive name for quadrilateral ABCD with vertices A(-3, -7) B(-9, 1) C(3, 9) D(9, 1) Step 3: compare the slopes of opposite sides to determine whether they are parallel. Slope AB = Slope CD (- 4/3) and Slope BC = Slope AD (2/3) SAME SLOPES  Opposite sides PARALLEL  Parallelogram! compare the slopes of consecutive sides to see if they are perpendicular. (slope AB)(slope BC) = (-4/3)(2/3) ≠ -1, so consecutive sides are not  OPPOSITE RECIPROCAL SLOPES  Consecutive sides PERPENDICULAR  Rectangle Step 4: We know the quadrilateral is not a rectangle, but it is a parallelogram. Find the slopes of the diagonals to see if it is a rhombus.

Sample Problem 1: What is the most descriptive name for quadrilateral ABCD with vertices A(-3, -7) B(-9, 1) C(3, 9) D(9, 1) Step 4: We now know the quadrilateral is not a rectangle, but IT IS A PARALLELOGRAM. Find the slopes of the diagonals to see if it is a rhombus. (-9, 1)B C(3, 9) A(-3, -7) (9, 1) D Slope BD = (1 – 1)/[9 – (-9)] = 0/18 = 0 Slope AC = [9 – (-7)] / [3 – (-3)] = 16/6 = 8/3 (Slope BD)(Slope AC) = (0)(8/3) ≠ -1 Since the slopes are not opposite reciprocals, the diagonals are not  ABCD is a PARALLELOGRAM In a rhombus, the diagonals are , 

Given: AB || CD ∡ ABC  ∡ ADC AB  AD PROVE: ABCD is a rhombus A C B D || lines  alt int ∡s , so ∡ABD  ∡CDB AB = AD was not used yet, because FIRST we must prove that ABCD is a parallelogram, then we will try to prove that it is a RHOMBUS!

Given: AB || CD ∡ ABC  ∡ ADC AB  AD PROVE: ABCD is a rhombus A C B D ∡CBD  ∡ADB Subtraction! ∡ABC - ∡ABD = ∡ CBD ∡ADC - ∡CDB = ∡ADB

Given: AB || CD ∡ ABC  ∡ ADC AB  AD PROVE: ABCD is a rhombus A C B D alt int ∡s   || lines BC || AD

Given: AB || CD ∡ ABC  ∡ ADC AB  AD PROVE: ABCD is a rhombus A C B D ABCD is a parallelogram If a quadrilateral has 2 pairs of || sides, then parallelogram

Given: AB || CD ∡ ABC  ∡ ADC AB  AD PROVE: ABCD is a rhombus A C B D ABCD is a rhombus If a parallelogram has at least 2 consecutive sides , then rhombus ! parallelogram

1. AB || CD 8. ABCD is a rhombus A C B D 6. ABCD is a parallelogram If a parallelogram has at least 2 consecutive sides , then RHOMBUS ! 3. ∡ ABC  ∡ ADC 7. AB  AD GIVEN 2. ∡ ABD  ∡ CDB|| lines  alt int ∡ s  4. ∡ ADB  ∡ CBD subtraction 5. BC || AD alt int ∡ s   || lines If a quad has 2 pairs of || sides, then parallelogram GIVEN Statements Reasons parallelogram then RHOMBUS !

Pp. 258 – 262 (1 – 6; 10; 12 – 14; 16, 17; 19 – 21; 24, 28, 29)