# Points, Lines, Planes, and Circles

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Points, Lines, Planes, and Circles
Lesson #1 Points, Lines, Planes, and Circles

Lesson Goals Students will be able to define a point, a line, a line segment, a ray, and a plane SWBAT define collinear points and concurrent lines SWBAT recognize collinear points and concurrent lines SWBAT define a circle

Do Now Draw a dot on a piece of paper.
Discuss with your neighbor what shape you might get if you drew a bunch of other dots all the same distance from your original dot. Is there another shape? Hint: think “outside the page.”

Points and lines Point Line Segment Ray Line

Planes Describe a plane in your own words and point out at least one plane in this room to your neighbor

Collinearity and Concurrency
When three or more points… When three or more lines… Why three or more?

Conclusion and Homework
Pg. 4 # 1-4 Pg. 6 # 1-4 Purchase compass, straightedge, textbook, and graph paper notebook from the school store

The Five Axioms of Geometry
Lesson #2 The Five Axioms of Geometry

Lesson Goals SWBAT copy a line segment, using only a compass and straightedge SWBAT name and understand Euclid’s five axioms

Do Now: Pick a segment, any segment
Draw a line segment on your page using your straightedge Assuming you couldn’t measure distances with your straightedge, how could you draw an identical line segment with just a straightedge and a compass?

Axioms: When you can’t prove it, just say it
Euclid (b. ~300 BC) Wrote Elements, which is sort of a much more impressive version of Introduction to Geometry Started with five basic axioms What’s an axiom? Euclid apparently looked like Santa

No?

Five Axioms Any two points can be connected by a straight line segment. Any line segment can be extended forever in both directions, forming a line. Given any line segment, we can draw a circle with the segment as a radius and one of the segment’s endpoints as the center. All right angles are congruent [the same measure]. Given any straight line and a point not on the line, there is exactly one straight line that passes through the point and never meets the first line.

Any two points can be connected by a straight line segment.

Any line segment can be extended forever in both directions, forming a line.

Given any line segment, we can draw a circle with the segment as a radius and one of the segment’s endpoints as the center.

All right angles are congruent.

Given any straight line and a point not on the line, there is exactly one straight line that passes through the point and never meets the first line.

Angles, and how to measure them
An angle is…

Make your own angle Draw two line segments on your page so that they share a common point Someone come up here and do the same… What tools do you have to measure an angle?

Some properties of angles
What if… We know <BAC and <CAD, but we want <BAD? We know <BAD and <BAC, but we want <CAD? B C A D

Acute, right, obtuse Acute is… Right is… Obtuse is…

Conclusion and HW Pg. 20 # 1,2 [Start Construction Worksheet]

Vertical Angles and Parallel Lines
Lesson #3 Vertical Angles and Parallel Lines

Lesson Goals SWBAT define a vertical angle, corresponding angles, alternate interior angles, alternate exterior angles, same-side interior angles, and same-side exterior angles SWBAT find the measure of these angles when given parallel lines and a transversal

Parallel Parking Draw a line segment in your notebook (this can be freehand) To “parallel park” your line segment, it needs to move its own length in the same direction, twice its length at a 45 degree angle to your original line segment, and then its length in a direction parallel to your original line. Give your car directions from its original location with your protractor and construct this series of line segments in your notebook.

Parallel Line Notation

Transversals

Why does a triangle have 180 degrees?

HW Pg. 30 # 1, 2, 3, 4, 5 Pg. 46 # 28, 31, 32, 33, 34

Lesson #4 Triangles

Lesson Goals SWBAT find the missing angle in a triangle given the measures of the other two angles SWBAT find missing angles in a triangle given an exterior angle and vice versa

Do Now Anyone complete the proof from yesterday?

Triangle Facts Triangles have _______ sides
The measures of the angles in a triangle sum to _________ There are different types of triangles: whether a triangle is _________, ___________, or ____________ tells you the biggest angle Special triangles you should know are __________, ___________, and _________

Finding the angles of a triangle
Using a protractor, you find that two of the angles in a triangle measure 30° Classify this triangle two different ways Find the third angle of this triangle

What if we only know how the angles relate to one another?
The triangular base of a see-saw is to be constructed such that one of its base angles is twice the degree measure of the other, and the third angle is 20° less than three times the smallest base angle. Find the exact measures of all three angles.

Exterior Angles A D C B If m<A=26°, and m<B=62°, find the m<ACD. What do you notice? Will this always be true?

HW Study for vocabulary quiz tomorrow Pg. 35 # 1, 2, 3, 4, 5

Parallel Lines Revisited
Lesson #5 Parallel Lines Revisited

Lesson Goals Students will understand what the converse of a statement is Students will prove the converse of the statement “If lines are parallel, then corresponding angles are congruent”

Do Now In the diagram below, what can we say about lines k and m? Why?

Statement converses If this, then that. P  Q If that, then this.
Q  P Are these logically equivalent? How do you know?

k m

HW Pg. 46 # 35-46

Lesson #6 Congruent Triangles

Lesson Goals SWBAT define congruence
SWBAT show triangles are congruent using SSS, SAS, ASA, and AAS SWBAT show that ASS does not imply triangles are congruent SWBAT use a CPCTC argument to make an argument about a congruent triangle

Do Now Triangle worksheet

Lesson #7 Congruent Triangles

Lesson Goals SWBAT show triangles are congruent using SSS, SAS, ASA, and AAS SWBAT show that ASS does not imply triangles are congruent SWBAT use a CPCTC argument to make an argument about a congruent triangle

Do Now Find <BAD. How do you know? (Be careful with your reasoning)
8 B A 3 3 32° C D 8

Proof Prove that if a radius of a circle bisects a chord of the circle that is not a diameter, then the radius must be perpendicular to the chord Hint: Use the triangle properties you now know and love.

SAS Prove that AB || CD B A 6 5 5 6 C D

Let’s fly a kite! Prove that DB is perpendicular to AC

HW Pg. 54 #1-4 Pg. 58 #1-4

Lesson #8 Congruent Triangles

Lesson Goals SWBAT show triangles are congruent using SSS, SAS, ASA, and AAS SWBAT show that ASS does not imply triangles are congruent SWBAT use a CPCTC argument to make an argument about a congruent triangle

Do Now: Prove it Given: AD=BC, AD || BC, E and F are on AC, <ADE=<CBF Prove: AB || CD AND Prove: DF=EB A B E F C D

Which of the following triangles are congruent?
16” 16” 28° 28° 55° 13.2” 97° 28° 16” 13.2” 16” 97°

HW Pg. 63 # 1 - 4 Pg. 75 #24, 26, 27, 28, 29, 30

Isosceles and Equilateral Triangles
Lesson #9 Isosceles and Equilateral Triangles

Lesson Goals SWBAT define isosceles and equilateral triangles
SWBAT to utilize properties of isosceles and equilateral triangles to solve problems SWBAT to utilize properties of isosceles and equilateral triangles for geometric proofs

Do Now: Equilateral Triangles
An equilateral triangle has three equal sides Prove that all its angles are equal too.

Definitions An isosceles triangle is a triangle with two equal sides (the legs) and one unequal side (the base) An isosceles triangle has base angles and a vertex angle An equilateral triangle has three equal sides

Problem solving with types of triangles
XY=XZ=14 and <X=42° Find the other two angles. X Y Z

Proofs with types of triangles

HW Pg. 71 #1-7

Lesson #10 Area and Perimeter

Lesson Goals SWBAT define area SWBAT define perimeter
SWBAT find the area of a grid-based shape SWBAT find the perimeter of a polygon

Guard the perimeter! Search the area!
What do these statements mean? Come up with a definition for perimeter and write it in your notes. Come up with a definition for area and write it in your notes.

Find the perimeter 14 18√2

Find the area

Word problems with perimeter
The length of each leg of an isosceles triangle is three times the length of the base of the triangle. The perimeter of the triangle is 91 cm. What is the length of the base of the triangle?

HW Pg. 83 #1-5 Pg. 88 #1-6 STUDY

Lesson #11 Area

Lesson Goals SWBAT find the area of a rectangle, square, and triangle

Do Now Mr. P would like to paint his classroom blue, his happy color. Unfortunately he does not know the area he needs to paint. Can you determine this, approximately and in square feet, using the classroom rulers?

Finding areas of various rectangles
What is a general formula for finding the area of a rectangle?

Area of a right triangle
What is a general formula for finding the area of a triangle? How does it relate to the formula for the area of a rectangle and why?

Problems with area The length of one side of a rectangle is 4 less than 3 times an adjacent side. The perimeter of the rectangle is 64. Find the area of the rectangle.

Areas of non-right triangles
Find the area of triangle DEF.

HW Pg. 88 #1-6 6 is hard.

Same Base, Same Altitude
Lesson #12 Same Base, Same Altitude

Lesson Goals Students will understand and solve problems using the same base principle Students will understand and solve problems using the same altitude principle

Do Now Determine [ABC]/[ACD] and [ABC]/[ABD].

Same altitude property
If two triangles share an altitude, then the ratio of their areas is the ratio of the bases to which that altitude is drawn This is particularly useful for problems in which two triangles have bases along the same line

Find the ratio Determine the ratio of [ABC]/[ABD] C D A B

Same base property If two triangles share a base, then the ratio of their areas is the ratio of the altitudes to that base

HW Pg. 91 # 1 – 5 Pg. 93 # 13, 14, 15, 16, 17

Lesson #13 Triangle Similarity

Lesson Goals SWBAT define the term “similarity”
SWBAT determine AA similarity in a triangle SWBAT solve for sides of a triangle based on AA similarity

Do Now The shapes below are identical, except one is a “blown-up” version Discuss with a partner: what do you think you can say about the angles and the sides in this shape?

Similar Shapes We call two figures similar if one is simply a blown-up, and possibly rotated and/or flipped, version of the other Similar figures will have IDENTICAL ANGLES and their SIDES WILL BE IN THE SAME PROPORTION to one another

AA Similarity Two triangles are similar if they have two identical angles Why not 3 identical angles?

Can you solve for the value of x?
12 x 5 10

Can you solve for x and y? 3 y 5 4 x

Another note on this diagram
x/w=y/z Why? z y x w

HW Pg. 101 # 1, 2 Pg. 108 # 1, 2 Extra Credit Offering (For 3 additional points on your last examination IF you scored below an 85 or 1 additional point IF you scored over an 85): Pg. 96 # 32, 34, 35, 38, 39

Lesson #14 SAS Similarity

Lesson Goals SWBAT find similar triangles using SAS
SWBAT use SAS similarity to solve for sides of a triangle

Do Now Solve for x. 4 7 9 x

What can you say about the triangles below?
10 5 5 2.5

SAS Similarity If two sides in one triangle are in the same ratio as two sides in another triangle, and the angles between these sides are equal, then the triangles are similar

A hard example! Given AC=4, CD=5, and AB=6 as in the diagram, find BC if the perimeter of BCD is 20 B 6 A D 4 C 5

HW Pg. 112 # 1-4

Lesson #15 SSS Similarity

Lesson Goals SWBAT find similar triangles using SSS
SWBAT use SSS similarity to solve for sides of a triangle

Do Now: Conjectures? What do you think we can say about the two triangles below? Why? (Be specific) 4 14 7 5 10 8

SSS Similarity SSS similarity tells us that if each side of one triangle is the same constant multiple of the corresponding side of another triangle, then the triangles are similar Corollary: SSS similarity tells us that their corresponding angles are equal

Using SSS Similarity Given the side lengths shown in the diagram, prove that AE || BC and AB || DE A 5 B 4 12 6 C 4 D E 10

HW Pg. 114 #1

Using Similarity in Problems
Lesson #16 Using Similarity in Problems

Lesson Goals SWBAT use their knowledge of AA, SAS, and SSS similarity to solve problems

Do Now In the diagram, DE || BC, and the segments have the lengths shown in the diagram. Find x, y, and z A z 45 E D 27 36 60 64 y C B x

Practice Problem #1 (a) Use similar triangles to find ratios of segments that equal EF/AB (b) Use similar triangles to find ratios of segments that equal EF/DC (c) Use one ratio from each of the first two parts and add them to get an equation you can solve for EF D A E 12 9 B C F 40

Areas and Similarity ABC ~ XYZ, AB/XY=4, and [ABC]=64. Draw this.
Let c be the altitude of ABC to AB and let z be the altitude of XYZ to XY. Draw this. What is c/z? Find [XYZ]. Can you make a general statement about the area of similar triangles?

Area and Similarity If two triangles are similar such that the sides of the larger triangle are k times the size of the smaller, then the area of the larger triangle is k2 times that of the smaller!

Proof with Similarity In the diagram, PX is the altitude from right angle QPR of right triangle PQR as shown. Show that PX2=(QX)(RX), PR2=(RX)(RQ), and PQ2=(QX)(QR). How does the transitive property come into play here? P R Q X

HW Pg. 115 # 16, 18 Pg. 120 # 1-4

Lesson #17

Lesson Goals SWBAT define the legs and hypotenuse of a right triangle
SWBAT to prove the Pythagorean Theorem (just one of the many proofs) SWBAT use the PT to find the sides of a right triangle

Do Now Prove that a2=cd Prove that b2=ce
Use the two statements above to show that a2+b2=c2 a b d e c

Key Vocabulary Pythagorean Theorem: a2+b2=c2, where a and b are the legs of a right triangle, and c is the hypotenuse of the same right triangle

Find the missing side 3 4

Find the missing side 6 10

Find the missing side 3 5 4 9 x

HW Pg. 139 # 1, 2, 4, 5

Two Special Right Triangles
Lesson #18 Two Special Right Triangles

Lesson Goals SWBAT find the side lengths of a triangle, given one side SWBAT find the side lengths of a triangle, given one side

Do Now: Using Pythagorean Theorem, show that side AB and BC must both be 1 A √2 B C

Side Note The √2 is called “irrational”
The Pythagorean who determined that it was irrational was killed

Find the length of x in each of the following: can you write a rule?
2 1 4 3

30-60-90 A 30-60-90 triangle will have sides in the ratio 1:√3:2
Here’s why…

Proof

Finding the sides of a 30-60-90 triangle
1 8 y y x x

HW Pg. 146 # 1, 2

Lesson #19 Pythagorean Triples

Lesson Goals SWBAT recognize Pythagorean triples
SWBAT generate an infinite number of Pythagorean triples based on a given {a,b,c} triple SWBAT generate an infinite number of Pythagorean triples using even numbers

Pythagorean Triples A Pythagorean triple is a set of three whole numbers (integers greater than 0) that satisfy the Pythagorean Theorem

Pythagorean Triple Contest!
Split into groups of 3 and write as many Pythagorean triples as you can in 5 minutes The winners shall be held up in the glory of the SUNSHINE CORNER and receive an additional 10 points on their next homework

Prove it Given {a,b,c} is a set of Pythagorean triples, prove that {na,nb,nc} is a set of Pythagorean triples for any whole number n

How to generate a massive amount of triples
Take any even number and call it a Divide by 2 Square it Call this number z Subtract 1 from z Call this number b Add 1 to z Call this number c {a,b,c} is a Pythagorean triple

Big Pythagorean triples to know
{3,4,5}  The Granddaddy of them all {5,12,13} {7,24,25} {8,15,17}

HW Finish problems from last lesson Pg. 151 # 1, 3, 4

Lesson #20 Congruence and Similarity Revisited (in the context of right triangles and the PT)

Lesson Goals SWBAT prove two right triangles congruent given two sides
SWBAT prove two right triangles similar given two sides

Do Now Prove that the two right triangles below are congruent. What can you say if you are given two right triangles with identical hypotenuses and one identical leg? 15 12 12 15

Hypotenuse-Leg Congruence
HL congruence states that if the hypotenuse and a leg of one right triangle equal those of another, then the triangles are congruent. Note you don’t need leg-leg congruence, because you already have it by SAS.

Hypotenuse-Leg Similarity
Prove the two triangles below are similar: 16 15 20 12

HL Similarity HL Similarity states that if the hypotenuse and a leg of one right triangle are in the same ratio as the hypotenuse and leg of another right triangle, then the two triangles are similar

If a radius of a circle bisects a chord of a circle…
The center of a circle is 4 units away from a chord PQ of the circle. If PQ=12, what is the radius of the circle?

HW Pg. 155 #1, 2, 4

Lesson #21 Heron’s Formula

Lesson Goals SWBAT state Heron’s formula
SWBAT apply Heron’s formula to find the area of a triangle, given three side lengths

Do Now Find the area of the two triangles below: 7 9 7 9 7 9

Heron’s Formula [Proof]
You will not need to prove Heron’s (it’s quite a lot of algebra), but you will need to be able to apply it Heron’s formula states that given three sides of a triangle, {a, b, c}, the area of the triangle is √(s(s-a)(s-b)(s-c)), where s=(a+b+c)/2

Applying Heron’s Formula
Use Heron’s Formula to find the area of the triangles below 7 7 11 9 7 7

HW Pg. 160 #1, 2 Spend 10 minutes (then stop if you are hitting a wall) going through the Heron’s formula proof, just for your own edification…

Perpendicular Bisectors of a Triangle
Lesson #22 Perpendicular Bisectors of a Triangle

Do Now From what two points must every point on k be equidistant?
From what two points must every point on m be equidistant?

Lesson Goals Students will be able to define concurrent, circumcenter, circumradius, circumcircle SWBAT evaluate the circumradius of a triangle for all triangles and the special case of the right triangle

Perpendicular bisectors of the sides of a triangle are CONCURRENT
Lines are concurrent if they all meet at a single point The point at which the perpendicular bisectors of a triangle meet is called the circumcenter The circle centered at the circumcenter that passes through the vertices of the original triangle is called the circumcircle, which is circumscribed about the triangle The circumradius is the radius of this circle

Circumcenter of a right triangle
Construct at least TWO right triangles in your books using a protractor and a straightedge Create perpendicular bisectors of all three sides for your triangles Where is the circumcenter in all of your triangles?

Circumcenter of a right triangle
The circumcenter of a right triangle is the midpoint of the hypotenuse The circumradius is ½ the length of the hypotenuse Therefore, the hypotenuse is the diameter of the circumcircle

How many points define a circle?
1, 2, 3, more?

Find the circumradius Find the circumradius of an equilateral triangle with side length 6

HW Pg. 177 #2, 3, 4

Angle Bisectors of a Triangle
Lesson #23 Angle Bisectors of a Triangle

Lesson Goals SWBAT define the incenter, inradius, and incircle of a triangle SWBAT derive and understand the angle bisector theorem SWBAT find the area of a triangle given its inradius and its side lengths

Do Now Construct a triangle and bisect two of its angles
What can you say about the bisector of the third angle?

Key Vocabulary The angle bisectors of a triangle are concurrent at a point called the incenter The common distance from the incenter to the sides of the triangle is called the inradius The circle inscribed in the triangle is called the incircle NOTE: Each triangle has only one incircle, whose center is the intersection of the angle bisectors of a triangle

The Angle Bisector Theorem
Given: Triangle ABC with BE its angle bisector Then: AB/AE=CB/CE B A C E

How to use Angle Bisector Theorem
Find AC in the diagram A 12 C 7 D B 6

Results from bisecting an angle
Distance from all three sides is equal at the incenter (note this was not the case with perpendicular bisectors) Therefore the incircle is tangent to each side of the triangle at just one point and is inscribed within the triangle

Finding the Area of a Triangle from its inradius
Recall that all the perpendicular lines drawn to the sides from the inradius are equal in length Can you write a formula for the area of the triangle given an inradius of length r and side lengths of a, b, and c?

Finding the Area of a Triangle from its inradius
The area of a triangle equals its inradius times its semiperimeter (s=(a+b+c)/2) Example: Find the radius of a circle that is tangent to all three sides of triangle ABC, given that the sides of ABC have lengths 7, 24, and 25

HW Pg. 182 #1, 2, 3, 5, 7

Lesson #24 Medians

Lesson Goals SWBAT define median, centroid, and medial triangle
SWBAT show that medians divide the triangle into 6 triangles of equal area SWBAT show that the centroid cuts each median into a 2:1 ratio SWBAT prove the midline theorem

New Vocabulary A median of a triangle is a segment from a vertex to the midpoint of the opposite side The medians of a triangle are concurrent at a point called the centroid of the triangle

Do Now Show that the medians of triangle ABC cut the triangle into six triangles of equal area A B C

Medians and ratios Show that the centroid of any triangle cuts each of the triangle’s medians into a 2:1 ratio, with the longer portion being the segment from the centroid to the vertex A B C

The Medial Triangle In ABC below, DEF is referred to as the medial triangle A B C E D F

Prove the four smaller triangles below are congruent
D F

The Midline Theorem DEF~ABC, DEF=FBD=AFE=EDC EF/BC=DE/AB=DF/AC=1/2
DF||AC, EF||BC, DE||AB A B C E D F

HW Pg. 187 #1, 2, 3

Lesson #25 Altitudes

Lesson Goals SWBAT define orthocenter
SWBAT prove that the lines containing the altitudes of any triangle are concurrent SWBAT solve problems involving the properties of altitudes

Do Now Where do you think the altitudes of a right triangle intersect? (Don’t prove this; just use a few examples)

Prove: Altitudes are concurrent
This is legitimately very hard… Draw a line parallel to BC through A, parallel to AB through C, and parallel to AC through B The intersections of these lines form another triangle, which we’ll call JKL Prove CAK=ACB Show that A, B, and C are the midpoints of KL, JL, and JK, respectively Describe the relationship of AD, BE, and CF to JKL What does this imply? C B A D E F

Vocabulary The altitudes of any triangle are concurrent at a point called the orthocenter

Example involving altitudes
Altitudes QZ and XP of XYZ intersect at N. Given that <YXZ=70° and <XZY=45°, find: m<ZXP m<XZQ m<YXP

Point of interest with orthocenters
The altitudes of ABC meet at point H. At what point do the altitudes of ABH meet? How about ACH? C B A D E F H

HW Pg. 192 #1, 2, 4 (this is a proof), 5 (this is a proof also)

Lesson Goals SWBAT define a quadrilateral by its sides, vertices, and angle measures SWBAT find the measures of angles of a quadrilateral

Do Now Write a full proof demonstrating how many degrees are in the sum of the angles of a convex quadrilateral (below).

Convex Concave >180 degrees!

Other major types Think/pair/share: Name some other quadrilaterals that you know.

Prove that any convex quadrilateral has angles of total measure 360 degrees

Finding angle measures in a quadrilateral
A quadrilateral has angles of measure x, 3x+20, 2x-20, and 6x+12. Find all the angles in the quadrilateral and sketch what it might look like. Is the quadrilateral concave or convex?

HW Pg. 208 #1, 2, 3

Lesson #27 Trapezoids

Lesson Goals SWBAT define a trapezoid and the special types of trapezoids SWBAT find angle measures in a trapezoid SWBAT find the area of a trapezoid

Finding the area of a trapezoid
Below is a trapezoid, a quadrilateral with (only) two parallel sides. Using what you know about the area of rectangles and triangles, find the area of the trapezoid 6 8 12

Area of a trapezoid If x and y are the lengths of the two bases and h is the height of a trapezoid ABCD, [ABCD]=(x+y/2)(h)=(the average of the base lengths)(height)

Finding the area of a trapezoid
Find the area of the below trapezoid 13 7 4

Angles in a trapezoid Find the base angles in the trapezoid below:
106°

Trapezoids and Parallel Lines
Most problems with trapezoids can be reduced to the facts about parallel lines and similar triangles we learned at the beginning of the year

Special Type of Trapezoids
Isosceles trapezoids have: Two equal-length legs Congruent base angles Equal-length diagonals ANY OF THESE DEFINES AN ISOSCELES TRAPEZOID!

HW Pg. 214 #1-4

Lesson #28 Parallelograms

Lesson Goals SWBAT define a parallelogram
SWBAT find that a shape is a parallelogram based on its diagonals

Do Now A parallelogram is a quadrilateral made up of two pairs of parallel sides Find x, y, and <C in the parallelogram below x+y 3x 30°

Prove: AE=CE Given: ABCD is a parallelogram A B E D C

The diagonals of a parallelogram
The diagonals of a parallelogram bisect one another, as you just proved

Area of a parallelogram
Find the area of the parallelogram below Hint: remember how we proved the area of a trapezoid 14 11 15

Area of a parallelogram
Easy! A=bh Just like a rectangle (intuitive geometric way of showing this?)

HW Pg. 218 #1, 2, 3, 5

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