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Transcript
```CCGPS Geometry
Unit 5: Intro to Geometry and Congruence
Notes
Unit 5: Basics of Geometry, Proofs, and Congruence
After completion of this unit, you will be able to…






Prove theorems about lines and angles and use them to find missing measures
Prove theorems about parallel lines and use them to find missing measures
Prove theorems about triangles and use them to finding missing measures
Use rigid motions to determine if two figure are congruent
Prove if two triangles are congruent using SAS, SSS, ASA, AAS, & HL
Prove theorems about parallelograms and use them to find missing measures
Timeline for Unit 2
Monday
Tuesday
Wednesday
Thursday
Friday
13
16
Day 1 –
Basics of Geometry
Review (cont’d)
23
Day 6 –
Proving Triangle
Congruency and
CPCTC
17
18
Day 2 –
Parallel Lines and
Transversals
24
19
Day 3 –
Geometric Proofs
25
Day 7 –
Proving
Day 4 –
Triangle Angle
Relationships
Quiz Days 1-3
26
Day 8 –
Segment
Relationships in
Triangles
Day 1 –
Basics of Geometry
Review
20
Day 5 –
Rigid Motions and
Congruent Triangles
27
Day 9 –
Review Day
Day 10 –
Unit 5 Test
1
CCGPS Geometry
Unit 5: Intro to Geometry and Congruence
Notes
Day 1 – Basics of Geometry
Naming Angles and Lines
Line Segment
Point
Points are named
with capital letters.
Two points are connected with a straight line. This line segment can be
named AB or BA .
Line
Ray
A line does not have a beginning or end point.
Lines are named using two points on the line.
one direction. Rays are named using its starting
point and one other point on the ray. The ray
This line can be named VW or WV .
Angle
can be named AB but NOT BA .
Angles are made up of two rays that have the same beginning point. The
point is called the vertex and the two rays are called the side of the angle.
Angles can be name in ways:
One Letter (if the vertex is not shared): A
Number (if given): 1
Three Letters (vertex is middle letter): MAS or SAM
Practice: Name the following angles:
a. Name the angle in four ways:
b. Name angle 1 as many ways as possible:
2
CCGPS Geometry
Unit 5: Intro to Geometry and Congruence
Notes
TYPES OF ANGLES
Acute Angles
Acute angles have measures between ____ & ____
Obtuse Angles
Obtuse Angles have measures between ____ & ____
Right Angles
Right Angles measure exactly ____
Straight Angles
Straight Angles measure exactly ____
Important Geometry Symbols

Angle

Triangle

Congruent (same shape & size)

Degrees

Perpendicular (90 degrees)
m
Measure of
Parallel
Congruent Angles
Similar
`
Congruent Segments
Using Geometry Terminology
A conditional statement (if-then) is a statement that contains a hypothesis (if) and conclusion (then).
Ex. If a student plays basketball, then they are an athlete.
A converse is a statement that has the hypothesis and conclusion switched around.
Ex. If a student is an athlete, then they play basketball. (Is this true?)
A postulate is a statement that is accepted as true without proof.
A theorem is a statement that must be proven before it can be accepted as true. We are going to prove many
theorems throughout this unit. We will prove a few of the following relationships on Day 3.
Practice: Take the following statement: I do my homework; I get my allowance, and write it in if-then form and
then write the converse of it.
3
CCGPS Geometry
Unit 5: Intro to Geometry and Congruence
Notes
Supplementary and Complementary Angles
Complementary Angles: Two or more angles whose sum of measures equals 90°.
40° and 50° angles are complementary angles because 40° + 50° = 90°.
Complementary Angles
Example: A 30° angle is called the complement of the 60° angle.
Similarly, the 60° angle is the complement of the 30° angle.
Practice: Find the complement of each angle.
a. 35°
b) Two angles, 2x° and 3x° are complementary. Find the value of x and each angle.
Supplementary Angles: Two or more angles whose sum of measures equals 180°.
60° and 120° angles are supplementary angles because 60° + 120° = 180°.
Supplementary Angles
Example: A 70° angle is called the supplement of the 110° angle.
Similarly, the 110° angle is the supplement of the 70° angle.
Practice: Find the supplement of each angle.
a.) 126°
b) Two angles, 4x° and 6x° are supplementary. Find the value of x and each angle.
4
CCGPS Geometry
Unit 5: Intro to Geometry and Congruence
Notes
Special Pairs of Angles
Linear Pair: Two adjacent (next to) angles whose noncommon sides are opposite rays. A linear pair also forms a
line (supplementary).
Linear Pair
a. Name all the linear pairs in the diagram below:
b. Solve for x:
Vertical Angles: Two nonadjacent angles that are formed by two intersecting lines. Vertical angles
are congruent.
Vertical Angles
a. Name all the vertical angles in the diagram below:
b. Find the measure of angles 1, 2, 3, and 4.
c. Solve for x. Then determine the measure of angle 1.
5
CCGPS Geometry
Unit 5: Intro to Geometry and Congruence
Notes
Angle Relationships
Angle Addition Postulate: If point D lies in the interior of
a. Find the measure of


ABC, then m  ABD + m  DBC = m  ABC.
PTM:
b. Given mÐQST = 135  , find mÐQSR .
Perpendicular: Two lines, rays, or segments that intersect to form a 90° angle.
a. Name all the angles you know are right angles.
Perpendicular
b. Solve for x.
6
CCGPS Geometry
Unit 5: Intro to Geometry and Congruence
Notes
Angle Bisector: A ray that divides an angle into two congruent angles (two angles with equal measure).
a.
QS bisects PQR . Find mPQS .
b.
KM bisects JKL . Find the value of x.
Angle Bisector
Segment Relationships
Segment Addition Postulate: If point B is on AC , and between points A and C, then AB + BC = AC .
a. Use the diagram to find EF .
b. Write an expression for AC.
c. Find the value of z.
7
CCGPS Geometry
Unit 5: Intro to Geometry and Congruence
Notes
Midpoint: Point that divides the segment into two congruent segments.
Midpoint
a. Find FM and MG .
b. Find YM and YZ .
c. T is the midpoint of QR . Solve for x.
Segment Bisector: A line, line segment, or ray that divides the line segment into two line segments of equal
length.
Segment Bisector
a. Find CB and AB .
b. Determine if you have enough information to determine if PC is the
segment bisector of AB . Explain why or why not.
Perpendicular Bisector: A line, line segment, or ray that intersects at the midpoint of a line segment at a 90
degree angle.
a. Determine if you have enough information to determine if WY is the
perpendicular bisector of ZX . Explain why or why not.
Perpendicular Bisector
8
CCGPS Geometry
Unit 5: Intro to Geometry and Congruence
Notes
Day 2 – Parallel Lines
Alternate Exterior Angles:
Alternate Interior Angles:
Same Side (Consecutive) Exterior Angles:
Same Side (Consecutive) Interior Angles:
Corresponding Angles:
Alternate Exterior Angles: Two angles in the exterior of the parallel lines and on opposite sides.
Alternate Interior Angles: Two angles in the interior of the parallel lines and on opposite sides.
Same Side Exterior Angles: Two angles in the exterior of the parallel lines and on same sides.
Same Side Interior Angles: Two angles in the interior of the parallel lines and on same sides.
Corresponding Angles: Two angles that lie in the same relative location.
A transversal is a line that intersects two or more coplanar lines at different points.
Postulates and Theorems Regarding Parallel Lines Cut by a Transversal:
Corresponding Angles Postulate
If 2 PARALLEL LINES are cut by a transversal,
then the pairs of corresponding angles are
congruent.
Alternate Interior Angles Theorem
If 2 PARALLEL LINES are cut by a
transversal, then the pairs of alternate
interior angles are congruent.
9
CCGPS Geometry
Unit 5: Intro to Geometry and Congruence
Notes
Alternate Exterior Angles Theorem
Consecutive Interior Angles Theorem
If 2 PARALLEL LINES are cut by a transversal,
then the pairs of alternate exterior angles
are congruent.
If 2 PARALLEL LINES are cut by a transversal,
then the pairs of consecutive interior
angles are supplementary.
Summary of Parallel Line Relationships
Relationships with Parallel & Non Parallel Lines
Angle Type
Alternate Exterior Angles
Parallel Lines
Non Parallel Lines
Alternate Interior Angles
Same Side Exterior Angles
Same Side Interior Angles
Corresponding Angles
Vertical Angles
Practice:
1. If the measure of angle 1 = 67  and a d, find the measure of all other angles.
10
CCGPS Geometry
Unit 5: Intro to Geometry and Congruence
Notes
2. Find the measure of the following:
a. Solve for x:
b. mECF
c. mDCE
3. Find the measure of the following:
a. Solve for x:
b. mEDG
c. mBDG
11
CCGPS Geometry
Unit 5: Intro to Geometry and Congruence
Notes
Day 3 – Intro to Proofs (Algebraic and Geometric)
1. Solve the following equation.
Justify each step as you solve it.
2(4x - 3) – 8 = 4 + 2x
2. Rewrite your proof so it is “formal” proof.
2(4x - 3) – 8 = 4 + 2x
When writing a geometric proof, you create a chain of logical steps that move from the hypothesis to the
conclusion of the conjecture you are proving. By proving the conclusion is true, you have proven the original
conjecture is true.
When writing a proof, it is important to justify each logical step with a reason. You can use symbols and
abbreviations, but they must be clear enough so that anyone who reads your proof will understand them.
Two Column Proofs

______________________________________________

______________________________________________

______________________________________________
12
CCGPS Geometry
Unit 5: Intro to Geometry and Congruence
Notes
Properties of Equality and Other Definitions
Property
General Example
Example 1
If a = b,
then a + c = b + c
If x – 4 = 8,
then x = 12
Subtraction Property
If a = b,
then a – c = b - c
If x + 5 = 7,
then x = 2
Multiplication Property
If a = b,
then ac = bc
If = 9,
then x = 18
Division Property
If a = b,
𝑎
𝑏
then =
If 2x = 10,
then x = 5
Distributive Property
If a(b + c),
then ab + ac
If 2 (x + 5),
then 2x + 10
Reflexive Property
a=a
5=5
̅̅̅̅
𝐴𝐵 ≅ ̅̅̅̅
𝐴𝐵
Symmetric Property
If a = b, then b = a
If 2 = x, then x = 2
If
∠𝐴 ≅ ∠𝐵, then
∠𝐵 ≅ ∠𝐴
Transitive Property
If a = b, and b = c,
then a = c
̅̅̅̅ ,
If ̅̅̅̅̅
𝑊𝑋 ≅ ̅̅̅̅
𝑋𝑌 and ̅̅̅̅
𝑋𝑌 ≅𝑌𝑍
̅̅̅̅̅
̅̅̅̅
then 𝑊𝑋 ≅𝑌𝑍
Substitution Property
If x = y, then y can be
substituted for x in any
expression
If ̅̅̅̅
𝐴𝐵 + ̅̅̅̅
𝐵𝐶 = ̅̅̅̅
𝑋𝑍 and ̅̅̅̅
𝐴𝐵 =
̅̅̅̅
̅̅̅̅
𝐶𝐷 , then 𝐶𝐷 + ̅̅̅̅
𝐵𝐶 = ̅̅̅̅
𝑋𝑍
𝑐
𝑐
Example 2
𝑥
2
Definition of Congruent
Segments
If mA  mB , then A  B .
Definition of Congruent
Angles
If mAB  mCD , then AB  CD .
If ∠A ≅ ∠B and ∠B ≅∠C,
then ∠A ≅ ∠C
If ∠A is supplementary
to ∠B and ∠B ≅ ∠C,
then ∠A is
supplementary to ∠C
Algebraic Proofs
Practice #1:
STATEMENTS
REASONS
PROVE: AB = 13
13
CCGPS Geometry
Practice #2:
Unit 5: Intro to Geometry and Congruence
STATEMENTS
Notes
REASONS
PROVE: x =
Practice: #3
STATEMENTS
REASONS
Prove: x =
14
CCGPS Geometry
Unit 5: Intro to Geometry and Congruence
Notes
Geometric Proofs
Prove the Linear Pair Theorem by filling in the blanks of a two column proof.
Given: Angle 1 and 2 form a linear pair.
Prove: Angle 1 and 2 are supplementary.
Statements
1.  1 and  2 form a linear pair.
Reasons
1. Given
2. BA and BC form a line.
2. __________________________________________
3. m  ABC = 180 
3. __________________________________________
4. ______________________________________
5. ______________________________________
5. Substitution
6.
 1 and  2 are supplementary
6. _________________________________________
Prove the Vertical Angles Theorem using a two column proof.
Given: Angle 4 and 1 are a linear pair.
Angle 1 and 2 are a linear pair.
Prove: 2  4
1.
Statements
Reasons
1. Given
2.
2. Given
3.
3. Linear Pair Theorem
4.
4. Linear Pair Theorem
5.
5. Definition of Supplementary Angles
6.
6. Definition of Supplementary Angles
7.
7. Substitution
8.
8. Subtraction Property
9.
9. Definition of Congruent Angles
15
CCGPS Geometry
Unit 5: Intro to Geometry and Congruence
Notes
Prove the Alternate Interior Angles are Congruent Theorem using a two column proof:
Given : l m
Pr ove : 2  3
Statements
1. ______________________________
Reasons
1. Given
2. ______________________________
2. Vertical Angles are Congruent
3. ______________________________
3. Corresponding Angles Postulate
4. ______________________________
4. Transitive Property
Prove the Right Angle Congruence Theorem using a two column proof.
Given: ACD and BCD are right angles
Prove: ACD  BCD
Statements
1. ___________________________
Reasons
1. Given
2. ___________________________
2. ___________________________
3. mACD  90
3. ___________________________
4. ___________________________
4. ___________________________
5. ___________________________
5. Transitive Property
6. ACD  BCD
6. ___________________________
Prove the Congruent Supplement Theorem using a two column proof:
Given: Angle 1 is supplementary to angle 2
Angle 3 is supplementary to angle 4
2  4
Prove: 1  3
Statements
1. ______________________________
Reasons
1. Given
2. ______________________________
2. Given
3. 2  4
3. ____________________________________
4. ______________________________
4. Definition of congruent angles
5. ______________________________
5. Defintion of supplementary angles
6. m3  m4  180
6. ____________________________________
7. ______________________________
7. Substitution Property
8. m1  m2  m3  m2
8. ____________________________________
9. ______________________________
9. Subtraction Property of Equality
10. ______________________________
10. ____________________________________
16
CCGPS Geometry
Unit 5: Intro to Geometry and Congruence
Notes
Day 4 – Angle Relationships in Triangles
A triangle is a figure formed when three noncollinear (not on the same line) points are connected by
segments.
The sides are:
Opposite Side of  F:
The vertices are:
Opposite Side of  E:
The angles are:
Opposite Side of  D:
Triangles can be classified by two categories: by Angles and by Sides.
Acute
All Acute Angles
Scalene
No Sides Congruent
ANGLES
Obtuse
One Obtuse Angle
Right
One Right Angle
SIDES
Isosceles
Equilateral
At Least 2 Sides Congruent
All Sides Congruent
Practice: Classify the triangles by sides and angles. Think About It: Check which triangles are possible.
Acute
Obtuse
Right
Scalene
Isosceles
Equilateral
17
CCGPS Geometry
Unit 5: Intro to Geometry and Congruence
Notes
Triangle Sum Theorem
Triangle Sum Theorem: The measures of the three interior angles in a triangle add up to be 180 
This means:
Corollary to Triangle Sum Theorem: The acute angles of a right triangle are complementary.
This means:
Proof of the Triangle Sum Theorem:
Statements
1. _____________________________
1. Given
2. _____________________________
2. Given
3. 4  1
3. ____________________________
4. _____________________________
4. Alt. Interior Angles Theorem
5. m4  m1
5. ____________________________
6. _____________________________
6. Def. of Congruent Angles
7. _____________________________
7. Linear Pair Postulate
8. mACD  m5  m3
8. ____________________________
9. _____________________________
9. Substitution Property
10. _____________________________
Examples: Find m  U.
Reasons
Find m  UPM
10. ____________________________
Find the measure of x.
18
CCGPS Geometry
Unit 5: Intro to Geometry and Congruence
Notes
Side Inequality Theorem
Side Inequality Theorem: If one side of a triangle is longer than the other side, then the angle opposite the
longer side has a greater measure than the angle opposite the shorter side.
This means: The largest angle of a triangle lies opposite the longest side. The smallest angle lies opposite the
shortest side. If two angles are equal, their side lengths will be equal.
Example: List the sides from shortest to longest for each diagram.
Exterior Angle Theorem
Exterior Angle Theorem: The measure of the exterior angle is equal to the sum of two remote interior angles.
Interpret:
What does exterior mean? __________________________________
What does interior mean? ___________________________________
What does remote mean? ___________________________________
Proof: Prove the exterior angle theorem:
Given:  1,  2, and  3 are interior angles.
Prove:  4 =  1 +  2
19
CCGPS Geometry
Examples:
A.
Unit 5: Intro to Geometry and Congruence
B.
Notes
C.
Solve for x using the Exterior Angle Theorem:
a.
b.
c.
Solve for x using the Triangle Sum Theorem:
a.
b.
c.
20
CCGPS Geometry
Unit 5: Intro to Geometry and Congruence
Notes
Isosceles Base Angle Theorem and Its Converse
Examples:
A. Find the value of x
D. Find the measure of <P.
 R = 30 
Base Angles Theorem:
If two sides of a triangle are
congruent, then the angles
opposite them are congruent.
Converse of Base Angles Theorem:
If two angles of a triangle are
congruent, then the sides opposite
of them are congruent.
B. Find the m  T
C. Find the value of x.
E. Find the measure of  Q
F. Find the value of x & y.
21
CCGPS Geometry
Unit 5: Intro to Geometry and Congruence
Notes
Day 5 – Rigid Motions and Congruent Triangles
Translations: Type of transformation in which a figure “slides” horizontally, vertically, or both.
Rule: (x, y)  (x + a, y + b)
a  left or right translations
b  up or down translations
A. Graph triangle ABC by plotting points A(8, 10), B(1, 2), and C(8, 2).
B. Translate triangle ABC 15 units to the left to form triangle A’B’C’ and write new coordinates.
C. Translate triangle ABC 12 units down to form triangle A’’B’’C’’ and write new coordinates.
Coordinates of
Triangle
ABC
Coordinates of
Triangle
A’B’C’
Coordinates of
Triangle
A’’B’’C’’
A (8, 10)
B (1, 2)
C(8, 2))
Observation: Did the figure change size or shape after each
translation?
Reflections: Type of transformation in which a figure “flips” over a line of symmetry.
x-axis: (x, y)  (x, -y)
y –axis (x, y)  (-x, y)
A. Graph triangle ABC by plotting points A(8, 10), B(1, 2), and C(8, 2).
B. Reflect triangle ABC over the x-axis to form triangle A’B’C’ and write new coordinates.
C. Reflect triangle ABC over the y-axis to form triangle A’’B’’C’’ and write new coordinates.
Coordinates of
Triangle
ABC
Coordinates of
Triangle
A’B’C’
Coordinates of
Triangle
A’’B’’C’’
A (8, 10)
B (1, 2)
C(8, 2))
Observation: Did the figure change size or shape after each
reflection?
22
CCGPS Geometry
Unit 5: Intro to Geometry and Congruence
Notes
Rotations: Type of transformation in which a figure “turns” about a fixed point at a given angle and direction.
90  clockwise or 270  counterclockwise:
(x, y)  (y, -x)
180  clockwise or 180  counterclockwise: (x, y)  (-x, -y)
270  clockwise or 90  counterclockwise:
(x, y)  (-y, x)
A. Graph triangle ABC by plotting points A(8, 10), B(1, 2), and C(8, 2).
B. Rotate triangle ABC 90  clockwise to form triangle A’B’C’ and write new coordinates.
C. Rotate triangle ABC 180  counterclockwise to form triangle A’’B’’C’’ and write new coordinates.
D. Rotate triangle ABC 90  counterclockwise to form triangle A’’’B’’’C’’’ and write new coordinates.
Coordinates
of Triangle
ABC
Coordinates
of Triangle
A’B’C’
Coordinates
of Triangle
A’’B’’C’’
Coordinates
of Triangle
A’’’B’’’C’’’
A (8, 10)
B (1, 2)
C(8, 2))
Observation: Did the figure change size or shape after each
rotation?
What you just discovered is that certain transformations preserve the size and shape of figures. Transformations
that preserve size and shape are called rigid motions or isometries. When two figures maintain their size and
shape, they remain congruent to each other. Any combination of translations, rotations, and reflections of a
figure will ALWAYS maintain congruency to the original figure.
Two triangles are congruent to each other if and only if their corresponding angles and sides are congruent.
Corresponding parts of triangles are the parts of the congruent triangles that “match.”
Example: If RST  XYZ , identify all pairs of congruent corresponding parts. Draw a picture and label the
congruent angles and sides.
Third Angle Theorem: If two angles on one triangle are congruent to two angles to another triangle, then the
third pair of angles are congruent.
Example: Find the measures of angles C and I.
23
CCGPS Geometry
Unit 5: Intro to Geometry and Congruence
Notes
Triangle Congruence Theorems
Side – Side – Side (SSS) Congruence Theorem
three sides of one triangle are congruent
to three sides of a second triangle
Angle – Side – Angle (ASA) Congruence Theorem
two angles and the included side of one
triangle are congruent to two angles and
the included side of a second triangle
Side – Angle – Side (SAS) Congruence Theorem
two sides and the included angle of one
triangle are congruent to two sides and
the included angle of a second triangle
Angle – Angle – Side (AAS) Congruence Theorem
two angles and a non-included side of one
triangle are congruent to two angles and a
non-included side of a second triangle
Hypotenuse – Leg (HL) Congruence Theorem (RIGHT TRIANGLES ONLY!)
In a right triangle, the hypotenuse and one
leg is congruent to the hypotenuse and leg
of another right triangle
Included Side
The side between two angles
Included Angle
The angle between two sides
Practice: Mark the included side in each triangle
Practice: Mark the included angle in each triangle.
24
CCGPS Geometry
Unit 5: Intro to Geometry and Congruence
Notes
Practice: Mark the appropriate sides and angles to make each congruence statement true by the stated
congruence theorem.
a. ASA
b. SSS
c. SAS
d. HL
e. AAS
Markings You Are Allowed to Add
Share a Side
Reason: Reflexive Property
Vertical Angles
Reason: Vertical Angles are Congruent
25
CCGPS Geometry
Unit 5: Intro to Geometry and Congruence
Notes
Practice: Determine whether there is enough information to conclude if the triangles are congruent. If so, state
the congruence theorem. If not, write not enough information.
A.
B.
C.
D.
E.
F. .
G.
H.
I.
J.
K.
L.
26
CCGPS Geometry
Unit 5: Intro to Geometry and Congruence
Notes
Day 6 – Proving Triangles Congruent (including CPCTC)
From yesterday, you learned that you only need 3 pieces of information (combination of angles and sides) to
determine if two triangles are congruent. Today, we are going to prove two triangles are congruent using two
column proofs.
A. Given: AB  CD, BC  AD
Prove: ABC  CDA ?
B.
C.
Given: AB CD and AE  CE
Pr ove : ABE  CDE?
Given : RF  BP and BF RP
Pr ove : RFP  BPF?
27
CCGPS Geometry
D. Given : WN HK
Pr ove : WNZ  HKZ?
Unit 5: Intro to Geometry and Congruence
E.
Given: JA  MY and YM bi sects JYA
Pr ove :JYM  AYM?
F.
Given: JA  MY and JY  AY
Pr ove :JYM  AYM?
Notes
28
CCGPS Geometry
Unit 5: Intro to Geometry and Congruence
Notes
CPCTC
Once you conclude two triangles are congruent, then you can also conclude that corresponding
parts of congruent triangles are congruent (CPCTC). CPCTC can be used as a justification after you
have proved two triangles are congruent. Look at the example proof below:
Statements
Reasons
1. SU  SK
1. Given
2. SR  SH
2. Given
3. S  S
3. Reflexive Property
4. SUH  SKR
4. SAS
5. U  K
5. CPCTC
Prove the Isosceles Base Angles Theorem
Given: GB  GD , GU bis ec ts DGB
Prove: B  D
Statements
Reasons
2. _________________________
1. Given
2. _________________________
3. _________________________
3. Definition of bisects
4. GU  GU
4. _________________________
5. _________________________
5. _________________________
6. B  D
6. _________________________
1. GB  GD
Prove the Converse of the Isosceles Base Angles Theorem:
Given: B  D , GU bis ec ts DGB
Prove: GB  GD
Statements
1. B  D
Reasons
2. _________________________
1. Given
2. _________________________
3. _________________________
3. _________________________
4. GU  GU
4. _________________________
5. _________________________
5. _________________________
6. GB  GD
6. _________________________
29
CCGPS Geometry
Unit 5: Intro to Geometry and Congruence
Notes
Proof:
Given: JHK  LHK, JKH  LKH
Prove: JK  LK
Proof:
Given: CW and SD bi sect each other
Prove: CS  WD
Proof:
Given: AB  CB, D is the midpoint of AC
Prove: A  C
30
CCGPS Geometry
Unit 5: Intro to Geometry and Congruence
Notes
Day 7 – Segment Relationships in Triangles
A midsegment of a triangle is a segment that joins the midpoints of two sides of the triangle. Every
triangle has three midsegments, which forms the midsegment triangle.
Triangle Midsegment Theorem: A midsegment of a triangle is parallel to a side of the triangle, and its
length is half the length of that side.
The Midsegment is:
 Parallel to one side of the triangle
 Is half the length of the parallel side
 Connects to the midpoints
Practice:
A. Solve for x:
D. Given CD = 14, GF = 8, and GC = 5,
Find the perimeter of BCD .
Midsegments:
Midsegment Triangle:
B. Solve for x:
C. Solve for x, y, and z:
E. Find the measure of the following:
JL
PM
MLK
31
CCGPS Geometry
Unit 5: Intro to Geometry and Congruence
Notes
Perpendicular Bisectors of Triangles
If you remember from Day 1, perpendicular bisectors are lines, line segments, or rays that intersect at
the midpoint of a line segment at a 90 degree angle.
Perpendicular Bisector Theorem: If a point is on the perpendicular bisector of a segment, then it is
equidistant from the endpoints of the segment.
If:
Then:
Converse of the Perpendicular Bisector Theorem: If a point is equidistant from the endpoints of the
segment, then it is on the perpendicular bisector of the segment.
Practice:
A. Find BC.
B. Find AD if AC is the perpendicular bisector to BD.
C. Find TU
32
CCGPS Geometry
Unit 5: Intro to Geometry and Congruence
Notes
Points of Concurrency
When three or more lines intersect at one point, the lines are said to be concurrent. The point of concurrency is
the point where they intersect. You will examine several different points of concurrency today.
Medians
The median of a triangle is a segment
whose endpoints are a vertex of the
triangle and the midpoint of the
opposite side.
Centroid
The centroid is where the three medians
of a triangle intersect.
The centroid is always located on the
inside of the triangle.
The centroid is also called the center of
gravity because it is the point where a
triangular region will balance.
Angle Bisectors
A triangle has three angle bisectors.
Angle bisectors divide an angle into two
congruent angles.
Incenter
The incenter is where the three angle
bisectors of a triangle intersect.
The incenter is always located inside the
triangle.
The incenter is located equidistant from
the sides of the triangle.
PX = PY = PZ
33
CCGPS Geometry
Unit 5: Intro to Geometry and Congruence
Perpendicular Bisectors
A triangle has three perpendicular
bisectors. They pass through the
midpoint on each side of a triangle at a
90  angle.
Notes
Circumcenter
The circumcenter is where the three
perpendicular bisectors of a triangle
intersect.
The circumcenter is located equidistant
from the vertices of the triangle.
PA = PB = PC
The circumcenter can be located inside
(acute), outside (obtuse) or on (right) a
triangle.
Altitudes
Orthocenter
A triangle has three altitudes. An altitude
is a perpendicular segment from a
vertex to the line containing the
opposite side.
The othrocenter is where the three
altitudes of a triangle intersect.
The orthocenter is located inside for an
acute triangle, outside for an obtuse
triangle, and on the vertex for a right
triangle.
A triangle’s altitude can be located in
three locations (inside, on, or outside)
the triangle.
34
CCGPS Geometry
Unit 5: Intro to Geometry and Congruence
Notes
The Incenter and Circumcenter also represent the center of a circle inscribed or circumscribed with a
triangle.
Incenter with an Inscribed Circle
(Inscribed – circle is inside the
triangle))
Circumcenter with a Circumscribed
Circle
(Circumscribed – circle is outside the
triangle)
How Can I Remember How the Points of Concurrency Are Created?
Angle Bisector – Incenter
Perpendicular Bisector – Circumcenter
Altitudes – Orthocenter
Medians – Centroid
Practice: Name the point of concurrency:
a.
b.
e. Use the diagram as shown; G is the
_____________________ of triangle ABC.
If GC = 2x – 7 and GB = -3x + 13, what
is the value of GA?
c.
d.
f. Use the diagram as shown; G is the
_____________________ of triangle ABC.
If GD = 4x + 8 and GE = 7x - 10, what
is the value of x?
35
CCGPS Geometry
Unit 5: Intro to Geometry and Congruence
Notes
Day 8 – Parallelograms
A parallelogram is a type of quadrilateral that has two pairs of opposite sides that are parallel. Parallelograms
are denoted by the symbol . If a quadrilateral has two pairs of opposite sides, then it can be classified as a
parallelogram.
There are 5 theorems associated with parallelograms:
 Opposite sides are congruent
 Opposite angles are congruent
 Consecutive angles are supplementary
 Diagonals bisect each other
 Diagonals form two congruent triangles
Theorem 1: Opposite sides are congruent
Theorem 2: Opposite angles are
congruent.
Theorem 3: Consecutive angles are
supplementary.
Find the value of x. Then find the
length of BC.
Find the value of x. Then find Angle Q.
Find the value of y. Then find the
measure of angle A and B.
36
CCGPS Geometry
Unit 5: Intro to Geometry and Congruence
Theorem 4: Diagonals bisect each other.
Notes
EFGH is a parallelogram. Find w and z.
Given: JKLM is a parallelogram
Prove: JKL  LMJ
Theorem 5: Diagonals form two
congruent triangles.
Practice: Apply the properties of parallelograms to solve for the specified variable.
a. Solve for x, y, and z.
b. Solve for x, y, and z.
c. Solve for x and z.
37
CCGPS Geometry
Unit 5: Intro to Geometry and Congruence
Notes
Proofs with Parallelograms
Determine if each quadrilateral must be a parallelogram. Explain why or why not.
a.
b.
c. Given: ABCD is a parallelogram
Statements
1. ABCD is a parallelogram
Reasons
1. ___________________________
2. AB  CD
2. ___________________________
3. DA  BC
3. ___________________________
4. ___________________________
4. ___________________________
5. ___________________________
5. SSS
6. ___________________________
6. ___________________________
d. Given: ABCD and AFGH are parallelograms
Prove: C  G
Statements
1. ABCD is a parallelogram
Reasons
1. ___________________________
2. ___________________________
2. Given
3. C  A
3. ___________________________
4. A  G
4. ___________________________
5. ___________________________
5. ____________________________
38
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