Download Geometry Unit 3 slides

Do Now
• I need a volunteer for the class job of pass-out
specialist (benefit, you get a pass on Do Nows
for the week)
1. Find the measure of angle b
2. Find the value of x.
Review
• Name the angle types and solve for b in each
angle.
Intro to Triangles
Students will find measures of angles
in triangles.
Video
• http://www.youtube.com/watch?v=vwrOqDBAvs&feature=related
Triangle Sum Theorem
• Triangle Sum Theorem – the sum of the
measures of the angles of a triangle is 180.
Example #1
• What is the measure of angle 1?
Example #1
• What is the measure of angle 1?
• <1 = 180 – 37 – 57 = 86
How many Triangles?
How many Triangles?
Answer: 3! ABD, BDC, and the big triangle ABC.
How many triangles?
Example #2
• Find the values of x, y, and z.
Example #2
• Find the values of x, y, and z.
• 43 + 59 + x = 180. x = 180 – 59 – 43. x = 78
• x + y = 180. 78 + y = 180. y = 180 – 78. y = 102
• y + z + 49 = 180. 102 + z + 49 = 180.
z = 180-49-102. z = 29.
Exterior Angle of a Polygon
• Exterior Angle of a Polygon – an angle formed
by a side and an extension of an adjacent side.
–1
• Remote Interior Angles – the two
nonadjacent interior angles
– 2 and 3
• Name the exterior angle of the polygon and
the remote interior angles in the diagram
below.
Triangle Exterior Angle Theorem
• The measure of each exterior angle of a
triangle equals the sum of its two remote
interior angles.
Example #1
• What is the measure of angle 1?
Example #1
• What is the measure of angle 1?
• <1 = 80 + 18
• <1 = 98
Example #2
• What is the measure of angle 2?
Example #2
• What is the measure of angle 2?
• <2 + 59 = 124
• <2 = 124 – 59
• <2 = 65
You Try
• On the back of your notes!
• Find the values of the variables x, y, and z.
You Try
• Find the values of the variables x, y, and z.
• y = 36, z = 90, x = 38
You Try
On the back of your notes: Find the values of
the variables and the measures of the angles.
You Try
On the back of your notes: Find the values of the
variables and the measures of the angles.
• (2x + 4) + (2x – 9) + x = 180
• x = 37
Exit Ticket
1. Find the value of <1 in the diagram to the right
2. Find the value of x, y, and z
3. Solve for x.
HOMEWORK!!!: Page 75 1,3,4,8,10,17,18
Do Now
1. Find 𝑚∠1
2. Find 𝑚∠3, 𝑚∠4, 𝑎𝑛𝑑 𝑚∠5
Exit Ticket 10/1
1. Find the value of <1 in the diagram to the right
2. Find the value of x, y, and z
3. Solve for x.
HOMEWORK!!!: Page 75 1,3,4,8,10,17,18
Congruent Figures
Students will be able to find
corresponding parts of congruent
figures
Congruent Figures
• Congruent figures have the same size and
shape.
Congruent Figures
• Congruent Polygons have congruent
corresponding parts - their sides and angles
match!!
Congruent Figures
• ***When naming congruent polygons, you
MUST list the corresponding vertices in the
SAME ORDER.
Video
• http://app.discoveryeducation.com/player/vie
w/assetGuid/A39E2AC4-E031-4E6A-8115FC59EF04BF76
Let’s Practice!
• WXYZ  JKLM
1. Line segment WX  _?_
2. Line segment KL  _?_
3. Line segment MJ  _?_
4. ∠𝑊 ≅ _?_
5. ∠𝐾 ≅ _?_
6. ∠𝑍 ≅ _?_
Let’s Practice
1. Complete the following statements: Given:
ΔNMK ≅ ΔVYZ
a) line segment MK ≅ line segment _?_
b) line segment VY ≅ line segment _?_
c) ∠𝑍 ≅ _?_
d) ∠𝑁 ≅ _?_
Let’s Practice!
Third Angles Theorem
You try!
• What is 𝑚∠𝐷?
You try
• ∆𝑁𝐽𝑇 ≅ ∆𝐷𝑂𝐺, 𝑚∠𝑁 = 55 𝑎𝑛𝑑 𝑚∠𝐺 =72.
What is 𝑚∠𝑂? (Draw a diagram to help you
answer the question. Think back to the last
problem)
Exit Ticket 10/2
1. Complete the following statements: Given: ΔDEF ≅
ΔGZT
a) line segment DE ≅ line segment _?_
b) ∠𝑍 ≅ _?_
2. ∆ABC  ∆LMN. Name all of the pairs of corresponding
congruent parts. (Draw a picture of the two triangles
on a separate sheet of paper to help you answer the
question.)
3. ∆𝑋𝑌𝑍 ≅ ∆𝑂𝑃𝑄, 𝑚∠𝑋 = 47 𝑎𝑛𝑑 𝑚∠𝑃 = 65. What is
𝑚∠𝑍? (Draw a diagram to help you answer the
question.)
Do Now
1. Complete the following statements: Given: ΔSML ≅
ΔTNY
a) line segment ML ≅ line segment _?_
b) ∠𝑌 ≅ _?_
2. ∆QRS  ∆TUV. Name all of the pairs of corresponding
congruent parts. (Draw a picture of the two triangles
on a separate sheet of paper to help you answer the
question.)
3. ∆𝐴𝐵𝐶 ≅ ∆𝐷𝐸𝐹, 𝑚∠𝐹 = 21 𝑎𝑛𝑑 𝑚∠𝑃 = 50. What
is 𝑚∠𝐵? (Draw a diagram to help you answer the
question.)
Exit Ticket 10/2
1. Complete the following statements: Given: ΔDEF ≅
ΔGZT
a) line segment DE ≅ line segment _?_
b) ∠𝑍 ≅ _?_
2. ∆ABC  ∆LMN. Name all of the pairs of corresponding
congruent parts. (Draw a picture of the two triangles
on a separate sheet of paper to help you answer the
question.)
3. ∆𝑋𝑌𝑍 ≅ ∆𝑂𝑃𝑄, 𝑚∠𝑋 = 47 𝑎𝑛𝑑 𝑚∠𝑃 = 65. What is
𝑚∠𝑍? (Draw a diagram to help you answer the
question.)
Yesterday we learned that…
• … two polygons were congruent if all sides
AND all angles were congruent.
• But that’s WAY more info than we need!!
Today we will learn…
• …how to prove that two triangles are
congruent by using:
1. 3 pairs of corresponding sides
2. 2 pairs of corresponding sides and 1 pair of
corresponding angles
3. 1 pair of corresponding sides and 2 pairs of
corresponding angles
Tick Marks and Curves
What do those red tick marks and curves mean?
You Try!
1. The single tick mark means line segment
NJ ≅ _?_
2. The double tick marks mean FR ≅ _?_
3. The curve means ∠𝑅 ≅ _?_
Side-Side-Side Postulate (SSS)
Side-Angle-Side Postulate (SAS)
Identifying Congruent Triangles
Look at the triangles:
1. How many congruent sides do we have (count the
sets of tick marks).
2. How many congruent angles do we have ?
3. Are the angles between the sides?
4. Are the triangles congruent? Justify.
Identifying Congruent Triangles
Look at the triangles:
1. How many congruent sides do we have (count the
sets of tick marks).
2. How many congruent angles do we have ?
3. Are the angles between the sides?
4. Are the triangles congruent? Justify.
Identifying Congruent Triangles
Would you use SSS or SAS to prove the triangles
congruent? If there is not enough information,
write not enough information.
Identifying Triangles with Funky
Shapes
Are the following triangles congruent? Justify.
What else do I need to know?
What other information do you need to prove
∆ABC ≅ ∆ADC by SAS? Explain your answer.
What else do I need to know?
What other information do you need to prove ∆𝑁𝐻𝐽 ≅
∆𝐷𝑅𝐹 by SAS? Explain your answer.
1. What does SAS mean?
2. What do I have currently?
3. What else do I need?
What else do I need to know?
What other information do you need to prove
∆ABC ≅ ∆ADC by SAS? Explain your answer.
1. Answer: NH ≅ DR
2. Explanation: Already know JH ≅ FR and ∠𝑅 ≅ ∠𝐻 ,
so with NH ≅ DR, we have a side, and angle in
between and
another side.
You Try
What other information do you need to
prove ∆ABC ≅ ∆ADC by SSS? Explain your
answer.
Exit Ticket
1. Are the triangles to the
right congruent? Justify.
2. What other information
do you need to prove
∆𝐴𝐵𝐶 ≅ ∆𝐿𝑀𝑁
Do Now
1. Are the triangles at the
right congruent? Justify
2. Are the triangles at the
right congruent? Justify
3. What are the two
postulates we learned
yesterday to prove two
triangles are congruent?
Exit Ticket from Yesterday
1. Are the triangles to the
right congruent? Justify.
2. What other information
do you need to prove
∆𝐴𝐵𝐶 ≅ ∆𝐿𝑀𝑁
Two more postulates
Angle-Side-Angle Postulate (ASA)
Angle-Angle-Side Theorem (AAS)
All Together Now
• 4 Ways to prove triangles congruent:
1. SSS – If two triangles have THREE congruent pairs of
sides, they are congruent by SSS
2. SAS – If two triangles have TWO congruent pairs of
sides and an angle BETWEEN them, they are
congruent by SAS
3. ASA – If two triangles have TWO congruent pairs of
ANGLES and a side BETWEEN them, they are
congruent by ASA
4. AAS – If two triangles have TWO congruent pairs of
ANGLES and a side NOT BETWEEN them, they are
congruent by AAS.
Which two triangles are congruent by
ASA?
• List the theorem/postulate that you would use
to prove the two triangles are congruent. If
none apply, write not enough information.
1.
2.
3.
4.
Exit Ticket
Do Now
Let’s Finish our Worksheets
• You will be turning in ASA and AAS today!
• SSS and SAS will be due at the beginning of
class Monday.
Worksheet
• Find EVERY vertical angle (“kissing Vs”) and
shared side and draw in angle marks or tick
marks
• Label EVERY angle with an A and side with an S
Worksheet
• Find EVERY vertical angle (“kissing Vs”) and
shared side and draw in angle marks or tick
marks
• Label EVERY angle with an A and side with an S
Worksheet Back Page
• You should have labeled all given angles and
sides
Worksheet Back Page
• You should have labeled all given angles and sides
• NOW find the angle or side that will prove
congruence by the theorem listed
Worksheet Back Page
• Finally, YOU MUST LIST THE NEW INFORMATION
(NEW SIDES/ANGLES)
• You will NOT receive credit unless you do this
∠𝐹𝐾𝐿 ≌ ∠𝐽𝐿𝐾
∠𝑉𝐻𝐺 ≌ ∠𝐼𝐺𝐻
Practice
∠𝐴𝐵𝐶 ≌ ∠𝐺𝐻𝐼
Justification:
𝐴𝐵 ≌ 𝐺𝐻
𝐵𝐶 ≌ 𝐻𝐼
CA ≌ 𝐼𝐺
Practice
Practice
• List the theorem/postulate that you would use
to prove the two triangles are congruent. If
none apply, write not enough information.
1.
2.
3.
4.