5.7 Proving Special Quadrilaterals

1. Advanced Geometry 5.7 Proving Special Quadrilaterals

2. Proving a quadrilateral is a RECTANGLE 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

3. Proving a quadrilateral is a RECTANGLE 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 rectangle if… All FOUR angles are right angles 2 3 Two isn’t enough! 1 4

4. Proving a quadrilateral is a KITE 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

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

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

7. Proving a quadrilateral is a SQUARE A QUADRILATERALis 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

8. Proving a quadrilateral is a SQUARE A QUADRILATERALis a square if… It is both a rectangle and a rhombus From Rhombus: ALL sides congruent Diagonals are perpendicular bisectors of each other Diagonals bisect opposite angles From Rectangle: 4 Right Angles Diagonals are congruent Both are Parallelograms, so: Opposite sides parallel Opposite angles congruent Consecutive angles supplementary Each diagonal creates two 45⁰-45⁰-90⁰ ISOSCELES, RIGHT TRIANGLES!

9. Proving that a trapezoid is ISOSCELES 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 √ √ √ Isosceles Trapezoid: √√√ Trapezoid : X XX

10. Sample Problems Pp 257 - 258 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) C(3, 9) 12 - 6 Step 1: Graph ABCD Step 2: Find the slope of each side. 3 – (-9) 3 - 9 9 - 1 8 9 - 1 8 Slope: AB  -8/6 = -4/3 BC  8/12 = 2/3 CD  8/-6 = -4/3 AD  -8/-12 = 2/3 (-9, 1)B (9, 1) D 1- (-7) - 8 -7- (1) - 8 -9- (-3) -3 – 9 6 -12 A(-3, -7)

11. Sample Problems (Pp 257 – 258) 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.

12. Sample Problems (Pp 257 – 258) 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. C(3, 9) Slope BD = (1 – 1)/[9 – (-9)] = 0/18 = 0 Slope AC = [9 – (-7)] / [3 – (-3)] = 16/6 = 8/3 ABCD is a PARALLELOGRAM (-9, 1)B (Slope BD)(Slope AC) = (0)(8/3) ≠ -1 Since the slopes are not opposite reciprocals, the diagonals are not  (9, 1) D In a rhombus, the diagonals are ,  A(-3, -7)

13. Sample Proof: Prove ABCD is a rhombus 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 C B PROVE: ABCD is a rhombus A D || lines  alt int ∡s  , so ∡ABD  ∡CDB

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

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

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

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

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

19. 5.7 Homework Pp. 258 – 262 (1 – 6; 10; 12 – 14; 16, 17; 19 – 21; 24, 28, 29)