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STANDARDS OF LEARNING

CONTENT REVIEW NOTES

GEOMETRY

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rd Nine Weeks, 2016-2017

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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.

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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).

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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.

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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?

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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

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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.

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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)

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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)

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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.

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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 )

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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.

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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.

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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.

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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

.

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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.

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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.

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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.

°

°

°

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Answers to the problems: Polygons

1.

2.

3.

Quadrilaterals 1.

2.

3. trapezoid

4. rhombus

Circles

1. 2.

3.

4. 5. 6. 7.