Lesson 5 Rhombi and Squares
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Prove a Quadrilateral is a Rhombus
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Quadrilateral Family Tree
Using the applet on the previous slide, determine the properties of rhombus. Since a rhombus is a parallelogram, it will have all SIX parallelogram properties. A rhombus will also have an additional THREE "special" properties that distinguishes them from other parallelograms
Both pairs of opposite sides are congruent
4 right angles (Equiangular quadrilateral)
Both pairs of opposite angles are congruent
Diagonals bisect its angles
Only one pair of opposite sides parallel
Both pairs of opposite sides are parallel
Diagonals are congruent.
4 congruent sides (Equilateral quadrilateral)
A diagonal divides the parallelogram into two congruent triangles
Diagonals bisect each other
Diagonals are perpendicular.
Consecutive angles are supplementary
Only one pair of opposite sides congruent
QRST is a rhombus. Which of the following statements are (ALWAYS) TRUE?
m
mmmTrue Statement
Not (necessarily) True
In rhombus QRST,
, ,, and are triangles.Explain your reasoning for your answer to the previous question.
In Rhombus QRST, which of the following statements MUST BE TRUE?
True Statement
When both diagonals of a rhombus are drawn, the diagonals form four rectangle into triangles.
List some right triangle applications, you might be able to apply when given a diagonal of a rhombus.
RSTV is a rhombus with diagonals
and intersecting at M.If
and , find ST.RSTV is a rhombus with diagonals
and intersecting at M.If m
and m , find m.m
RSTV is a rhombus with diagonals
and intersecting at M.If m
and m , find m.m
RSTV is a rhombus with diagonals
and intersecting at M.If m
, classify by sides and angles.and
RSTV is a rhombus with diagonals
and intersecting at M.Name three angles congruent to
RSTV is a rhombus with diagonals
and intersecting at M.Name three angles congruent to
RSTV is a rhombus with diagonals
and intersecting at M.Name three angles congruent to
RSTV is a rhombus with diagonals
and intersecting at M.Name three triangles congruent to
QRST is a rhombus with diagonals
and intersecting at F.If m
, find:m
m
m
m
m
In Rhombus QRST,
and Find:Perimeter of QRST =
m
m
*Round answers to nearest tenth when necessary.
In Rhombus WXYZ,
and Find:Perimeter of WXYZ =
m
m
*Round answers to nearest tenth when necessary.
When using coordinate geometry to prove that a quadrilateral is a rhombus:
STEP 1 - Find the LENGTH (distance formula) of all 4 sides.
STEP 2 - Prove that quad is a parallelogram by showing that both pairs of opposite sides are congruent (2 sets of equal distances/lengths)
STEP 3 - Prove that parallelogram is a rhombus by showing that one pair of consecutive sides are congruent
Quadrilateral NATS has coordinates N(−4,−3), A(1,2), T(8,1), and S(3,−4). Prove quadrilateral NATS is a rhombus. [The use of the set of axes below is optional.]
Quadrilateral MATH has vertices M(−7,−2), A(0,4), T(9,2), and H(2,−4). Prove that parallelogram MATH is a rhombus. [The use of the set of axes below is optional.]
Quadrilateral MIKE has vertices with coordinates M(−1,−3), I(−3,3), K(5,4), and E(7,−2). Prove MIKE is a parallelogram, and prove MIKE is not a rhombus. [The use of the set of axes below is optional.]
Using the applet on the previous slide, determine the properties of square. Since a square is a parallelogram, it will have all SIX parallelogram properties. Since a square is a rectangle , it will have the TWO rectangle properties. Since a square is a rhombus, it will also have the THREE rhombus properties.
Both pairs of opposite sides are congruent
Both pairs of opposite sides are parallel
4 right angles (Equiangular quadrilateral)
Diagonals bisect its angles
Consecutive angles are supplementary
Diagonals bisect each other
Both pairs of opposite angles are congruent
A diagonal divides the parallelogram into two congruent triangles
Only one pair of opposite sides parallel
Only one pair of opposite sides congruent
Diagonals are congruent.
Diagonals are perpendicular.
4 congruent sides (Equilateral quadrilateral)
QRST is a square. Which of the following statements are (ALWAYS) TRUE?
m
mmmTrue Statement
Not (necessarily) True
In Square QRST,
, ,, and are triangles.Explain your reasoning for your answer to the previous question.
In Square QRST, which of the following statements MUST BE TRUE?
True Statement
ABCD is a square.
If
and , thenABCD is a square.
If
, find:
ABCD is a square.
Find:
m
m
m
m
m
ABCD is a square.
and are triangles.ABCD is a square. If
find:Perimeter of ABCD =
ABCD is a square. If
find:Perimeter of ABCD =
