NYS COMMON CORE MATHEMATICS CURRICULUM Unit 1 · 2018-08-10 · 5 Module 1 NYS COMMON CORE MATHEMATICS CURRICULUM Unit 1 Exploratory Challenge 1 You will need a compass and a straightedge

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NYS COMMON CORE MATHEMATICS CURRICULUM Module 1 Unit 1

GEOMETRY

Congruence, Proof and Constructions

NAME ____________________________________

DATE Lesson # Page(s) Topic Practice Assignment

9/7 1 2-3 Vocab and postulates Problem Set Unit 1 Lesson 1

9/8 2 4-6 Construct Equilateral Triangles Problem Set Unit 1 Lesson 2

9/11 3 7-9 Copy Angles and Construct Angle Bisectors Problem Set Unit 1 Lesson 3

9/12 3 Finish Lesson 3 QUIZ

None

9/13 4 10-11 Construct Perpendicular Bisectors None

9/14 4 Finish lesson 4 and In Class assignment Finish Class work

9/15 5 12-13 Constructions of Parallel and Perpendicular lines

Problem Set Unit 1 Lesson 5

9/18 6 14-15 Points of Concurrency Problem Set Unit 1 Lesson 6

9/19 QUIZ None

9/20 7 16-18 Angles and Lines at a point Problem Set Unit 1 Lesson 7

9/21 8 19-21 Transversals

Problem Set Unit 1 Lesson 8

9/22 9 22-23 Auxiliary Lines Problem Set Unit 1 Lesson 9

9/25 10 24 Sum of Angles in A Triangle Problem Set Unit 1 Lesson 10

9/26 11 25-26 Angles in Triangles Problem Set Unit 1 Lesson 11

9/27 Review TICKET IN- Study!!!!

9/28 TEST None

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Lesson 1: Vocab and Postulates with Planes and Lines Term and definition Link to explore

definition Diagram

Point- A precise location or place on a plane. Usually represented by a dot.

http://www.mathopenref.com/point.html

Line- A geometrical object that is straight, infinitely long and infinitely thin.

http://www.mathopenref.com/line.html

Segment- A straight line which links two points without extending beyond them.

http://www.mathopenref.com/linesegment.html

Ray- A portion of a line which starts at a point and goes off in a particular direction to infinity.

http://www.mathopenref.com/ray.html

Collinear- Points that lie on the same straight line.

http://www.mathopenref.com/collinear.html

Plane- A flat surface that is infinitely large and with zero thickness.

http://www.mathopenref.com/plane.html

Coplanar- Objects are coplanar if they all lie in the same plane.

http://www.mathopenref.com/coplanar.html

Circle- The set of points that are all equidistant from a fixed point, the center point.

http://www.mathopenref.com/circle.html

Radius- A line from the center of a circle to a point on the circle.

http://www.mathopenref.com/radius.html





























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Lesson 2: Construct an Equilateral Triangle II

Watch the short animation on constructing an equilateral triangle:

http://www.mathsisfun.com/geometry/construct-equitriangle.html

Based on the animation, list the precise steps used to create any equilateral triangle:

_________________________________________________________________________________________________________

_________________________________________________________________________________________________________

_________________________________________________________________________________________________________

_________________________________________________________________________________________________________

_________________________________________________________________________________________________________

_________________________________________________________________________________________________________

_________________________________________________________________________________________________________

_________________________________________________________________________________________________________

TRY IT! Construct an equilateral triangle using AB as one side of the triangle.

http://www.mathsisfun.com/geometry/construct-equitriangle.html

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Exploratory Challenge 1

You will need a compass and a straightedge

a.) Using the skills you have practiced, construct three equilateral triangles, where the first and second triangles share a common

side, and the second and third triangles share a common side. Clearly and precisely list the steps needed to accomplish this

construction.

CONSTRUCTION:

b.) Continuing with the process same process, construct a regular hexagon (think about how many equilateral

triangles you would need to join together)

c.) Draw a circle such that the hexagon is inscribed inside the circle.

Inscribe:__________________________________________________________________________

d.) Connect three points on the hexagon to form an equilateral triangle. Explain how we would know that the

triangle would have to be equilateral:

e.) What other shapes could you make by joining equilateral triangles?

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Exploratory Challenge 2:

Construct two isosceles triangles one acute isosceles triangle and one obtuse isosceles triangle where AB is the base of the triangle.

Define: Isosceles triangle-_____________________________________________________________

Acute triangle- _______________________________________________________________

Obtuse triangle- _______________________________________________________________

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Lesson 3: Angle Vocab, Copy and Bisect an Angle Term and definition Link to

explore definition

Diagram

Angle- The union of two rays sharing the same endpoint.

http://www.mathopenref.com/angle.html

Degree- 1/360th of a circle. http://www.mathopenref.com/degrees.html

Acute Angle- An angle whose measure is greater than 0o and less than 90o.

http://www.mathopenref.com/angleacute.html

Obtuse Angle- An angle whose measure is greater than 90o and less than 1800

http://www.mathopenref.com/angleobtuse.html

Reflex Angle- An angle whose measure is greater than 180o

http://www.mathopenref.com/anglereflex.html

Adjacent Angles- Two angles that share a common side and common vertex.

http://www.mathopenref.com/anglesadjacent.html

Complementary Angles- Two angles that add to 90o. Often times complementary angles are adjacent in which case they form a right angle.

http://www.mathopenref.com/anglecomplementary.html

Supplementary Angles- Two angles that add to 180o. When the two angles are adjacent they are called linear pairs.

http://www.mathopenref.com/linearpair.html




http://www.mathopenref.com/degrees.html























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Angle Bisector- A ray that divides an angle into two equal angles.

http://www.mathopenref.com/bisectorangle.html

Midpoint- A point on a segment that is equidistant from each endpoint.

http://www.mathopenref.com/midpoint.html

Segment Bisector-A line, ray or segment which cuts another line segment into two equal parts.

http://www.mathopenref.com/bisectorline.html

Video: Watch the video Angles and Trim

While watching the video make note of how to bisect an angle.

Experiment with the angles below to determine the correct steps for the construction.

List the steps you used to bisect the angles:










http://youtu.be/EBP3I8O9gIM

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Critical thinking:

- Constructing an ______________ is fundamentally the same as constructing a whole circle.

- The angle bisector could also be called the line of ________________________ since the same procedure was done to

both sides of the angle.

Example 2: Investigate How to Copy an Angle

You will need a compass and a straightedge.

Together with a partner, copy the angle onto the blank space below by following the steps given:

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A B

Lesson 4: Construct a Perpendicular Bisector

Term and definition Link to explore

definition

Diagram

Equidistant-A point P is equidistant from others if it is the same distance from them.

http://www.mathopenref.com/eq

uidistant.html

Perpendicular-A line is perpendicular to another if it meets or crosses it at right angles (90°)

http://www.mathopenref.com/perpendicular.html

Perpendicular Bisector-A line which cuts a line segment into two equal parts at 90°.

http://www.mathopenref.com/bisectorperpendicu

lar.html

Construct a perpendicular bisector of a line segment using a compass and straightedge. Using what you know about the

construction of an angle bisector,

Precisely describe the steps you took to bisect the segment.

http://www.mathopenref.com/equidistant.html






http://www.mathopenref.com/bisectorperpendicular.html




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A B

C

D

E

Discussion:

Watch the animation of a perpendicular bisector here (http://www.mathsisfun.com/geometry/construct-

linebisect.html). Are the steps taken in the animation the same or different than the steps above?

Now that you are familiar with the construction of a perpendicular bisector, we must make one last observation. In

the diagram CE is the perpendicular bisector of AB. Using your compass, examine the following pairs of segments:

I. 𝐴𝐶̅̅ ̅̅ , 𝐵𝐶̅̅ ̅̅

II. 𝐴𝐷̅̅ ̅̅ , 𝐵𝐷̅̅ ̅̅

III. 𝐴𝐸̅̅ ̅̅ , 𝐵𝐸̅̅ ̅̅

Based on your findings, fill in the observation below.

Observation:

Any point on the perpendicular bisector of a line segment is

_____________________ from the endpoints of the line segment.

What kind of triangles are triangles ACB, ADB, and AEB? ______________________________________________

http://www.mathsisfun.com/geometry/construct-linebisect.html

http://www.mathsisfun.com/geometry/construct-linebisect.html

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A

Lesson 5: Constructions of Perpendicular and Parallel lines

Mathematical Modeling Exercise

You know how to construct the perpendicular bisector of a segment. Now you will investigate how to construct a perpendicular

to a line ℓ from a point 𝐴 not on ℓ. Think about how you have used circles in constructions so far and why the perpendicular

bisector construction works the way it does. Watch the animation (http://www.mathsisfun.com/geometry/construct-

perpnotline.html) to see the construction, complete the construction on your own and write the steps.

Steps:

ℓ

http://www.mathsisfun.com/geometry/construct-perpnotline.html

http://www.mathsisfun.com/geometry/construct-perpnotline.html

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Discussion:

What other shapes can we construct now that we know how to make parallel and perpendicular lines?

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Lesson 6: Points of Concurrency

Discussion

1.) When three or more lines intersect in a single point, they are _____________________, and the point of intersection is the

point of _____________________________.

2.) The point of concurrency of the three perpendicular bisectors is the _________________________________________ of the

triangle.

EXPLORE: We will use http://www.mathopenref.com/trianglecircumcenter.html to explore what happens when the triangle is

right or obtuse. Sketch the location of the circumcenter on the triangles below:

3.) The circumcenter of △ 𝐴𝐵𝐶 is shown below as point 𝑃.

Mark all of the right angles in the diagram.

Mark the congruent segments in the diagram.

http://www.mathopenref.com/trianglecircumcenter.html

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EXAMPLE 2

Use the triangle below to construct the angle bisectors of each angle in the triangle.

The construction of the three angle bisectors of a triangle also results in a point of concurrency, which we call the

_________________________________.

EXPLORE: We will use http://www.mathopenref.com/triangleincenter.html to explore what happens when the triangle is right or

obtuse. Sketch the location of the incenter on the triangles below:

The triangle below shows an incenter. Using the incenter, draw a circle inscribed inside the triangle.

A

B C

http://www.mathopenref.com/triangleincenter.html

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Lesson 7: Solve for Unknown Angles—Angles and Lines at a Point

Opening Exercise- Determine the measure of the missing angle in each diagram.

Types of Angles Link Write an equation using the angles:

Vertical Angles- A pair of non-adjacent

angles formed by the intersection of

two straight lines

http://www.mathopenref.com/angles

vertical.html

Angles Forming a Right angle-

Angles Forming a Straight line-

Angles around a Point-



http://www.mathopenref.com/intersection.html

http://www.mathopenref.com/anglesvertical.html

http://www.mathopenref.com/anglesvertical.html

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Example 1

Find the measures of each labeled angle. Give a reason for your solution.

Angle Angle

measure Reason

a

b

c

d

e

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Exercises

In the figures below, 𝐴𝐵̅̅ ̅̅ , 𝐶𝐷̅̅ ̅̅ , and 𝐸𝐹̅̅ ̅̅ are straight line segments. Find the measure of each marked angle or find the unknown

numbers labeled by the variables in the diagrams. Give reasons for your calculations. Show all the steps to your solution.

1. 2.

3. 4.

5. 6.

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Lesson 8: Solve for Unknown Angles—Transversals

Given a pair of lines 𝐴𝐵 ⃡ and 𝐶𝐷 ⃡ in a plane (see the diagram below), a third line 𝐸𝐹 is called a transversal if it intersects 𝐴𝐵 at a

single point and intersects 𝐶𝐷 ⃡ at a single but different point.

Follow the link to explore the definitions and fill in the chart:

http://www.mathopenref.com/tocs/paralleltoc.html

Pairs of Angles

Identify a pair of angles in the above diagram for each term and write an equation using the pair of angles

Corresponding Angles-

Alternate Interior Angles-

Alternate Exterior Angles-

Same Side Interior Angles (See Interior Angles of a Transversal)

EXAMPLE 1

http://www.mathopenref.com/tocs/paralleltoc.html

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EXAMPLE 2

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LESSON 9: AUXILARY LINES and angles in triangles

Sometimes adding a __________________ line to a picture can be helpful.

In this case we call the line an _______________________________________.

In this figure, we use an auxiliary line to find the measures of ∠𝑒 and ∠𝑓

(how?), then add the two measures together to find the measure of ∠𝑊.

What is the measure of ∠𝑊?

Note: An auxiliary line can be drawn anywhere in the diagram needed but is usually most helpful when drawn parallel

to and between the two given parallel lines.

Exercises

In each exercise below, find the unknown (labeled) angles. Give reasons for your solutions.

1. 2.

3. 4.

5. 6.

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7.

8.

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Lesson 10: Sum of Angles in a Triangle

Exercises

In each figure, determine the measures of the unknown (labeled) angles. Give reasons for your calculations.

1.

2.

3.

4.

5. 6.

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Lesson 11: Angles in Triangles

Exercises:

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NYS COMMON CORE MATHEMATICS CURRICULUM Unit 1 · 2018-08-10 · 5 Module 1 NYS COMMON CORE MATHEMATICS CURRICULUM Unit 1 Exploratory Challenge 1 You will need a compass and a straightedge