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Focus – Prove the Triangle Sum Theorem Triangle Sum Theorem Focus - Prove the Triangle Sum Theorem
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Focus – Prove the Triangle Sum Theorem
Triangle Sum Theorem
Focus - Prove the Triangle Sum Theorem
Focus – Prove the Triangle Sum Theorem
If you add up the measures of the interior angles of a
triangle, What is the total?
180º
25º
65º
Focus – Prove the Triangle Sum Theorem
How many lines can be drawn through point B that would
become parallel to M�������������� �
M
B
Focus – Prove the Triangle Sum Theorem
Unique Parallel line Postulate – Through a point not on a given line, there is one and only one line parallel to the
given line.
m
b
This postulate will be used in the proofon the next slide.
Focus – Prove the Triangle Sum Theorem
Given:Prove: 1. ABC is a triangle2. Draw helping line through B so that
3.
4.5.6.7.
1. Given2. Unique Parallel Postulate
(pg. 439)3. If the sides of an angle form
a straight line, then the angle is a straight angle with a measure of 180º
4. Whole-Parts Postulate5. Substitution6. Alternate Interior Angle
Theorem7. Substitution Property
ABC1 2 3 180m m m
||DE AC���������������������������������������� ���DE�������������� �
B
CA
2
1 3
D
180m DBE
4 2 5m DBE m m m
E4 5
4 2 5 180m m m
1 4; 3 5m m m m
1 2 3 180m m m
Focus – Prove the Triangle Sum Theorem
Quadrilateral Theorem
• The sum of the measures of the angles of a quadrilateral are:
• What is the sum of the measures of a rectangle, which is a quadrilateral?
360
Focus – Prove the Triangle Sum Theorem
Theorem:Opposite Angles of
QuadrilateralsIf both pairs of opposite angles of a
quadrilateral are equal in measure, …
then the quadrilateral is a parallelogram.
Focus – Prove the Triangle Sum Theorem
What can we find out?
• The ladder is leaning on the ground at a 75º angle. At what angle is the top of the ladder touching the building?
75
Focus – Prove the Triangle Sum Theorem
What can we find out?
• The ladder is leaning on the ground at a 75º angle. At what angle is the top of the ladder touching the building?
75
75 90 180
165 180
15
x
x
x
15
Focus – Prove the Triangle Sum Theorem
Find the missing angle.
47º
Focus – Prove the Triangle Sum Theorem
Find the missing angle.
47º
43
Focus – Prove the Triangle Sum Theorem
Find the missing angles.
120º
xº 55º 25º yº
wº
120x
35y
35w
Order: y w x
Focus – Prove the Triangle Sum Theorem
Exterior Angle Theorem• An exterior angle of a triangle…• … is equal in measure to the sum of
the measures of its two remote interior angles.
Focus – Prove the Triangle Sum Theorem
Find the missing angles’ measurements (L and J).
120º
55º 25º
12035 35J L
90 55L 120 35J
remote interior angles
Focus – Prove the Triangle Sum Theorem
What is the measure of Angle U?
120º
55º 25º
12035 35J L
145L 155J
60
?
T
U
T U
To find the measure of angle T, use the definitionof supplemental angles.To find the measure of angle U, use the verticalangle theorem.
Focus – Prove the Triangle Sum Theorem
In today’s lesson we…
• Proved the Triangle Sum Theorem (180 degrees)
• Found Quadrilaterals’ Interior Angles Theorem to state the total degrees is 360.
• Used the above theorems and the Exterior Angle Theorem to find measurements of a Triangle’s exterior angle.
Focus – Prove the Triangle Sum Theorem
Assignment: pg. 443: 2-9, 13
Focus – Prove the Triangle Sum Theorem
For Warmup
70°
y°
z° w°
D E50° x°
(x-30)°
x°
70°
60°
50° 110°
D E50° 70°
(80-30)°
80°
Focus – Prove the Triangle Sum Theorem
From tdn.com last night.OLYMPIA, Wash. – The state Senate has made good on a
promise to help students graduate from high school by passing a bill that would eliminate one diploma obstacle.
The measure would eliminate the requirement to keep taking the math Washington Assessment of Student Learning every year until graduation.
High school students who fail the 10th grade math test, would still be required to keep earning math credits, however.
The bill, which passed on a vote of 43-4, still requires approval in the House. It would affect the class of 2009, if it passes both houses and is signed by the governor.
• The bill is Senate Bill 5498.• * Legislature (weblink) <http://www.leg.wa.gov>
Focus – Prove the Triangle Sum Theorem
Want to spend class time preparing for math WASL
questions?• Group Work• Group Points• Three sections will vote.• 60% will show favorable response.
Focus – Prove the Triangle Sum Theorem
Pg. 446: #17 Proof of Theorem
If both pairs of opposite angles of a quadrilateral are equal in measure, then the
quadrilateral is a parallelogram.
Focus – Prove the Triangle Sum Theorem
Given: JKLM is a quadrilateral
Prove: JKLM is a parallelogram 1. JKLM is a quadrilateral2. 3.4.5.6.7. are supplementary
angles
8.9.10. are
supplementary angles
11.12. JKLM is a
parallelogram
1. Given2. Quadrilateral Sum Theor.3. Given4. Substitution5. Distributive Property6. Division Property of Equality7. Def. of Supplementary
angles8. Converse of Co-Interior Th.9. Substitution10. Def. of Supplemental
Angles11. Chuck Taylor Co-Interior Th.
12. Def. of Parallelogram
360m J m K m L m M
M
;m J m L m K m M
360m J m M m J m M 2( ) 360m J m M
180m J m M &m J m M
m J m L
J
L
K
m K m M
||JK ML
&m J m K
180m J m K
||JM KL
statementsjustifications
Focus – Prove the Triangle Sum Theorem
Pg. 447: #34 Proof of Theorem
An Exterior Angle of a Triangle is equal in measure to the sum of its two remote
interior angles.
Focus – Prove the Triangle Sum Theorem
Given: with exterior Prove: 1. ABC is a triangle
with exterior angle 4.
2.
3.4.5.6.7.
1. Given
2. Straight Angle Postulate
3. Whole-Parts Postulate4. Substitution5. Triangle Sum
Theorem6. Substitution Property7. Subtraction Prop. Of
Equality
ABC4 1 2m m m
180m BCD
C D
3 4m BCD m m
3 4 180m m
1 2 3 180m m m
3 4 1 2 3m m m m m
4 1 2m m m
4A
B2
1
3 4
statements justifications
Focus – Prove the Triangle Sum Theorem
Assignment: pg. 444-448: 14-18a, 26-34, 37-40
