The Angle Sum of a Triangle...angle sum of a triangle is 180 A B C x y z Title...
Transcript of The Angle Sum of a Triangle...angle sum of a triangle is 180 A B C x y z Title...
![Page 1: The Angle Sum of a Triangle...angle sum of a triangle is 180 A B C x y z Title tdt_G_twoproofsanglesumoftriangle.pps Author AAMT Created Date 20130906071751Z ...](https://reader034.fdocuments.net/reader034/viewer/2022042612/5f5ae25e4ce60631f63d9f3b/html5/thumbnails/1.jpg)
The Angle Sum of a Triangle
![Page 2: The Angle Sum of a Triangle...angle sum of a triangle is 180 A B C x y z Title tdt_G_twoproofsanglesumoftriangle.pps Author AAMT Created Date 20130906071751Z ...](https://reader034.fdocuments.net/reader034/viewer/2022042612/5f5ae25e4ce60631f63d9f3b/html5/thumbnails/2.jpg)
Prove that x + y + z = 180°
A
B
C
x
y
z
![Page 3: The Angle Sum of a Triangle...angle sum of a triangle is 180 A B C x y z Title tdt_G_twoproofsanglesumoftriangle.pps Author AAMT Created Date 20130906071751Z ...](https://reader034.fdocuments.net/reader034/viewer/2022042612/5f5ae25e4ce60631f63d9f3b/html5/thumbnails/3.jpg)
Construct PQ through A so that PQ||BC
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B
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Q
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x
y
z
![Page 4: The Angle Sum of a Triangle...angle sum of a triangle is 180 A B C x y z Title tdt_G_twoproofsanglesumoftriangle.pps Author AAMT Created Date 20130906071751Z ...](https://reader034.fdocuments.net/reader034/viewer/2022042612/5f5ae25e4ce60631f63d9f3b/html5/thumbnails/4.jpg)
∠PAB = y° (alternate angles, PQ||BC)
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![Page 5: The Angle Sum of a Triangle...angle sum of a triangle is 180 A B C x y z Title tdt_G_twoproofsanglesumoftriangle.pps Author AAMT Created Date 20130906071751Z ...](https://reader034.fdocuments.net/reader034/viewer/2022042612/5f5ae25e4ce60631f63d9f3b/html5/thumbnails/5.jpg)
∠QAC = z° (alternate angles, PQ||BC)
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B
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Q
P
x
y
z
z y
![Page 6: The Angle Sum of a Triangle...angle sum of a triangle is 180 A B C x y z Title tdt_G_twoproofsanglesumoftriangle.pps Author AAMT Created Date 20130906071751Z ...](https://reader034.fdocuments.net/reader034/viewer/2022042612/5f5ae25e4ce60631f63d9f3b/html5/thumbnails/6.jpg)
Now PAQ is a straight line
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B
C
Q
P
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y
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z y
![Page 7: The Angle Sum of a Triangle...angle sum of a triangle is 180 A B C x y z Title tdt_G_twoproofsanglesumoftriangle.pps Author AAMT Created Date 20130906071751Z ...](https://reader034.fdocuments.net/reader034/viewer/2022042612/5f5ae25e4ce60631f63d9f3b/html5/thumbnails/7.jpg)
x + y + z = 180° (PAQ is a straight line)
A
B
C
Q
P
x
y
z
z y
![Page 8: The Angle Sum of a Triangle...angle sum of a triangle is 180 A B C x y z Title tdt_G_twoproofsanglesumoftriangle.pps Author AAMT Created Date 20130906071751Z ...](https://reader034.fdocuments.net/reader034/viewer/2022042612/5f5ae25e4ce60631f63d9f3b/html5/thumbnails/8.jpg)
x + y + z = 180°
A
B
C
x
y
z
![Page 9: The Angle Sum of a Triangle...angle sum of a triangle is 180 A B C x y z Title tdt_G_twoproofsanglesumoftriangle.pps Author AAMT Created Date 20130906071751Z ...](https://reader034.fdocuments.net/reader034/viewer/2022042612/5f5ae25e4ce60631f63d9f3b/html5/thumbnails/9.jpg)
The angle sum of a triangle is 180°
A
B
C
x
y
z
![Page 10: The Angle Sum of a Triangle...angle sum of a triangle is 180 A B C x y z Title tdt_G_twoproofsanglesumoftriangle.pps Author AAMT Created Date 20130906071751Z ...](https://reader034.fdocuments.net/reader034/viewer/2022042612/5f5ae25e4ce60631f63d9f3b/html5/thumbnails/10.jpg)
A different proof for the same result
![Page 11: The Angle Sum of a Triangle...angle sum of a triangle is 180 A B C x y z Title tdt_G_twoproofsanglesumoftriangle.pps Author AAMT Created Date 20130906071751Z ...](https://reader034.fdocuments.net/reader034/viewer/2022042612/5f5ae25e4ce60631f63d9f3b/html5/thumbnails/11.jpg)
Prove that x + y + z = 180°
A
B
C
x
y
z
![Page 12: The Angle Sum of a Triangle...angle sum of a triangle is 180 A B C x y z Title tdt_G_twoproofsanglesumoftriangle.pps Author AAMT Created Date 20130906071751Z ...](https://reader034.fdocuments.net/reader034/viewer/2022042612/5f5ae25e4ce60631f63d9f3b/html5/thumbnails/12.jpg)
Produce BC to P A
B
C P
x
y
z
![Page 13: The Angle Sum of a Triangle...angle sum of a triangle is 180 A B C x y z Title tdt_G_twoproofsanglesumoftriangle.pps Author AAMT Created Date 20130906071751Z ...](https://reader034.fdocuments.net/reader034/viewer/2022042612/5f5ae25e4ce60631f63d9f3b/html5/thumbnails/13.jpg)
At C construct CQ||BA A
B
C
Q
P
x
y
z
![Page 14: The Angle Sum of a Triangle...angle sum of a triangle is 180 A B C x y z Title tdt_G_twoproofsanglesumoftriangle.pps Author AAMT Created Date 20130906071751Z ...](https://reader034.fdocuments.net/reader034/viewer/2022042612/5f5ae25e4ce60631f63d9f3b/html5/thumbnails/14.jpg)
∠ACQ = x° (alternate angles, PQ||BC)
A
B
C
Q
P
x
y
z x
![Page 15: The Angle Sum of a Triangle...angle sum of a triangle is 180 A B C x y z Title tdt_G_twoproofsanglesumoftriangle.pps Author AAMT Created Date 20130906071751Z ...](https://reader034.fdocuments.net/reader034/viewer/2022042612/5f5ae25e4ce60631f63d9f3b/html5/thumbnails/15.jpg)
∠PCQ = y° (corresponding angles, PQ||BC)
A
B
C
Q
P
x
y
z y x
![Page 16: The Angle Sum of a Triangle...angle sum of a triangle is 180 A B C x y z Title tdt_G_twoproofsanglesumoftriangle.pps Author AAMT Created Date 20130906071751Z ...](https://reader034.fdocuments.net/reader034/viewer/2022042612/5f5ae25e4ce60631f63d9f3b/html5/thumbnails/16.jpg)
Now BCP is a straight line A
B
C
Q
P
x
y
z y x
![Page 17: The Angle Sum of a Triangle...angle sum of a triangle is 180 A B C x y z Title tdt_G_twoproofsanglesumoftriangle.pps Author AAMT Created Date 20130906071751Z ...](https://reader034.fdocuments.net/reader034/viewer/2022042612/5f5ae25e4ce60631f63d9f3b/html5/thumbnails/17.jpg)
A
B
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Q
P
x
y
z y x
x + y + z = 180° (BCP is a straight line)
![Page 18: The Angle Sum of a Triangle...angle sum of a triangle is 180 A B C x y z Title tdt_G_twoproofsanglesumoftriangle.pps Author AAMT Created Date 20130906071751Z ...](https://reader034.fdocuments.net/reader034/viewer/2022042612/5f5ae25e4ce60631f63d9f3b/html5/thumbnails/18.jpg)
x + y + z = 180° A
B
C
x
y
z
![Page 19: The Angle Sum of a Triangle...angle sum of a triangle is 180 A B C x y z Title tdt_G_twoproofsanglesumoftriangle.pps Author AAMT Created Date 20130906071751Z ...](https://reader034.fdocuments.net/reader034/viewer/2022042612/5f5ae25e4ce60631f63d9f3b/html5/thumbnails/19.jpg)
The angle sum of a triangle is 180°
A
B
C
x
y
z