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Polygons – Rhombuses and Trapezoids
Rhombus - four congruent sides
Polygons – Rhombuses and Trapezoids
Rhombus - four congruent sides
- opposite angles are congruent
Polygons – Rhombuses and Trapezoids
Rhombus - four congruent sides
- opposite angles are congruent
Polygons – Rhombuses and Trapezoids
Rhombus - four congruent sides
- opposite angles are congruent
- diagonals bisect the angles at the vertex
A
C
B
D
2
2A
CAD
ACAB
Polygons – Rhombuses and Trapezoids
Rhombus - four congruent sides
- opposite angles are congruent
- diagonals bisect the angles at the vertex
- diagonals bisect each other and are perpendicular
A
C
B
D
E ACBD
ECAE
EDBE
Polygons – Rhombuses and Trapezoids
Rhombus - four congruent sides
- opposite angles are congruent
- diagonals bisect the angles at the vertex
- diagonals bisect each other and are perpendicular
A
C
B
D
E14
EXAMPLE : If AD = 14, what is the measure of EB ?
60°
Polygons – Rhombuses and Trapezoids
Rhombus - four congruent sides
- opposite angles are congruent
- diagonals bisect the angles at the vertex
- diagonals bisect each other and are perpendicular
A
C
B
D
E14
EXAMPLE : If AD = 14, what is the measure of EB ?
SOLUTION : With angle ADE = 60 degrees we have a 30 – 60 – 90 triangle.
So segment EB = Segment ED which is half of AD.
60°
Polygons – Rhombuses and Trapezoids
Rhombus - four congruent sides
- opposite angles are congruent
- diagonals bisect the angles at the vertex
- diagonals bisect each other and are perpendicular
A
C
B
D
E14
EXAMPLE : If AD = 14, what is the measure of EB ?
SOLUTION : With angle ADE = 60 degrees we have a 30 – 60 – 90 triangle.
So segment EB = Segment ED which is half of AD. ED = 7
60°
Polygons – Rhombuses and Trapezoids
Rhombus - four congruent sides
- opposite angles are congruent
- diagonals bisect the angles at the vertex
- diagonals bisect each other and are perpendicular
A
C
B
D
E14
EXAMPLE : What is the measure of angle ECD ?
60°
Polygons – Rhombuses and Trapezoids
Rhombus - four congruent sides
- opposite angles are congruent
- diagonals bisect the angles at the vertex
- diagonals bisect each other and are perpendicular
A
C
B
D
E14
EXAMPLE : What is the measure of angle ECD ?
SOLUTION : Again we have a 30 – 60 – 90 triangle. So angle DAC = 30 degrees.
60°
Polygons – Rhombuses and Trapezoids
Rhombus - four congruent sides
- opposite angles are congruent
- diagonals bisect the angles at the vertex
- diagonals bisect each other and are perpendicular
A
C
B
D
E14
EXAMPLE : What is the measure of angle ECD ?
SOLUTION : Again we have a 30 – 60 – 90 triangle. So angle DAC = 30 degrees.
So angle ECD would also be 30 degrees.
60°
Polygons – Rhombuses and Trapezoids
Trapezoid - two parallel sides that are not congruent
D
A B
C
CDAB ║
CDAB
Polygons – Rhombuses and Trapezoids
Trapezoid - two parallel sides that are not congruent
D
A B
C
CDAB ║
CDAB - these parallel sides are called bases
- non-parallel sides are called legs
base 1
base 2
leg leg
Polygons – Rhombuses and Trapezoids
Trapezoid - two parallel sides that are not congruent
D
A B
C
CDAB ║
CDAB - these parallel sides are called bases
- non-parallel sides are called legs
base 1
base 2
leg leg
- there are two pairs of base angles
Polygons – Rhombuses and Trapezoids
Trapezoid - two parallel sides that are not congruent
D
A B
C
CDAB ║
CDAB - these parallel sides are called bases
- non-parallel sides are called legs
base 1
base 2
leg leg
- there are two pairs of base angles
- diagonal base angles are supplementary
Polygons – Rhombuses and Trapezoids
Trapezoid - two parallel sides that are not congruent
D
A B
C
CDAB ║
CDAB - these parallel sides are called bases
- non-parallel sides are called legs
base 1
base 2
leg leg
- there are two pairs of base angles
- diagonal base angles are supplementary
- base angles that share a leg are also supplementary
Polygons – Rhombuses and Trapezoids
Isosceles Trapezoid - has all the properties of a trapezoid
- legs are congruent
- base angles are congruent
D
A B
C
DC
BA
Polygons – Rhombuses and Trapezoids
Isosceles Trapezoid - has all the properties of a trapezoid
- legs are congruent
- base angles are congruent
- diagonals have the same length
D
A B
C
DC
BA
BDAC
Polygons – Rhombuses and Trapezoids
Median of a Trapezoid
- parallel with both bases
- equal to half the sum of the bases
- joins the midpoints of the legs
D
A B
C
2
21 basebase
X Y
Polygons – Rhombuses and Trapezoids
Let’s try some problems…
D
A B
C
EXAMPLE : What is the median length ?
20
28
Polygons – Rhombuses and Trapezoids
Let’s try some problems…
D
A B
C
EXAMPLE : What is the median length ?
20
28
242
48
2
2820
2
21
basebase
24
Polygons – Rhombuses and Trapezoids
Let’s try some problems…
D
A B
C
EXAMPLE : If AD = 18, what is the measure of AX ?
18 X Y
Polygons – Rhombuses and Trapezoids
Let’s try some problems…
D
A B
C
EXAMPLE : If AD = 18, what is the measure of AX ?
18 X Y
92
18 The median joins the midpoints of the legs
Polygons – Rhombuses and Trapezoids
Let’s try some problems…
D
A B
C
EXAMPLE : ABCD is an isosceles trapezoid. If angle DAB = 110°, what is the measure of angle ABC ?
Polygons – Rhombuses and Trapezoids
Let’s try some problems…
D
A B
C
EXAMPLE : ABCD is an isosceles trapezoid. If angle DAB = 110°, what is the measure of angle ABC ?
110° - base angles are congruent in an isosceles trapezoid
Polygons – Rhombuses and Trapezoids
Let’s try some problems…
D
A B
C
EXAMPLE : What is the length of side AB?
?
50
YX 40
Polygons – Rhombuses and Trapezoids
Let’s try some problems…
D
A B
C
EXAMPLE : What is the length of side AB?
?
50
YX 40
2
25040
base
Polygons – Rhombuses and Trapezoids
Let’s try some problems…
D
A B
C
EXAMPLE : What is the length of side AB?
?
50
YX 40
22
250240
base2
25040
base
Polygons – Rhombuses and Trapezoids
Let’s try some problems…
D
A B
C
EXAMPLE : What is the length of side AB?
?
50
YX 40
230
25080
22
250240
base
base
base
2
25040
base