Download 6.3: Types of Quadrilaterals

6.3: Types of Quadrilaterals
Brinkman
Geometry
1. Warm up proof!
A
Given: ∆𝐴𝐵𝐶 𝑖𝑠 𝑖𝑠𝑜𝑠𝑐𝑒𝑙𝑒𝑠.
Prove: ∠5 ≅ ∠4.
Conclusion
1
Justification
5
2
3 4
C
∆𝐴𝐵𝐶 𝑖𝑠 𝑖𝑠𝑜𝑠𝑐𝑒𝑙𝑒𝑠.
Given
∠2 ≅ ∠3
Isosceles Triangle Base Angles Theorem
𝑚∠3 + 𝑚∠4 = 180
Definition of Linear Pair
𝑚∠2 + 𝑚∠5 = 180
Definition of Linear Pair
𝑚∠3 + 𝑚∠5 = 180
Substitution Property of Equality
∠5 ≅ ∠4
Addition Property of Congruence
B
Define and draw the following Quadrilaterals:
1.
Parallelogram: Opposite sides of the quadrilateral are parallel.
2. Rhombus: All four sides of a quadrilateral are congruent.
3. Rectangles: A quadrilateral comprised of 4 right angles.
4. Square: A quadrilateral with all four sides congruent and four right angles.
5. Kite: A quadrilateral with 2 distinct pairs of congruent consecutive sides.
6. Trapezoid: A quadrilateral with at least one pair of parallel sides.
7. Isosceles Trapezoid: A quadrilateral with a pair of base angles that have the same measure and
the sides opposite of the base angles are congruent.
Time to think!
6. Is a parallelogram a rhombus? Explain! No, there must be four equal sides!
7. Is a rectangle a rhombus? Explain! No, not all four sides have to be congruent.
8. Is a rhombus a rectangle? Explain! No, there needs to be 4 right angles.
9. Is a square a rectangle? Explain! Yes, there are four right angles.
10. Is a rectangle a square? Explain! No, only if all four sides are congruent.
11. Is a rectangle a parallelogram? Explain! Yes, there are two pairs of parallel sides.
Proof time!
Given: 𝑃𝑂//𝐴𝑅 , and 𝑃𝐴 ≅ 𝑂𝑇.
∆𝑂𝑇𝑅 𝑖𝑠 𝑖𝑠𝑜𝑠𝑐𝑒𝑙𝑒𝑠 𝑤𝑖𝑡ℎ 𝑣𝑒𝑟𝑡𝑒𝑥 𝑂.
Prove: 𝑃𝑂𝑅𝐴 is an isosceles trapezoid.
Conclusion
1. 𝑃𝑂//𝐴𝑅 𝑎𝑛𝑑 𝑃𝐴 ≅ 𝑂𝑇
Justification
Given
2. ∆𝑂𝑇𝑅 𝑖𝑠 𝑖𝑠𝑜𝑠𝑐𝑒𝑙𝑒𝑠 𝑤𝑖𝑡ℎ Given
𝑣𝑒𝑟𝑡𝑒𝑥 𝑂.
3. OT = OR
Definition of Isosceles Triangle
4. PA = OR
Transitive Property of Equality
5. PORA is an isosceles
trapezoid
Definition of Isosceles Trapezoid
12. Draw the hierarchy of quadrilaterals below.
6.3 Worksheet
Given below is a chart with five columns: rectangle, parallelogram, square, rhombus, and isosceles
trapezoid. Which of the 11 properties listed below are true for the five quadrilaterals? Fill in the following
chart with yes or no for each of the 11 properties.
1
2
3
4
5
6
7
8
9
10
11
6. At least one pair
of opposite sides is
congruent.
1. Opposite sides
are parallel
5. Has four right
angles.
3. Opposite sides
are congruent.
8. When sides are
congruent, diagonals
bisect each other.
2. Opposite
angles are ≅.
4. All sides are
congruent.
10. Diagonals are
angle bisectors.
9. Diagonals are
perpendicular.
7. Diagonals are
congruent.
11. Diagonals are
lines of symmetry.