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Geometry
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CHAPTER 2
Geometry
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CHAPTER 2
Table of Contents
Date
Topic
Description
Page
Common Core State Standards
G.MG.3 Apply geometric methods to solve problems.
Student Learning Targets
1. Students will be able to identify and use basic postulates
about points, lines and planes.
2. Students will be able write paragraph proofs.
Name:_______________________________________
Date:________________
Topic:_____________________________________
Section 2.5 Notes: Postulates and Paragraph Proofs
Vocabulary
Definition
Real Life Example
Postulate
Postulate
Points, Lines, and Planes Postulates
Definition
2.1
2.2
2.3
2.4
2.5
Intersection of Lines and Planes
2.6
2.7
Picture
Example 1: Explain how the picture illustrates that the statement is true. Then state the postulate that can be used to show the
statement is true.
a) Lines n and l intersect at point K.
b) Planes P and Q intersect in line m.
c) Points D, K, and H determine a plane.
d) Point D is also on the line n through points C and K.
e) Points D and H are collinear.
f) Points E, F and G are coplanar.
You can use postulates to explain your reasoning when analyzing statements.
Example 2: Determine whether the following statement is always, sometimes, or never true. Explain.
a) If plane T contains EF and EF contains point G, then plane T contains point G.
b) GH contains three noncollinear points.
Vocabulary
Definition
Proof
Theorem
The Proof Process
Step 1:
Step 2:
Step 3:
Step 4:
Step 5:
Real Life Example
Example 3: Use the steps above to prove the given statement. Given AC intersects CD , write a paragraph proof to show that A, C,
and D determine a plane.
The conjecture in Example 3 is known as the Midpoint Theorem.
Vocabulary
Definition
Picture
Midpoint Theorem
Summary
Explain how the figure illustrates that each statement is true. Then state the postulate that can be used to show each statement is true.
1. Planes P and Q intersect in line r.
2. Lines r and n intersect at point D.
3. Lines n contains points C, D, and E.
4. Plane P contains the points A, F, and D.
5. Line n lies in plane Q.
6. line r is the only line through points A and D.
Determine whether each statement is always, sometimes or never true. Explain your reasoning.
7. The intersection of three planes is a line.
8. Line r contains only point P.
9. Through two points, there is exactly one line.
mastery connect – section 2.5
1. In the figure, points A, B, and C lie in plane Z. Which of the following
postulates can be used to show that A and B are collinear?
a. If two lines intersect, then their intersection is exactly one point.
b. If two planes intersect, then their intersection is a line.
c. Through any two points, there is exactly one line.
d. Through any three points not on the same line, there is exactly one plane.
For questions 2 and 3, determine whether the statement is always, sometimes, or never true.
2. Plane A and plane B intersect in exactly one point
.
a. Always
b. Sometimes
c. Never
3. Point N lines in plane X and point R lies in plane Z. You can draw only one line that contains both points N and R.
a. Always
b. Sometimes
c. Never
4. Given ̅̅̅̅
𝑅𝑇 ≅ ̅̅̅̅
𝑇𝑌 , S is the midpoint of ̅̅̅̅
𝑅𝑇 , and X is the midpoint of ̅̅̅̅
𝑇𝑌 , a paragraph proof to show that
provided with one reason missing. Choose the best reason to complete the proof.
a. Definition of Midpoint
c. Definition of Congruent Segments
b. Segment Addition Postulate
d. Substitution
̅𝑆𝑇
̅̅̅ ≅ ̅̅̅̅
𝑇𝑋 has been
Name: _____________________________________
Section 2.5 Worksheet
For numbers 1 and 2, explain how the figure illustrates that each statement is true. Then state the postulate that can be used to show
each statement is true.
1. The planes J and K intersect at line m.
2. The lines l and m intersect at point Q.
For numbers 3 and 4, determine whether the following statements are always, sometimes, or never true. Explain.
3. The intersection of two planes contains at least two points.
4. If three planes have a point in common, then they have a whole line in common.
For numbers 5 and 6, state the postulate that can be used to show that each statement is true. In the figure, line m and TQ lie in plane
𝒜.
5. Points L, and T and line m lie in the same plane.
6. Line m and ST intersect at T.
7. In the figure, E is the midpoint of AB and CD , and AB = CD. Write a paragraph proof to prove that AE  ED.
8. ROOFING Noel and Kirk are building a new roof. They wanted a roof with two sloping planes that meet along a curved arch. Is
this possible?
Common Core State Standards
G.CO.9 Prove theorems about lines and angles.
Student Learning Targets
1. Students will be able to use algebra to write two-column
proofs.
2. Students will be able to use properties of equality to write
geometric proofs.
Name:_______________________________________
Date:________________
Topic:_____________________________________
Section 2.6 /2.7 Notes : Algebraic Proofs & Proving Segment Relationships
Let’s refresh our memories about properties of real numbers before we start talking Geometry:
Properties of Real Numbers
Addition Property of Equality
Subtraction Property of Equality
Multiplication Property of Equality
Division Property of Equality
Reflexive Property of Equality
Symmetric Property of Equality
Transitive Property of Equality
Substitution Property of Equality
Distributive Property of Equality
Example 1: Use the above properties to justify each step when solving the following equation: 2(5 – 3a) – 4(a + 7) = 92.
Vocabulary
Definition
Picture
Ruler Postulate
Segment Addition Postulate
Definition of Congruent
Definition of Congruent Segments
Example 2: Prove that if AB  CD, then AC  BD.
Statements
Reasons
1.
1.Given
2. AB = CD
2.
3.
3.Reflexive
4.AC + BC = AC
4.
5. CD+BC = AC
5.
6. CD + BC = BD
6.
7.
7. Substitution
8.
8.Definition of Congruent
Example 3: Prove the following
Given: AC = AB
AB = BX
CY = XD
Prove: AY = BD
Statements
Reasons
1.
1. Given
2.
2. Given
3. AC = BX
3.
4.
4.Given
5.AC + CY = BX + XD
5.
6. AC + CY = AY
6.
7. BX + XD = BD
7.
8.
8. Substitution
Properties of Segment Congruence
Reflexive Property of Congruence
Symmetric Property of Congruence
Transitive Property of Congruence
Example 4: BADGE Jamie is designing a badge for her club. The length of the top edge of the badge is equal to the length of the left
edge of the badge. The top edge of the badge is congruent to the right edge of the badge, and the right edge of the badge is congruent
to the bottom edge of the badge. Prove that the bottom edge of the badge is congruent to the left edge of the badge.
Given: WY = YZ
YZ  XZ
XZ  WX
Prove: WX  WY
Statements
Reasons
1.
1. Given
2.
2. Definition of Congruent Segments
3.
3. Given
4.
4. Transitive Property
5.
5.Given
6.
6.Transitive Property
Example 5: Prove the following.
Given: GD  BC
BC  FH
FH  AE
Prove: AE  GD
Statements
Reasons
1.
1. Given
2.
2. Given
3.
3. Transitive Property
4.
4. Given
5.
5. Transitive Property
6.
6. Symmetric Property
Mastery connect – section 2.7
For questions 1 – 3, use the proof below.
1. For the proof shown above, provide statement 2.
a. AB = BC, DE = EF
b. AC = DE, AB = DF
c. AB = DE
2. For the proof shown above, provide the reason for part c.
a. Subtraction Property of Equality b. Division Property of Equality
c. Addition Property of Equality
d. Multiplication Property of Equality
3. For the proof shown above, provide the reason for part f.
a. Symmetric Property of Equality b. Definition of congruent segments
c. Definition of Addition
d. Definition of a line
d. AC = DF, BC = EF
For questions 4 - 6, use the proof below.
4. For the proof shown above, provide a reason for part b.
a. Definition of Addition b. Symmetric Property (=)
c. Definition of a line
d. Definition of congruent segments
5. For the proof shown above, provide statement 3.
a. AB = BA
b. AB = EF
C. AB = CD
6. For the proof shown above, provide a reason for part d.
a. Definition of Addition b. Definition of congruent segments
c. Definition of a line
d. Symmetric Property (=)
d. CD = EF
Section 2.7 Worksheet
Name: _____________________________________
1. Given: AB  DE
B is the midpoint of AC .
E is the midpoint of DF .
Prove: BC  EF
Statements
Reasons
1.
1. Given
2. AB = DE
2.
3. B is the midpoint of AC .
3.
4.
4. Definition of Midpoint
5. E is the midpoint of DF .
5.
6.
6. Definition of Midpoint
7. BC = DE
7.
8. BC = EF
8.
9.
9.
2. TRAVEL Refer to the figure. DeAnne knows that the distance from Grayson to
Apex is the same as the distance from Redding to Pine Bluff. Prove that the distance
from Grayson to Redding is equal to the distance from Apex to Pine Bluff.
3. FAMILY Maria is 11 inches shorter than her sister Nancy. Brad is 11 inches shorter than his brother Chad. If Maria is shorter than
Brad, how do the heights of Nancy and Chad compare? What if Maria and Brad are the same height?
4. DISTANCE Martha and Laura live 1400 meters apart. A library is opened between
them and is 500 meters from Martha. How far is the library from Laura?
5. LIGHTS Five lights, A, B, C, D, and E, are lined up in a row. The middle light is the midpoint of the second and fourth light and
also the midpoint of the first and last light.
a) Draw a figure to illustrate the situation.
b) Complete this proof.
Given: C is the midpoint of BD and AE.
Prove: AB = DE
Statement
Reason
1. C is the midpoint of BD and AE.
1. Given
2. BC = CD and _________________
2.
3. AC = AB + BC,
CE = CD + DE
3.
4. AB = AC – BC
4.
5.
5. Substitution Property
6. DE = CE – CD
6.
7.
7.
Common Core State Standards
G.CO.9 Prove theorems about lines and angles.
Student Learning Targets
1. Students will be able to write proofs involving
supplementary and complementary angles.
2. Students will be able to write proofs involving congruent
and right angles.
Name:_______________________________________
Date:________________
Topic:_____________________________________
Section 2.8 Notes: Proving Angle Relationships
Vocabulary
Definition
Picture
Angle Addition Postulate
Example 1: CONSTRUCTION Using a protractor, a construction worker measures that the angle a beam makes with a ceiling is
42°. What is the measure of the angle the beam makes with the wall?
Vocabulary
Definition
Picture
Supplement Theorem
Complement Theorem
Example 2: TIME At 4 o’clock, the angle between the hour and minute hands of a clock is 120º. When the second hand bisects the
angle between the hour and minute hands, what are the measures of the angles between the minute and second hands and between the
second and hour hands?
Properties of Angle Congruence
Reflexive Property of Congruence
Symmetric Property of Congruence
Transitive Property of Congruence
Vocabulary
Definition
Picture
Congruent Supplement Theorem
Congruent Complement Theorem
Example 3: In the figure, 1 and 4 form a linear pair, and m3 + m1 = 180°.
Prove that 3 and 4 are congruent.
Statements
Reasons
1.
1. Given
2. 1 and 4 form a linear pair
2.
3. 1 and 4 are supplementary
3.
4.
4. Def. of Supplementary Angles
5.
5.
Vocabulary
Definition
Picture
Vertical Angles Theorem
Example 4: If 1 and 2 are vertical angles and m1 = (d – 32)° and m2 = (175 – 2d)°, find m1 and m2. Justify each step.
Statements
Reasons
1.
1.Given
2. 1 ≅ 2
2.
3. m1 = m 2
3.
4.
4. Substitution
5.
5. Addition Property
6.
6.
7. d = 69
7.
8.
8.
Theorem
2.9
2.10
2.11
2.12
Right Angle Theorems
Definition
Picture
2.13
Summary
Mastery connect – section 2.8
1. All right angles are ________________.
a. congruent
b.180°
c. transitive
d. complementary
2. If ∠3 and ∠4 are vertical angles and m∠3 = 9𝑥 + 5 and m∠4 = 5𝑥 + 33,
find m∠4.
a. 68°
b. 72°
c. 43°
d. 23°
3. Which statement explains why m∠3 + m∠4 = 180°?
a. If the sum of the measures of two angles is 180°, then they are supplementary.
b. Angles supplementary to the same angle are congruent.
c. If two angles form a linear pair, then they are supplementary.
d. All right angles are congruent.
4. In the picture, ∠1 and ∠3 are vertical. If m∠1 = 162°, then what is m∠3?
a. 90°
b. 81°
c.28°
d. 162°
5. If ∠1 and ∠2 form a linear pair and m∠1 = 72°, find m∠2.
a. 108°
b. 18°
C. 144°
d. 72°
6. If ∠1 and ∠2 form a linear pair and m∠1 = 2x + 16 and m∠2 = 50x + 60, find m∠2.
a. 160°
b. 80°
c. 180°
d. 20°
7. Two angles are congruent if they are ___________________.
a. vertical
b. supplementary c. adjacent
d. a linear pair
Name: _____________________________________
Section 2.8 Worksheet
For numbers 1 – 3, find the measure of each numbered angle and name the theorems that justify your work.
1. m1 = (x + 10)°
m2 = (3x + 18)°
2. m4 = (2x – 5)°
m5 = (4x – 13)°
4. Write a two-column proof.
Given: 1 and 2 form a linear pair.
2 and 3 are supplementary.
Prove: 1  3
5. STREETS Refer to the figure. Barton Road and Olive Tree Lane form a right
angle at their intersection. Tryon Street forms a 57° angle with Olive Tree Lane.
What is the measure of the acute angle Tryon Street forms with Barton Road?
6. TUBES A tube with a hexagonal cross section is placed on the floor.
What is the measure of 1 in the figure given that the angle at one corner
of the hexagon is 120°?
7. PAINTING Students are painting their rectangular classroom ceiling. They want to
3. m6 = (7x – 24)°
m7 = (5x + 14)°
paint a line that intersects one of the corners as shown in the figure. They want the painted
line to make a 15° angle with one edge of the ceiling. Unfortunately, between the line and
the edge there is a water pipe making it difficult to measure the angle. They decide to
measure the angle to the other edge. Given that the corner is a right angle, what is the
measure of the other angle?
8. BUILDINGS Clyde looks at a building from point E. AEC has the same measure as BED.
a) The measure of AEC is equal to the sum of the measures of AEB and what other angle?
b) The measure of BED is equal to the sum of the measures of CED and what other angle?
c) Is it true that mAEB is equal to mCED?