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

2. 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

3. 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.”

4. Points and lines
Point
Line Segment
Ray
Line

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

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

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

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

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

10. 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?

11. 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

12. No?

13. 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.

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

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

16. 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.

17. All right angles are congruent.

18. 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.

19. Angles, and how to measure them
An angle is…

20. 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?

21. 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

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

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

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

25. 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

26. 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.

27. Parallel Line Notation

28. Transversals

29. Why does a triangle have 180 degrees?

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

31. Lesson #4
Triangles

32. 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

33. Do Now
Anyone complete the proof from yesterday?

34. 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 _________

35. 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

36. 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.

37. 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?

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

39. Parallel Lines Revisited
Lesson #5
Parallel Lines Revisited

40. 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”

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

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

43. Proof by contradictionk m

44. HW
Pg. 46 # 35-46

45. Lesson #6
Congruent Triangles

46. 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

47. Do Now
Triangle worksheet

52. Lesson #7
Congruent Triangles

53. 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

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

55. 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.

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

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

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

59. Lesson #8
Congruent Triangles

60. 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

61. 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

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

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

64. Isosceles and Equilateral Triangles
Lesson #9
Isosceles and Equilateral Triangles

65. 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

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

67. 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

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

69. Proofs with types of triangles
Given: m<ABD=m<ACD, m<BAD=1/2m<ABD, m<BAD=m<CAD
Prove: ABC is an equilateral triangle A B C D

70. HW
Pg. 71 #1-7

71. Lesson #10
Area and Perimeter

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

73. 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.

74. Find the perimeter 14
18√2

75. Find the area

76. 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?

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

78. Lesson #11
Area

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

80. 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?

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

82. 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?

83. 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.

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

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

86. Same Base, Same Altitude
Lesson #12
Same Base, Same Altitude

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

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

89. 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

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

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

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

93. Lesson #13
Triangle Similarity

94. 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

95. 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?

96. 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

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

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

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

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

101. 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

102. Lesson #14
SAS Similarity

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

104. Do Now
Solve for x. 4 7 9 x

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

106. 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

107. 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

108. HW
Pg. 112 # 1-4

109. Lesson #15
SSS Similarity

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

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

112. 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

113. 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

114. HW
Pg. 114 #1

115. Using Similarity in Problems
Lesson #16
Using Similarity in Problems

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

117. 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

118. 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

119. 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?

120. 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!

121. 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

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

123. Lesson #17

124. 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

125. 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

126. 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

127. Find the missing side 3 4

128. Find the missing side 6 10

129. Find the missing side 3 5 4 9 x

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

131. Two Special Right Triangles
Lesson #18
Two Special Right Triangles

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

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

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

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

136. 30-60-90 A 30-60-90 triangle will have sides in the ratio 1:√3:2
Here’s why…

137. Proof

138. Finding the sides of a 30-60-90 triangle1 8 y y x x

139. HW
Pg. 146 # 1, 2

140. Lesson #19
Pythagorean Triples

141. 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

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

143. 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

144. 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

145. 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

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

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

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

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

151. 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

152. 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.

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

154. 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

155. 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?

156. HW
Pg. 155 #1, 2, 4

157. Lesson #21
Heron’s Formula

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

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

160. 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

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

162. 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…

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

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

165. 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

166. 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

167. 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?

168. 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

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

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

171. HW
Pg. 177 #2, 3, 4

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

173. 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

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

175. 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

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

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

178. 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

179. 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?

180. 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

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

182. Lesson #24
Medians

183. 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

184. 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

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

186. 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

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

188. Prove the four smaller triangles below are congruentD F

189. 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

190. HW
Pg. 187 #1, 2, 3

191. Lesson #25
Altitudes

192. 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

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

194. 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

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

196. 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

197. 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

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

199. Introduction to Quadrilaterals
Lesson #26
Introduction to Quadrilaterals

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

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

202. Two types of quadrilaterals
Convex
Concave
>180 degrees!

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

204. Angles in a quadrilateral
Prove that any convex quadrilateral has angles of total measure 360 degrees

205. 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?

206. HW
Pg. 208 #1, 2, 3

207. Lesson #27
Trapezoids

208. 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

209. 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

210. 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)

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

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

213. 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

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

215. HW
Pg. 214 #1-4

216. Lesson #28
Parallelograms

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

218. 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°

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

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

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

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

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