geometry unit 2 note sheets (segments, lines &...

Geometry Unit 2 Note Sheets (Segments, Lines & Angles)

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Date Name of Lesson

1.5 Angle Measure

1.4 Angle Relationships

3.6 Perpendicular Bisector (with Construction)

1.4 Angle Bisectors (Construct and Measurements of

Angle Bisector)

Quiz

3.1 Transversal Measurements

3.1 Parallel Lines with Transversal

3.2 Interior Angles

Quiz

3.2 Exterior Angles

3.2 Corresponding Angles

Class Activity

Quiz

2.6 Algebraic Proofs

2.7, 2.8 Mini Proofs

OC 1.7/3.5 Proofs about Parallel and Perpendicular Lines

Proofs Activity

Quiz

Practice Test

Unit Test


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1.5 Angle Measure Notes

Ray

Opposite Rays

Angle

Sides

Vertex

Naming an Angle

Points on a Plane with an Angle

Guided Practice

Use the map of a high school shown to answer the following.

1. Name all angles that have B as a vertex.

2. Name the sides of ∠3.

3. What is another name for ∠𝐺𝐻𝐿?

4. Name a point in the interior of ∠𝐷𝐵𝐾.


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Your Turn

5. Name all angles that have B as a vertex.

6. Name the sides of ∠5.

7. Write another name for ∠6.

Degree

Classify Angles

right angle acute angle obtuse angle


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Guided Practice

Classify each angle as right, acute, or obtuse. Then use a protractor to measure the angle to the nearest degree.

8. ∠𝑀𝐽𝑃

9. ∠𝐿𝐽𝑃

10. ∠𝑁𝐽𝑃

Your Turn

11. ∠𝑇𝑌𝑉

12. ∠𝑊𝑌𝑇

13. ∠𝑇𝑌𝑈


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1.4 Angle Relationships Notes

Special Angles Pairs

Name and Definition Examples Nonexamples

Adjacent Angles

Linear Pair

Vertical Angles

Guided Practice

Name an angle pair that satisfies each condition.

1. two acute adjacent angles

2. two obtuse vertical angles

Your Turn

3. two angles that form a linear pair

4. two acute vertical angles


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Angle Pair Relationships

Vertical Angles

Complementary Angles

Supplementary Angles

Linear Pair

Guided Practice

5. Find the measures of two supplementary angles if the measures of one angles is 6 less than five times

the measure of the other angle.

Your Turn

6. Find the measures of two supplementary angles if the difference in the measures of the two angles is 18.


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Perpendicular Lines

Guided Practice

7. Find x and y so that 𝑃𝑅 ⃡ and 𝑆𝑄 ⃡ are perpendicular.

Your Turn

8. Find x and y so that 𝐾𝑂 ⃡ and 𝐻𝑀 ⃡ are perpendicular.


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Interpreting Diagrams

CAN be Assumed

CANNOT be Assumed

All points shown are coplaner Perpendicular lines: 𝐻𝑀 ⊥ 𝐻𝐿

G, H, J are collinear

Congruent angles

∠𝐽𝐻𝐾 ≅ ∠𝐺𝐻𝑀

𝐻𝑀 , 𝐻𝐿 , 𝐻𝐾 ,𝐺𝐽 ⃡ intersect at H

∠𝐽𝐻𝐾 ≅ ∠𝐾𝐻𝐿

H is between G and J ∠𝐾𝐻𝐿 ≅ ∠𝐿𝐻𝑀

L is in the interior of ∠𝑀𝐻𝐾

Congruent segments

𝐺𝐻̅̅ ̅̅ ≅ 𝐻𝐽̅̅̅̅

∠𝐺𝐻𝑀 and ∠𝑀𝐻𝐿 are adjacent

angles

𝐻𝐽̅̅̅̅ ≅ 𝐻𝐾̅̅ ̅̅

∠𝐺𝐻𝐿 and ∠𝐿𝐻𝐽 are a linear pair 𝐻𝐿̅̅ ̅̅ ≅ 𝐻𝐺̅̅ ̅̅

∠𝐽𝐻𝐾 and ∠𝐾𝐻𝐺 are

supplementary

𝐻𝐾̅̅ ̅̅ ≅ 𝐻𝐿̅̅ ̅̅

Guided Practice

Determine whether each statement can be assumed from the figure. Explain.

9. ∠𝐾𝐻𝐿 and ∠𝐺𝐻𝑀 are complementary

10. ∠𝐺𝐻𝐾 and ∠𝐽𝐻𝐾 are a linear pair

11. 𝐻𝐿 is perpendicular to 𝐻𝑀

Your Turn

12. 𝑚∠𝑉𝑌𝑇 = 90

13. ∠𝑇𝑌𝑊 and ∠𝑇𝑌𝑈 are supplementary

14. ∠𝑉𝑌𝑊 and ∠𝑇𝑌𝑆 are adjacent angles


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3.4 ext. Perpendicular Bisector Notes

Bisector - _________________________________________________________________________________

__________________________________________________________________________________________

Instructions to Construct a Perpendicular Bisector

1. Place your compass point on A and stretch the compass MORE THAN half way to point B, but not

beyond B.

2. With this length, swing a large arc that will go BOTH above and below 𝐴𝐵̅̅ ̅̅ . (If you do not wish to make

one large continuous arc, you may simply place one small arc above 𝐴𝐵̅̅ ̅̅ and one small arc below 𝐴𝐵̅̅ ̅̅ .)

3. Without changing the span on the compass, place the compass point on B and swing the arc again. The

two arcs you have created should intersect.

4. With your straightedge, connect the two points of intersection.

5. This new straight line bisects 𝐴𝐵. Label the point where the new line and 𝐴𝐵 cross as C.

Guided Practice

1.

Your Turn

2. 3.


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Construct a perpendicular bisector and then name all the relationships that we know about the figure, and what

values we cannot state specific relationships about.

Relationships

What cannot be assumed

Geometry Unit 2 Note Sheets

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1.4 Angle Bisectors Notes

Angle Bisector

Guided Practice

1. In the figure, 𝐾𝐽 and 𝐾𝑀 are opposite rays, and 𝐾𝑁 bisects ∠𝐽𝐾𝐿. If 𝑚∠𝐽𝐾𝑁 = 8𝑥 − 13 and

𝑚∠𝑁𝐾𝐿 = 6𝑥 + 11, find 𝑚∠𝐽𝐿𝑁.

Your Turn

2. In the figure, 𝐵𝐴 and 𝐵𝐶 are opposite rays, and 𝐵𝐻 bisects ∠𝐸𝐵𝐶. If 𝑚∠𝐴𝐵𝐸 = 2𝑛 + 7 and

𝑚∠𝐸𝐵𝐹 = 4𝑛 − 13, find 𝑚∠𝐴𝐵𝐸.


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Steps to Bisecting an Angle

1. Start with angle PQR that we will bisect.

2. Place the compasses' point on the angle's vertex Q.

3. Adjust the compasses to a medium wide setting. The exact width is not important.

4. Without changing the compasses' width, draw an arc across each leg of the angle.

5. The compasses' width can be changed here if desired. Recommended: leave it the same.

6. Place the compasses on the point where one arc crosses a leg and draw an arc in the interior of the angle.

7. Without changing the compasses setting repeat for the other leg so that the two arcs cross.

8. Using a straightedge or ruler, draw a line from the vertex to the point where the arcs cross.

This is the bisector of the angle ∠PQR.

Guided Practice

Construct an angle bisector.

3.

Your Turn

4. 5.


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3.1 Transversal Measurements Notes

Use a protractor to measure all of the angles below.

Angle Measurements

∠1 _______________ ∠2 _______________

∠3 _______________ ∠4 _______________

∠5 _______________ ∠6 _______________

∠7 _______________ ∠8 _______________

What kind of relationships did you discover?


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Which examples do you think have parallel lines? Why?


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3.1 Parallel Lines with Transversal Notes

Transversal - _____________________________________________________________________________

in figure transversal is line _____.

In the figure line r and line s are parallel which make specific rules about angles given below.

Interior Angles – lie in the region between two lines that are not the transversal. In figure interior angles are

_____, _____, _____, and _____.

Consecutive Interior (AKA Same Side Interior) Angles are ________________________________.

Consecutive Interior Angles in figure are _____ and _____, and also _____ and _____.

Alternate Interior Angles are __________________________________. Alternate Interior Angles in figure

are _____ and _____, and also _____ and _____.

Exterior Angles – lie in the region not between the two lines that are not the transversal. In figure exterior

angles are _____, _____, _____, and _____.

Alternate Exterior Angles are _________________________________. Alternate Exterior Angles in figure


Corresponding Angles lie on the same side of the transversal and on the same side of a line. Corresponding

Angles are ______________________.

Corresponding Angles in the figure are _____ and _____, _____ and _____, _____ and _____, and also _____

and _____.


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Identify each pair of angles as corresponding, alternate interior, alternate exterior, or consecutive interior.


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3.2 Interior Angles Notes

Review:

Interior Angles – lie in the region between two lines that are not the transversal. In figure interior angles are

_____, _____, _____, and _____.

Consecutive Interior (AKA Same Side Interior) Angles are ________________________________.

Consecutive Interior Angles in figure are _____ and _____, and also _____ and _____.

Alternate Interior Angles are __________________________________. Alternate Interior Angles in figure


If 𝑚∠4 = 70° find the following:

𝑚∠3 =_____________

𝑚∠5 =_____________

𝑚∠6 =_____________

If 𝑚∠𝐵 = 143° find the following:

𝑚∠𝐶 =_____________

𝑚∠𝐸 =_____________

𝑚∠𝐻 =_____________

If 𝑚∠5 = 37° find the measure of the other interior angles

𝑚∠_______ =_____________

𝑚∠_______ =_____________

𝑚∠_______ =_____________


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For each of the following find the value of the variable and the measures of the two angles.


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3.2 Exterior Angles Notes

Review:

Exterior Angles – lie in the region not between the two lines that are not the transversal. In figure exterior

angles are _____, _____, _____, and _____.

Alternate Exterior Angles are _________________________________. Alternate Exterior Angles in figure


Use the information provided to find the numbered angle measure.

If 𝑚∠1 = 60° find the following:

𝑚∠2 =_____________

𝑚∠7 =_____________

𝑚∠8 =_____________

If 𝑚∠6 = 87° find the measure of the other exterior angles

𝑚∠_______ =_____________

𝑚∠_______ =_____________

𝑚∠_______ =_____________


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For each of the following find the value of the variable and the angle measure.


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3.2 Corresponding Angles Notes

Review:

Corresponding Angles lie on the same side of the transversal and on the same side of a line. Corresponding

Angles are ______________________.

Corresponding Angles in the figure are _____ and _____, _____ and _____, _____ and _____, and also _____

and _____.

If 𝑚∠4 = 55° find

𝑚∠8 =_____________

If 𝑚∠2 = 35° find

𝑚∠6 =_____________

If 𝑚∠𝐵 = 162° find

𝑚∠𝐹 =_____________

If 𝑚∠𝐺 = 76° find

𝑚∠𝐶 =_____________

If 𝑚∠7 = 37° find the measure of the corresponding angle

𝑚∠_______ =_____________

𝑚∠5 = 139° find the measure of the corresponding angle

𝑚∠_______ =_____________


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For each of the following find the value of the variable and the measures of the two angles.


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2.6 Algebraic Proofs Notes

Property Name Property Description

Addition Property of Equality If 𝑎 = 𝑏, then 𝑎 + 𝑐 = 𝑏 + 𝑐.

Subtraction Property of Equality If 𝑎 = 𝑏, then 𝑎 − 𝑐 = 𝑏 − 𝑐.

Multiplication Property of Equality If 𝑎 = 𝑏, then 𝑎𝑐 = 𝑏𝑐.

Division Property of Equality If 𝑎 = 𝑏 and 𝑐 ≠ 0, then 𝑎

𝑐=

𝑏

𝑐.

Reflexive Property of Equality 𝑎 = 𝑎

Symmetric Property of Equality If 𝑎 = 𝑏, then 𝑏 = 𝑎.

Transitive Property of Equality If 𝑎 = 𝑏 and 𝑏 = 𝑐, then𝑎 = 𝑐.

Substitution Property of Equality If 𝑎 = 𝑏, then b can be substituted for a in any

expression.

Guided Practice

1. Prove that if −5(𝑥 + 4) = 70, then 𝑥 = −18. Write justification for each step.

Your Turn

2. Solve 2(5 − 3𝑦) − 4(𝑦 + 7) = 92. Write a justification for each step.


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Guided Practice

3.

Your Turn

4.

5.


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2.7, 2.8 Mini Proofs Notes

A mini-geometry proof deals with knowing definitions for geometry terms and using them to show why

something is the way it is.

A few more properties to know for proofs. You will also need to know definitions from previous lessons.

Name Description

Segment Addition Postulate

Reflexive Property of

Congruence

Symmetric Property of

Congruence

Transitive Property of

Congruence

Angle Addition Postulate

Supplement Theorem If two angles form a linear pair, then they are supplementary.

Complement Theorem If two noncommon sides of two adjacent angles form a right angle , then

the angles are complimentary.

Congruent Supplements

Theorem

Congruent Complements

Theorem

Vertical Angles Theorem

Guided Practice

1.

2.

Your Turn

3.


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4.

5.

6.

Guided Practice

7.

8.

Your Turn

9.

10.

11.

12.


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OC 1.7/3.5 Proofs about Parallel and Perpendicular Lines Notes

2 PROOF Alternate Interior Angles Theorem If two parallel lines are cut by a transversal, then the pairs of alternate interior angles have the same measure.

Given: p ║ q

Prove: m∠3 = m∠5

Complete the proof by writing the missing reasons. Choose from the following reasons. You may use a reason

more than once.

Statements Reasons

1. p ║ q 1.

2. ∠3 and ∠6 are supplementary. 2.

3. m∠3 + m∠6 = 180° 3.

4. ∠5 and ∠6 are a linear pair. 4.

5. ∠5 and ∠6 are supplementary. 5.

6. m∠5 + m∠6 = 180° 6.

7. m∠3 + m∠6 = m∠5 + m∠6 7.

8. m∠3 = m∠5 8.

REFLECT

2a. Suppose m∠4 = 57° in the above figure. Describe two different ways to determine m∠6.

_________________

_________________

2b. In the above figure, explain why ∠1, ∠3, ∠5, and ∠7 all have the same measure.

_________________

_________________ 2c. In the above figure, is it possible for all eight angles to have the same measure? If so, what is that measure?

_________________


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3 PROOF Corresponding Angles Theorem If two parallel lines are cut by a transversal, then the pairs of corresponding angles have the same measure.

Given: p ║ q

Prove: m∠1 = m∠5

Complete the proof by writing the missing reasons.

Statements Reasons

1. p ║ q 1.

2. m∠3 = m∠5 2.

3. m∠1 = m∠3 3.

4. m∠1 = m∠5 4.

REFLECT

3a. Explain how you can you prove the Corresponding Angles Theorem using the Same-Side Interior Angles

Postulate and a linear pair of angles.

_________________

_________________

_________________

In the diagram, suppose p ║ q and line t is perpendicular to line p. Can you conclude that line t is perpendicular

to line q? Explain.

_______

_______

geometry unit 2 note sheets (segments, lines &...

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