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Name: __________________________

Geometry Period _______

MIDTERM REVIEW BOOKLET

M-1: Unit 1 and Unit 2



In this unit you must bring the following materials with you to class every day:

Calculator

Pencil

A compass

This Booklet

A device

Headphones!

Please note:

You will be receiving additional “Mock Midterms” to work on to get mixed

practice for your midterm exam. The keys for these will be online.

Each section of this booklet must be completed the day it is assigned

Answer keys will be posted as usual for each daily assignment on our website

Geometry Midterm Information Date: Thursday; 1/24

When: 12 pm (ARRIVE EARLY) – Starts at 12:15

Where: TBD – We’ll tell you in class

What to bring: Calculator, straightedge, compass, pen and pencil

The Test: The midterm is 2 hours, you are allowed to leave after 1.5 hours

MULTIPLE CHOICE:

Part 1: 16 questions - 2 points (SCANTRON)

SHORT ANSWER:

Part 2: 3 questions - 2 points



You must answer ALL parts of the test; you aren’t allowed to leave unless you answer/attempt to answer all questions.

This means no questions blank!

There is some scrap graph paper for your use in your exam booklet.

Other important information:

Please leave all books, book bags, and papers in your locker.

DEVICE POLICY- All Students are PROHIBITED from bringing cell phones and all other devices into a

classroom or other location where a State test is being administered. This includes Apple Watch/smart

watch

Get a good night’s sleep and a get something to eat before the exam! If you prepare during our

review sessions, you will be fine!

Prepare, PRACTICE AND CHECK as much as you can!

Good Luck! your proud Geometry Teachers

M-1 Unit 1 and Unit 2 Review

Midterm Review: Unit 1 (Rigid Motions) AND Unit 2 (Constructions)

Unit 1- Rigid Motions

1-1 Rigid Motions and transformations o Rigid Motions produce congruent figures. o Translation, Rotation, Reflections are all rigid motions o Rigid Motions preserve size, shape and angle measure, they only change the position of a figure

1-2 Translations

o Ta,b

o a →how to move your pre-image left/right o b →how to move your pre-image up/down o Vectors are drawn from pre-image to image and show distance and direction of the slide.

1-3 Reflections Notation: r Steps: 1. Graph line of reflection 2. Count how far away each point is on the line and count the opposite going the other way

1-4 Special reflections o Point Reflections o YOU MUST MEMORIZE ry=x and ry=-x

1-5 Special compositions o Perform Compositions from right to left! o Composition of reflections over parallel lines are the same as one translation

1-7 Rotations and more special compositions Notation: R o Either know your rules, or Rotate paper! o Rotate counterclockwise for positive angles, and clockwise for negative angles! o Composition of reflections over perpendicular lines are the same as one rotation

1-8 Rotational Symmetry

Order (How many “clicks” in one full revolution)

360

𝑛 to find least amount of degrees to map onto itself- Same formula to fins one exterior angle

3 Transformations are Rigid Motions:

________________ ________________ ________________

Notation: Notation: Notation:

Switch coordinates

Switch and negate coordinates

1. What is the image of A(-4,0), B (-1,3) , C(-4,3) under ry= - x? Show your work!

2. Graph segment 𝑅𝑆̅̅̅̅ with vertices R(4, 1) and S(6, 3) and its image after a 270° rotation about the origin. Show your work!

3. Consider the following composition: rx-axis ∘ T2,-3. Explain how you would perform this composition.

(You may want to bullet/number your steps to be more organized)!

3. a) Write the rule using appropriate detailed composition notation that describes the composition shown

below.

b) Is ∆𝐴𝐵𝐶 ≅ ∆𝐴"B"𝐶"? Use your knowledge of rigid motions to fully describe your answer.

5. Consider the regular octagon below:

a) Does the following figure have rotational symmetry?

b) What is the least amount of degrees you must rotate the octagon so that it maps onto itself?

Unit 2 Constructions

2-1 Construct an Inscribed Regular Hexagon and Inscribed equilateral triangle. -Measuring radius distance to make arcs. -Properties of equilateral triangle- All interior angles measure 60 degrees.

Construction looks like:


2-2

Construct an equilateral triangle GIVEN side length.

Construct a perpendicular bisector GIVEN side length. -Measuring side length to make congruent segments. Construct an inscribed Square. -Properties of squares- Diagonals are perpendicular Bisectors. -Diagonal of a square is the diameter of the circle around it.




2-3 Constructing a Circumscribed circle ( or inscribed triangle) -Circumcenter – Point of concurrence where 2 perpendicular Bisectors cross -Properties of the circumcenter (Where is it if triangle is acute, right, Obtuse?). Equidistant from vertices.


2-4 Perpendicular lines through points off and on the line. Construct an Altitude -Create a SEGMENT first, then perpendicular bisector. -Definition of Altitude use construction of perp. line through a point off the line to help you construct an ALTITUDE. Constructing a square with given side length. -Extend a side perpendicular Bisector Measure lengths.



2-5 Constructing an angle bisector. -Construct a 30 degree angle. -Construct a 45 degree angle.


2-6 Constructing an inscribed circle.

-Properties of Incenter ( Equidistant from sides, formed by angle bisector. -Construct the incenter (center of the circle), construct a line perp. to a side through incenter (radius length-incenter to midpt)


Support video links

Inscribed Hexagon

Perpendicular Bisector Construction

Angle Bisector Construction

Square of given side length

Equilateral triangle of given side length

Perpendicular through a Point NOT on a Line

*note, these videos are also availabe on edpuzzle from earlier in the year

Practice Constructions

Construct an Equilateral Triangle whose side length is the length of AB

Construct an Equilateral triangle whose sides are the same length as AC on the line below.

Construct a line Perpendicular to the given line through point p

Construct an Altitude in the following triangle through vertex B

Construct a square inscribed in circle Construct the midpoint of side AC, label it M

Construct an inscribed Hexagon Construct an Inscribed Equilateral Triangle

Using this triangle, construct a 30° angle with its vertex at A. (Leave all construction marks.) LABEL THIS ANGLE!

Using a compass and a straightedge, construct a 45° angle with its vertex at the midpoint of segment 𝐴𝐵̅̅ ̅̅ . (Leave all construction marks.) LABEL THIS ANGLE!


3-1

Writing equations of lines.

Determining slope and y intercept given an equation 𝑦 = 𝑚𝑥 + 𝑏

Writing the equation of a line given a graph.

Graphing Linear Equation

3-2 Point Slope Form y − y1 = m(x − x1)

3-3 Using slope and y intercept to determine if line are parallel, perpendicular, or the same line.

Same slope Parallel

Opposite reciprocal slope Perpendicular

Same slope and y-intercept same line

3-4 Writing equations of lines.

Parallel lines Same slope as original, use one point on the line then sub into 𝑦 − 𝑦1 = 𝑚(𝑥 − 𝑥1)

Perpendicular lines opposite reciprocal slope of original, use point on the line then sub into 𝑦 − 𝑦1 =𝑚(𝑥 − 𝑥1)

3-5 Slope of a line using slope formula

𝒎 = 𝒚𝟐−𝒚𝟏

𝒙𝟐−𝒙𝟏

Watch out for common mistakes!

1. Y’s are on top!

2. Double negatives! Use your CALCULATOR!

3. Simplify fractions!

3-6 Distance and Midpoint

Distance or length: 𝒅 = √(𝒙𝟐 − 𝒙𝟏)𝟐 + (𝒚𝟐 − 𝒚𝟏)𝟐

Midpoint Formula: 𝑴 = ( 𝒙𝟏+𝒙𝟐

𝟐,

𝒚𝟏+𝒚𝟐

𝟐)

Watch out for common mistakes!

1. Take it slow, show all steps.

2. Don’t forget parentheses and comma for Midpoint!

3-7 Writing equation of perpendicular Bisector.

Find slope of given line, take the opposite reciprocal. Calculate Midpoint of given line. Sub the new slope and

midpoint into 𝑦 − 𝑦1 = 𝑚(𝑥 − 𝑥1) make sure you sub in MIDPOINT. Not the points on line

3-8 QUIZ- Make sure you look at all your mistakes!

Simplify Radicalsmake prime factor treepairs go outside multiply by anything already outside.

3-9 Angles and notation

Vertical angles Across from each other, always congruent

Angles at a point Make a full circle add up to 360 degrees.

Linear pairs Adjacent and supplementary add up to 180 degrees.

Supplementary angles two angles that add up to 180 degrees.

Complementary anglestwo angles that add up to 90 degrees.

3-10 Angles on a transversal

Alt. Interior, Alt. Exterior, Corresponding angles are ≅ if the lines are ∥.

Same-side Int., Same side Ext. are supplementary if the lines are ∥.

3-11 Justifying parallel lines

When justifying parallel lines you need to include 1. Type of angle pair 2.Relationship 3. Conclusion

3-12 Construction of parallel lines (with justification)

-Construction works because we are copying an angle into the “corresponding” position on the new line.

You MUST know

the big 3

formulas!

Unit 4 Triangle Theorems and Rules Review

Part 1: Triangle Theorems and Rules

Name of relationship In words/ Symbols Diagrams/ Hints/ Techniques

1. Side angle relationship

The longest side is across from the largest angle.

The medium length side is across from the medium-sized angle.

The shortest side is across from the smallest angle

1. Solve for ALL angles in the triangle. 2. Draw arrows!

2. Triangle inequality Theorem

The sum of the lengths of the two smaller sides of a triangle is greater than the length of the largest side.

To find a range of possible sides, add two given sides, subtract them.

Add up the two smaller sides and compare to the largest side. If the sum is greater, it’s a triangle! 2,3,4 (2+3) > 4 ? yes!

3. Pythagorean Theorem to find a missing side. Pythagorean Inequality to Classify triangles

a) c2 = a2 + b2 C is longest side ( hypotenuse-across from the right angle)

b) If c2 < a2 + b2 it is acute

If c2 > a2 + b2 it is obtuse

If c2 = a2 + b2 it is right

Remember, c2 must be on the left of =

If you see a right angle, it’s a right triangle! use Pythagorean Theorem to solve for a missing side. WATCH OUT! If asked “does this make a triangle” you must use Theorem # 2- NOT PYTHAGOREAN. ERROR ALERT: you must SQUARE (power of 2) a, b and c, DIFFERENT FROM Theorem #2 seen above

4.Isosceles triangle Theorem

The base angles of an isosceles triangle are equal in measure. The sides opposite the base angles in an isosceles triangle (called legs) are equal in length.

Angles opposite are congruent! If you see expressions, make them equal to each other!

5. Exterior angle theorem

The measure of an exterior angle of a triangle is equal to the sum of the measures of the two remote interior angles of the triangle

IN + IN =OUT

Unit 4 Triangle Theorems and Rules Review Continued

Part 2- Segment in a triangle and Angle Relationships

Name of relationship In words/ Symbols Diagrams/ Hints/ Techniques 1. Exterior angles in a polygon. a). ONE Exterior b) Sum of Exterior

a) 360

𝑛 where n is the # of sides

b) Sum is AlWAYS 360 degrees( NO MATH NEEDED)

Exterior angles and formed by extending a side of the triangle.

2. Interior angles in a

polygon. a) ONE Interior b) Sum of interior

a) The supplement of on Exterior angle-they are linear pairs!--> Exterior + Interior = 180 b) Number of △ ′𝑠 times 180. ( n-2)180

Remember! One exterior and one interior angle add up to 180 degrees!

3. Segments in a triangle: Medians- Goes to the midpoint of the opposite side creating two equal segments Altitudes-Are perpendicular to the opposite side creating right angles Perpendicular bisectors- Goes to the midpoint of opposite side and is perpendicular to it. Angle bisectors- bisects the angle at the vertex it goes through making 2 congruent angles. ** In Isosceles and Equilateral triangles these segments coincide!

4. Points of concurrence.

2 or more: medians Centroid : Always inside the triangle. Cuts each median into a 2:1 ratio Altitudes Orthocenter: Inside for acute triangles, on the triangle for right triangles and outside for obtuse triangles. angle bisectors Incenter: Always inside the triangle. perpendicular bisectors Circumcenter: Inside for acute triangles, on the triangle for right triangles and outside for obtuse triangles.

ALL OF MY CHILDREN ARE BRING IN PEANUT BUTTER

COOKIES.

5.Centroid and Ratios Centroid cuts every median into a 2:1 ratio. Use this ratio to set up equation 2x + 1x= whole length of median.

Read carefully- What is the segment they want? Sometimes you need to substitute back in!

6. Auxiliary Lines Use with non-traditional diagrams.

Use with non-traditional diagrams.

Extend lines to help you find: Linear pairs, special angle pair relationships, vertical angles, triangles.

Unit 3 Practice:

1. Write the equation of the perpendicular bisector of GH, given that G(2,-1) and H(10, -3).

2. Solve for x.

3. The midpoint of AB is M 4,2( ) . If the coordinates of A are 6,-4( ) , what are the coordinates of B?

4. State whether the lines represented by the equations 𝑦 = 1

2𝑥 + 11 and 224 xy are parallel,

perpendicular, neither, or the same line. Justify your answer.

Unit 4: Triangles Questions

5. Solve for x

6. In the diagram below of DACE , medians AD ,EB and CF intersect at G. The length of FG is 12 cm.

What is the length, in centimeters, of GC?

7. Solve for the measure of g.

8. In the accompanying diagram, isosceles ∆𝐴𝐵𝐶 is congruent to isosceles ∆𝐷𝐸𝐹. 𝑚∠𝐶 = 5𝑥, 𝑚∠𝐷 = 2𝑥 + 18.

Find 𝑚∠𝐵 and 𝑚∠𝐵𝐴𝐺

9. Use the diagram below to answer the following:

a) Solve for and label in diagram the angles of ∆SAT.

State and determine the largest side of ∆SAT.

b) Solve for and label in diagram the angles of ∆RSA.

State and determine the smallest side of ∆RSA.

c) explain why RA||ST

10. Given the rectangle at the right, with diagonal 19 inches and height 10 inches. Find the width of the

rectangle to the nearest inch.


UNIT 5: TRIANGLE CONGRUENCY PROOFS - Concept Sheet

Concept Key Ideas/Tips

Beginning a Proof

*mark your picture *annotate question (givens)

*use the tools from your tool box *1st write your givens

*last step = what you're trying to prove *make a plan

*use a checklist for your shortcuts! *# your steps!

*All of your givens and markings should be a step in your proof!

Triangle Congruency Shortcuts

*WE CANNOT USE AAA or SSA!!! *

Proving Parts are Congruent

*To prove triangles are congruent, you need to use a shortcut. *To prove parts (sides and angles) of triangles are congruent, we: 1st: prove triangles are congruent 2nd: use CPCTC (Corresponding Parts of Congruent Triangles are Congruent)

Proving Definitions

Addition Postulate

With Substitution

Subtraction Postulate

With Substitution

Transitive Property

AB = 5; 𝐴𝐵̅̅ ̅̅ ≅ 𝐷𝐸 ̅̅ ̅̅ ̅ ; 𝐷𝐸̅̅ ̅̅ ≅ 𝐺𝐻̅̅ ̅̅ So, AB = GH (by the TRANSITIVE PROPERTY)

Using Supplements

<1 ≅ < 4

So, <2 ≅ <3 (since 2 is the supplement of <1 and <3 is the supplement of < 4)

IF AB = EF

CD = CD (Reflexive Property)

Then, AB + CD = EF + CD (Addition Postulate)

AD = ED (Substitution Postulate)

=

IF AC = BD,

BC = BC (Reflexive Property)

Then, AC- BC = BD - BC (Subtraction Postulate)

AB = CD (Substitution Postulate)

Unit 5 Practice 1. As shown in the diagram below, AC bisects ∠𝐵𝐴𝐷 𝑎𝑛𝑑 ∠𝐵 ≅ ∠𝐷. Which method could be used to prove ∆𝐴𝐵𝐶 ≅ ∆𝐴𝐷𝐶?

1) SSS 2) AA 3) SAS 4) AAS

2. Given: Triangle ABC is isosceles; BD bisects ∠𝐴𝐵𝐶. Prove: 𝐴𝐷 ≅ 𝐶𝐷.

3. Given diagram below and CE≅ BF , DE // AB , and AB ≅ DE .

Prove that : ∆𝐴𝐵𝐶 ≅ ∆𝐷𝐸𝐹

4. In the diagram of quadrilateral ABCD below ∆𝐴𝐵𝐷 ≅ ∆𝐷𝐶𝐴

Determine which statements are true and which are false.

𝑎) < 𝐴𝐵𝐶 ≅ < 𝐷𝐶𝐵 true false

b) < 𝐴𝐵𝐷 ≅ < 𝐷𝐶𝐴 true false

c)𝐷𝐶 ≅ 𝐶𝐵 true false

d) 𝐷𝐵 ≅ 𝐴𝐶 true false

Reflecting on the:

unit 6 Quadrilaterals test

1. Make a study guide on the questions you got wrong/big ideas you find important from our most recent

unit, Unit 6: Quadrilaterals.

2. Additionally, make test corrections in this space as well.

Add any papers onto this booklet for notes to yourself!

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