Download Accelerated Geometry Name Quadrilaterals

Accelerated Geometry
Quadrilaterals – Notes
Name __________________________
Date____________
Polygons are closed, plane figures. The sides of a polygon consist of segments. Each side intersects
exactly 2 other sides. The intersections are called vertices.
Determine whether the following shapes are polygons. If it is not a polygon, explain why.
1.
2.
3.
4.
5.
6.
Naming polygons – we name a polygon by starting at any vertex and then proceeding either clockwise or
counterclockwise. Give 3 different ways to name the polygon below.
Convex polygon – a polygon in which each interior angle has a measure less than 180.
Concave polygon – a polygon in which at least one interior angle is greater than 180. A concave polygon
appears “caved in.”
Determine whether the polygons below are convex or concave.
Accelerated Geometry
Quadrilaterals – Notes
Name __________________________
Date____________
Diagonal – any segment that connects two nonconsecutive (nonadjacent) vertices of a polygon.
Draw in all of the diagonals in the polygon below.
Quadrilateral – a four-sided polygon.
Properties of quadrilaterals:
1. The 4 angles of a quadrilateral add up to 360 degrees
2. Quadrilaterals have exactly 2 diagonals.
Parallelogram – a quadrilateral with both pairs of opposite sides parallel.
Parallelogram Property:
The opposite sides of a parallelogram are congruent.
B
A
C
D
In the parallelogram ABCD, AB ≅ CD and CB ≅ AD .
Place the proper markings in the diagram above, then solve the following problem.
Example #1.) If BC = 45, CD = 3y, AD = x + y, and AB = x, Find the value of x and y and the perimeter of
ABCD.
Accelerated Geometry
Quadrilaterals – Notes
Name __________________________
Date____________
Parallelogram Property:
B
A
The Opposite angles of a parallelogram are congruent.
C
D
In parallelogram ABCD, ∠A ≅ ∠C and ∠B ≅ ∠D
Place the proper markings in the diagram above, then solve the following problem.
1
Example #2.) If ∠A = 72°, ∠C = x − 48 , ∠B = 108°, ∠D = 7y + 59. Find the value of x and y.
3
Parallelogram Property:
Consecutive angles of a parallelogram are supplementary.
B
A
C
D
In parallelogram ABCD, ∠A and ∠B are supplementary, ∠B and ∠C are supplementary
∠C and ∠D are supplementary, ∠D and ∠A are supplementary.
Example #3.) If ∠C = 2 x + 10 y + 5 , ∠D = − 13 x + 3 y + 16 , ∠B = 4 x + 15 y + 5 , Find the value of x and y
and the value of ∠A.
Accelerated Geometry
Quadrilaterals – Notes
Parallelogram Property:
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Date____________
The Diagonals of a Parallelogram bisect each other
B
C
E
A
D
In parallelogram ABCD, AE ≅ CE and EB ≅ ED .
Place the proper markings in the diagram above, then solve the following problem.
Example #4.) If BE = 4x – 2 and BD = x 2 − x − 26 , find the value(s) of x and the length of ED
Rectangle – a parallelogram in which at least one angle is a right angle
Properties of rectangles:
1. All the properties of a parallelogram apply.
2. All angles are right angles.
3. The diagonals are congruent.
Example #5.) If quadrilateral JKLM is a rectangle, find the value of x. KP = x2, PJ = 7x – 10.
Example #6.) ABCD is a rectangle. Fill in all the missing measurements if you know AD = 14, DB = 50
and m∠ECD = 20 .
Accelerated Geometry
Quadrilaterals – Notes
Name __________________________
Date____________
Rhombus – a parallelogram in which at least two consecutive sides are congruent.
Properties of a rhombus:
1. All the properties of a parallelogram apply.
2. All sides are congruent.
3. The diagonals bisect the angles.
4. The diagonals are perpendicular.
5. The diagonals divide the rhombus into four congruent right triangles.
Example #7.) Use rhombus BCDE and the given information to solve each problem.
a. If m∠EBC = 132.6 , find m∠EBD .
b. If m∠BDC = 25.9 , find m∠EDC .
c. If m∠BEC = 2 x + 10 and m∠CED = 5 x − 20 , find x.
d. If m∠CBD = 2 x + 24 and m∠EBD = x 2 , find x.
Square – a parallelogram that is both a rectangle and a rhombus.
Properties of a square:
1. All the properties of a parallelogram apply.
2. All the properties of a rectangle apply.
3. All the properties of a rhombus apply.
4. The diagonals form four isosceles right triangles.
Example #8.) WXYZ is a square. Fill in all the missing measurements if you know WX = 2 2 .
Accelerated Geometry
Quadrilaterals – Notes
Name __________________________
Date____________
Kite – a quadrilateral with exactly two distinct pairs of congruent consecutive sides.
Properties of kites:
1. The diagonals are perpendicular
2. One diagonal bisects the other. (Usually the longer diagonal bisects the shorter one)
3. One pair of opposite angles is equal.
4. One diagonal bisects the angles.
Example #9.) KX = 12, KE = 24, m∠KEX = 30, m∠XKE = 60 and m∠XIT = 50. Find the following
measures.:
a.) ∠KXI =
b.) ∠KIX =
c.) ∠IKX =
d.) ∠XTE =
e.) ∠XTI =
f.) ∠XET =
g.) ∠EXT =
h.) XE =
i.) TE =
k.) KT =
Trapezoid – a quadrilateral with exactly one pair of parallel sides. The parallel sides are called bases. The
non parallel sides are called legs.
base
3
1
leg
leg
4
2
base
Property of a trapezoid: The leg angles on the same side are supplementary. That means in the trapezoid
above, ∠1 + ∠2 = 180° and ∠3 + ∠4 = 180° .
Accelerated Geometry
Quadrilaterals – Notes
Name __________________________
Date____________
Isosceles Trapezoid – a trapezoid in which the legs are congruent.
1
3
2
4
Properties of an isosceles trapezoid:
1. Any lower base angle is supplementary to any upper base angle.
2. Both pairs of base angles are congruent. So ∠1 ≅ ∠3 and ∠2 ≅ ∠4
3. The diagonals are congruent. So XZ ≅ WY (Draw them in.)
Example #10.) Trapezoid STUV has bases ST and UV . ∠U = x 2 − 2 x , ∠T = x + 108. Find all possible
values of x, m∠U and m∠T .
Median of a Trapezoid =
Now you know everything there is to know about quadrilaterals!