2
OVERVIEW
Geometry Content Review Notes are designed by the High School Mathematics Steering Committee as a
resource for students and parents. Each nine weeks’ Standards of Learning (SOLs) have been identified and a
detailed explanation of the specific SOL is provided. Specific notes have also been included in this document
to assist students in understanding the concepts. Sample problems allow the students to see step-by-step models
for solving various types of problems. A “ ” section has also been developed to provide students with the
opportunity to solve similar problems and check their answers.
The document is a compilation of information found in the Virginia Department of Education (VDOE)
Curriculum Framework, Enhanced Scope and Sequence, and Released Test items. In addition to VDOE
information, Prentice Hall Textbook Series and resources have been used. Finally, information from various
websites is included. The websites are listed with the information as it appears in the document.
Supplemental online information can be accessed by scanning QR codes throughout the document. These will
take students to video tutorials and online resources. In addition, a self-assessment is available at the end of the
document to allow students to check their readiness for the nine-weeks test.
To access the database of online resources scan this QR code
The Geometry Blueprint Summary Table is listed below as a snapshot of the reporting categories, the number of
questions per reporting category, and the corresponding SOLs.
5
Polygons G.10 The student will solve real-world problems involving angles of polygons. Polygons A convex polygon is defined as a polygon
with all its interior angles less than 180°.
This means that all the vertices of the polygon will point
outwards, away from the interior of the
shape.
A non-convex (concave) polygon is
defined as a polygon with one or more interior
angles greater than 180°. It looks like a vertex has been 'pushed in' towards the inside of the polygon.
A regular polygon is a polygon that is
equiangular (all angles are equal in measure)
and equilateral (all sides have the same
length).
A regular polygon is a polygon that is equiangular (all
angles are equal in measure) and
equilateral (all sides have the same
length).
6
Interior and
Exterior Angles
n - # of sides
Sum of measures
of interior s
Each Interior Angle
Sum of
the Exterior Angles
Each Exterior Angle
REGULAR
POLYGONS
IRREGULAR POLYGONS
Will vary based on the
algebraic or numerical
expressions
Supplementary to
each of the
corresponding
interior angles
Example 1: Given a regular nonagon (9 sided convex polygon), what are the following measures?
a. The sum of the interior angles
b. Each interior angle
c. The sum of the exterior angle
d. Each exterior angle
Example 2: What are the values of x and y ?
a.
7
The interior angle, (5x + 5), and exterior angle, y, are supplementary. Therefore,
Example 3: Each interior angle of a regular polygon is . How many sides does the polygon have?
Polygons
1. What are the interior and exterior angle measures of a regular heptagon?
2. Given the 8-sided convex polygon, What is the value of n ?
3. Each interior angle of a regular polygon is . How many sides does the polygon have?
8
Quadrilaterals G.9 The student will verify characteristics of quadrilaterals and use properties of
quadrilaterals to solve real-world problems. Example 1: ABCD is a parallelogram, solve for y.
Given:
Properties of Quadrilaterals
Quadrilateral Properties
Parallelogram
Opposite Sides are Congruent
Consecutive Angles are Supplementary
Opposite Angles are Congruent
Diagonals Bisect Each Other
Rhombus
A parallelogram with 4 congruent sides
Diagonals are perpendicular
Each diagonal bisects opposite angles
Rectangle A parallelogram with 4 right
angles
Diagonals are congruent
Square A parallelogram with 4 congruent
sides and 4 right angles
Trapezoid
Exactly one pair of parallel sides
Midsegment is parallel to bases
Length of the midsegment is the average of the lengths of the bases
Isosceles Trapezoid
Legs are congruent
Base angles are congruent
Diagonals are congruent
Diagonals of a parallelogram bisect each other. Therefore
9
Example 2: Based on the given information, can you prove that DEFG is a parallelogram? Example 3: Find the measure of the numbered angles in the rhombus.
Quadrilaterals 1. What value of x will make the figure at the right a rectangle? 2. Janet is making a garden in the shape of a rhombus. One pair of opposite angles each measure 70°. What measure does each of the other opposite pair of angles measure? (Hint: Draw a picture.)
The diagonals of a rhombus are perpendicular.
1 + 4 + 32° = 180° 90° + 4 + 32° = 180°
4 = 58°
Triangle Angle-Sum Theorem
3 = 4
4 = 58°
Alternate Interior Angles are Congruent
2 = 32° Diagonals of a rhombus bisect opposite angles.
You can show that by Angle Side Angle.
Because corresponding parts of congruent triangles are congruent you
can show that
. Once you show that both pairs of
opposite sides are congruent, you can say that DEFG is a parallelogram.
Scan this QR code to go to a
video tutorial on Quadrilaterals.
10
It is often easier to classify geometric figures when they are drawn in the coordinate plane. Using slopes, distances and midpoints can help you with this.
Formula Example
Distance Formula
Midpoint Formula
Slope Formula
Find distance from A to B.
A (-2, -1) B (6, 3)
Find the midpoint of AB.
A (-2, -1) B (6, 3)
Find the slope of AB.
A (-2, -1) B (6, 3)
11
Example 4: Is figure TRAP an isosceles trapezoid? Example 5: Is figure GRAM a square?
All of the sides meet at right angles because 1 and -1 are negative reciprocals of each other.
All of the sides will also have the same length ( ), therefore GRAM is a square.
In order to be an isosceles trapezoid, the legs must
be the same length. Therefore must equal .
Use the distance formula to determine if this is true.
Find distance from T to P.
T (-1, 3) P (-3, -2)
Find distance from R to A.
R (4, 3) A (5, -2)
These distances are not the same therefore TRAP is NOT an isosceles trapezoid.
To be a square we must show that all sides are the same length, and that all sides meet at right angles
(are perpendicular to one another). Remember that for two sides to be perpendicular,
their slopes must be negative reciprocals.
Find the slope of AR.
A (1, -3) R (6, 2)
Find the slope of MA.
M (-4, 2) A (1, -3)
Find the slope of GR.
G (1, 7) R (6, 2)
Find the slope of GM.
G (1, 7) R (-4, 2)
12
Quadrilaterals 3. What figure is formed by the points (-1, -3), (1, 2), (7, 3) and (5, 5) 4. What figure is formed by connecting the midpoints of figure RECT? Circles G.11 The student will use angles, arcs, chords, tangents, and secants to
a) investigate, verify, and apply properties of circles; b) solve real-world problems involving properties of circles; and c) find arc lengths and areas of sectors in circles.
The measure of a minor arc is equal to the measure of its corresponding central angle. You can add adjacent arc measures to find the measure of combined arc.
Scan this QR code to go to a video tutorial on Coordinate
Geometry.
You name a circle by its center. This is Circle X (ʘ X).
is a diameter
is a radius
is a chord
is a central angle (an angle whose vertex
is the center of a circle)
is a semicircle (an arc that is half of a circle)
is a minor arc (an arc that is less than a
semicircle)
is a major arc (an arc whose measure is
greater than 180° (a semicircle))
You name a minor arc by its endpoints.
You name a semicircle or major arc by its
endpoints and another point on the arc.
13
Example 1: What is the measure of ?
The circumference of a circle is the measure of the distance around the outside of the circle. The formula for finding the circumference of a circle is
Use the circumference along with arc measure to find the length of a given arc.
Example 2: What is the length of , given ?
because it is a semicircle
therefore
therefore
You could have also found the measure of
by (because )
14
Circles
The area of a circle can be found using the formula .
The sector of a circle is the region that is bounded by two radii. To find the area of the
sector of a circle use the formula
.
Example 3: Find the area of sector BOC. Leave your answer in terms of .
1. Given that and are diameters of , find the measures of all minor arcs of .
2. Given that , find the length of . Express in terms of .
3. What is the circumference of ?
To find the area of a sector we will use the formula
The measure of the arc is 90°, and the radius is 6 in.
15
Sometimes you will be asked to find the area of a segment of a circle. A segment is
made by joining the endpoints of an arc as shown in the picture below the shaded area
is the segment of the circle.
To find the area of the segment, use the radii from its endpoints to form a triangle. Example 4: Find the area of the shaded segment.
Scan this QR code to go to a video tutorial on Areas of
Circles and Sectors.
Let’s start by finding the area of the sector that
includes the shaded segment.
To find the area of the shaded region we need to
subtract the area of the triangle from the area
of the sector.
Area of a triangle is found by the formula
.
The base and height of this triangle are both 10.
The area of the shaded segment is the area of the sector with the area of the triangle subtracted.
16
In the picture below is tangent to . This means that is in the same plane as and intersects the circle in exactly one place. This place is point , and is called the point of tangency. If a line is tangent to a circle, then that line is perpendicular to the radius of the circle.
Example 5: Is tangent to at ?
If is tangent to at then must
be a right triangle.
Use the Pythagorean Theorem to determine
if is a right triangle.
Side is
Since is a right triangle, that means
that is tangent to at .
Scan this QR code to go to a
video tutorial on Tangent Lines.
17
Circles 4. Find the area of the shaded region. Round to the nearest hundredth. 5. Find the radius of .
Theorems about Chords and Arcs
Within a circle, or in congruent circles, congruent central angles have
congruent arcs.
The converse is also true.
Within a circle or in congruent circles, congruent central angles have
congruent chords.
The converse is also true.
Within a circle or in congruent circles, congruent chords have congruent
arcs.
The converse is also true.
°
If , then
.
If , then
.
If , then
.
If , then
.
If , then
If , then
.
18
Within a circle or in congruent circles, chords equidistant from the center or
centers are congruent.
The converse is also true.
In a circle, if a diameter is perpendicular to a chord, then it
bisects the chord and its arc.
In a circle, if a diameter bisects a chord (that is not a diameter), then it
is perpendicular to the chord.
In a circle, the perpendicular bisector of a chord contains the center of a
circle.
Example 6: Given , and . How can you show that
Because the circles are congruent, you can
say that their radii are congruent. Because
the two congruent angles are across from
these radii, you can say that the other
angles across from the radii ( )
are also congruent. If you subtracted the two
“known” angles from 180° you would have
the angle measure of the central angle.
These would have to be the same.
If , then
If , then
.
If is a diameter and
Then and
.
If is a diameter and
.
Then .
If is the
perpendicular bisector
of
Then contains the
center of the circle.
19
Theorems about Angles and Segments
The measure of an inscribed angle is half the measure of its intercepted arc.
The measure of an angle formed by a tangent and a chord is half the measure
of the intercepted arc.
The measure of an angle formed by two lines that intersect inside a circle is half
the sum of the measure of the intercepted arcs.
The measure of an angle formed by two lines that intersect outside of a circle is
half the difference of the measures of the intercepted arcs.
For a given point and circle, the product of the lengths of the two segments from the point to the circle is constant along
any line through the point and circle
Case I Case II Case III
Example 7: Find the value of each variable.
Angle c is a vertical angle with the third angle in the
triangle that includes ’s a and b.
20
Example 8: Find the value of x.
Circles 6. What is the 7. What is the value of x?
Scan this QR code to go to a video tutorial on Angle Measures
and Segment Lengths.
°
°
°