Name: _________________________________________
Date: _______________
Regents Review #4
Polygons:
Polygon – a closed figure that is the union of
line segments in a plane.
Convex Polygon - a polygon in which each of the
interior angles measures less than 180 degrees.
Concave Polygon- a polygon in which at least one
interior angle measures more than 180 degrees.
Regular Polygon – All sides and all angles are
congruent
n-gon: A polygon with n number of sides.
Interior Angle Sum:
One Interior Angle (for regular polygons):
Exterior Angle Sum: 360
One Exterior Angle (for regular polygons):
Properties of a Parallelogram:
 Opposite sides are congruent
 Opposite sides are parallel
 Opposite angles are congruent
 Consecutive angles are supplementary
 Diagonals divide the parallelogram into 2 congruent triangles
 Diagonals bisect each other
Properties of a Rectangle:
 Has all the properties of a parallelogram
 Has 4 right angles (equiangular)
 Diagonals are congruent
Properties of a Rhombus:
 Has all the properties of a parallelogram
 Is equilateral (all sides are congruent)
 Diagonals are perpendicular to each other
 Diagonals bisect the angles
Properties of a Square:
 Has all the properties of a rectangle
 Has all the properties of a rhombus
Properties of a Trapezoid:
 Two and only two sides are parallel
 The measure of two same side interior angles are supplementary
Properties of an Isosceles Trapezoid:
 Has all the properties of a trapezoid
 Base angles are congruent
 Legs are congruent
 Bases are parallel
 Diagonals are congruent
base
leg
leg
base
The median of a trapezoid is a line segment whose endpoints are the midpoints of the
nonparallel sides of the trapezoid.
 Parallel to both bases
 Has the length equal to the
average of the length of the
bases
Ways to Prove Parallel Lines:
Euclidean:
 A pair of alternate interior angles are congruent
 A pair of corresponding angles are congruent.
 A pair of same side interior angles are supplementary
 Both lines are perpendicular to the same line
 Both lines are parallel to the same line.
Coordinate: slopes are equal
Ways to Prove Perpendicular Lines:
Euclidean:
 The two lines intersect to form right angles.

The two lines intersect to form congruent adjacent angles.

Each of two points on one line is equidistant from the endpoints of another line
segment
Coordinate: slopes are negative reciprocals
Proving Quadrilaterals:
Methods for proving a Parallelogram:
1. Both pairs of opposite sides are congruent
2. Both pairs of opposite sides are parallel
3. One pair of opposite sides is congruent and parallel
4. Both pairs of opposite angles are congruent
*5. Diagonals bisect each other
Methods for proving a Rhombus:
First you must prove that the quadrilateral is a parallelogram:
1. with two consecutive sides congruent
2. with one diagonal bisecting an angle
*3. with diagonals perpendicular to each other
Methods for proving a Rectangle:
First you must prove that the quadrilateral is a parallelogram:
*1. with one right angle
2. with a pair consecutive angles congruent
3. with diagonals congruent
Steps to prove a quadrilateral is a Square:
1. Prove it’s a parallelogram
2. Prove it’s a rhombus
3. Prove it’s a rectangle
Methods for proving a Trapezoid:
1. One and only one pair of opposite sides are parallel
Methods for proving an Isosceles Trapezoid:
First you must prove that the quadrilateral is a trapezoid:
*1. with nonparallel sides congruent
2. with base angles congruent
3. with diagonals congruent
*Easiest/Recommended method for coordinate geometry proofs.
Coordinate Geometry Proofs:
Use the midpoint, slope and distance formulas.
What you need to Prove:
2 line segments bisect each other
2 lines are perpendicular to each other
2 lines are parallel to each other
2 line segments are congruent
1 line is a perpendicular bisector of a given line
segment
Isosceles Triangle
Right Triangle
Congruent Triangles
Similar Triangles
Slope Formula: m 
y 2  y1
x 2  x1
 x  x 2 y1  y 2 
,

Midpoint Formula: M   1
2 
 2
Distance Formula:
d
x2  x1 2   y2  y1 2
What formula do we use?
Midpoint formula (twice)
Slope (twice) [negative reciprocals]
Slope (twice) [same slope]
Distance (twice)
Midpoint and slope
Distance (3 times)
Slope (3 times)
Distance (6 times)
Distance (6 times)
Part I:
_____1) How many degrees are in one interior angle of a decagon?
(1) 144°
(2) 1,080°
(3) 1,440°
(4) 180°
_____2) Find the measure of one exterior angle of a regular octagon.
(1) 30°
(2) 45°
(3) 60°
(4) 135°
_____3) If the sum of two exterior angles of a triangle equal 249°, then what is the
measure of the third exterior angle?
(1) 111°
(3) 249°
(2) 180°
(4) 360°
_____4) In parallelogram ABCD, the ratio of the measures of angles A and B is 3:7. Find
the measure of A.
(1) 26°
(2) 54°
(3) 60°
(4) 126°
_____5) In parallelogram PQRS, <PQR = 76°. If <RSP is opposite <PQR, then what is the
measure of <RSP?
(1) 76°
(3) 180°
(2) 104°
(4) 360°
_____6) The diagonals of a parallelogram are congruent and they bisect the angles. Which
type of special parallelogram is it?
(1) rectangle
(2) rhombus
(3) square
(4) trapezoid
_____7) In rectangle ABCD, the diagonals intersect at E. If AE = 3x + y, BE = 4x - 2y, and
CE = 20, find x and y.
(1) x = 3, y = 11
(2) x = 7, y = -1
(3) x = 7, y = 4
(4) x = 6, y = 2
_____8) If one base of a trapezoid is 15 inches and the other base is 25 inches, what is
the length of the median?
(1) 20 inches
(2) 30 inches
(3) 40 inches
(4) 50 inches
_____9) Which of the following is not sufficient to prove a trapezoid is isosceles?
(1) The diagonals are congruent.
(2) The legs are congruent.
(3) Consecutive angles are supplementary.
(4) Base angles are congruent.
_____10) Quadrilateral QUAD has diagonals that bisect each other, are congruent, and
perpendicular. What type of quadrilateral is QUAD?
(1) kite
(2) rectangle
(3) square
(4) rhombus
_____11) The diagonals of a rhombus are perpendicular and bisect
(1) 2 opposite interior sides
(2) 2 opposite exterior angles
(3) 2 adjacent angles
(4) 2 opposite angles
_____12) Which of the following formulas would be used to prove that the diagonals of a
square are perpendicular?
(1) Distance Formula
(3) Slope Formula
(2) Midpoint Formula
(4) Slope-Intercept Formula
Part II:
13) The diagonals of rectangle RSTV intersect at Q. If VQ = (3x – 3) and RT = (5x – 1),
find the value of x.
14) In the accompanying diagram, isosceles trapezoid CDEF has bases of lengths 8 and 18
and an altitude of 12. Find CD.
D
8
E
12
C
18
F
Part III:
17) Quadrilateral MIKE has vertices M(-1,0), I(5, -2), K(3,4), and E(-3,6). Prove that
quadrilateral MIKE is a rhombus and not a square.
18) Given quadrilateral ABCD, A(-3, 0), B(9,0), C(6, 6), D(0, 6). Prove that quadrilateral
ABCD is an isosceles trapezoid.
Part IV:
19) In the diagram ABD  CDB. Prove, that ABCD is a parallelogram.
A
B
D
REASON
STATEMENT
REASON
C
20) Given: ABCD is a parallelogram, 1  2
Prove: AECD is an isosceles trapezoid
D
C
1
A
STATEMENT
B
2
E