Congruent Triangles

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Congruent Triangles in the World
By Sierra Smith
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• Anna Johnson is going on a trip with her family
  traveling all around the world! What triangles
  will she see in different countries and cities of
  the world?

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4.1 Triangles and Angles Hong Kong

• The first place Anna stops is Hong Kong, where
  she sees the Bank of China, the most famous Hong
  Kong skyscraper in the world! The first thing she
  notices are all the different triangles!
  Immediately she sees that all the triangles are
  isosceles triangles, triangles that have at least
  two congruent sides. And she quickly notices two
  right triangles, triangles with one right angle,
  in the pattern as well.
• And what other types of triangles are there that
  are not in the building?
• Acute Triangle
• Obtuse Triangle
• Scalene Triangle
• Equilateral Triangle
• Equiangular Triangle

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• Theorems 4.1 and 4.2
• 4.1- Triangle Sum Theorem-
• The sum of the measures of the interior angles
  of a triangle are 180 degrees.
• mltA mltB mltC 180
• 4.2- Exterior Angle Theorem-
• The measure of an exterior angle of a triangle
  is equal to the sum of the measures of the two
  nonadjacent interior angles.
• mltD mltA mltB
• Corollary

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4.2 Congruence and Triangles the Netherlands
Anna is on the plane to the Netherlands and she
flies over Bourtange and notices a triangular
pattern in the center of the city. She knows from
her studies of vertical angles that lt1 is
congruent to lt2, and she knows that triangle A is
congruent to triangle B. She also knows that mlt4
mlt3. Using the Third Angles Theorem, theorem
4.3, she concludes that the third angles of the
triangles are congruent. Whats the other
theorem in lesson 4.2? You guessed it! Its the
Properties of Congruent Triangles, Theorem
4.4. Reflexive Property of Congruent
Triangles- Every triangle is congruent to
itself. Symmetric Property of Congruent
Triangles- If triangle ABC is congruent to
triangle DEF, then triangle DEF is congruent to
triangle ABC. Transitive Property of Congruent
Triangles- If triangle ABC is congruent to
triangle DEF and triangle DEF is congruent to
triangle GHI, then triangle ABC is congruent to
triangle GHI.

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1
2
3
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4.3 Proving Triangles are Congruent SSS and SAS
Turkey

• Anna stays in a hotel in Aksu, Turkey, near
  Antayla, and notices two triangles made by a
  building behind the hotel and its reflection,
  sharing the edge of the lake as another side. She
  sees that the angles opposite the shared side are
  congruent, and the bases are congruent. Using the
  reflexive property, she knows the shared side
  between them is congruent to itself. Recently she
  learned the Side-Angle-Side postulate in
  geometry, stating that if two sides and the
  included angle of a triangle are congruent to two
  sides and the included angle of another triangle,
  then the two triangles are congruent, and so she
  can conclude that the two triangles are
  congruent.
• And whats the Side-Side-Side postulate? It
  states that if three sides of a triangle are
  congruent to three sides of a second triangle,
  then the two triangles are congruent.

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4.4 Proving Triangles are Congruent ASA and AAS
Egypt

• Annas next stop was Egypt, the home of the
  famous ancient Pyramids. As Anna stood looking
  at the Pyramids, she noticed two congruent angles
  between two triangular faces at the top of the
  pyramid and two congruent angles between the same
  faces at the bottom of the pyramid. Using the
  reflexive property, Anna knew the included side
  shared by both triangles was congruent to itself.
  Thinking back on her geometry lesson before the
  left, she realized the two triangles were
  congruent by the Angle-Side-Angle postulate.
• Shed also learned the Angle-Angle-Side Theorem
  4.5 that day that stated if two angles and a
  nonincluded side of one triangle are congruent to
  two angles and the corresponding nonincluded side
  of a second triangle, then the two triangles are
  congruent.

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4.5 Using Congruent Triangles Denmark
Annas next stop was Denmark. As she was walking
through the streets, she saw this little shop and
became curious of the patterns on the side. If
triangles 1 and 2 were congruent and their sides
measured 5 feet, 4 feet and 3 feet, and ABCD was
a perfect rectangle, how could she find out what
the perimeter was for triangle ABD? She decided
to use geometry to figure it out... If triangles
1 and two were congruent, then by CPCTC,
Corresponding Parts of Congruent Triangles are
Congruent, DB would be congruent to DA. She
already knew the lengths of the hypotenuses was 5
feet (she always had her ever-ready ruler) and
she knew the length of CD and DE were 3 feet.
Since ABCD was a rectangle, AB had to be
congruent to CE by definition of a rectangle, and
by Segment Addition Postulate, CE was 6 feet,
making AB 6 feet. Therefore the perimeter of ABD
is 16 feet (5 5 6).
A
B
2
1
C
D
E
Hey! I used congruent triangles for that!
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4.6 Isosceles, Equilateral and Right Triangles
Greece

• Annas next stop is Greece, her favorite country
  in her travels. As soon as the plane lands in
  Athens, Anna begs her mother to take a bus to the
  Acropolis to see the Parthenon, the most famous
  ancient temple in Greece. While studying the
  Parthenon, Anna notices that the triangle at the
  top of the temple is an isosceles triangle. Using
  the Base Angles Theorem 4.6, she could conclude
  that the two angles opposite the congruent sides
  are also congruent. And if she only knew the two
  angles were congruent, she could use the Base
  Angles Converse Theorem 4.7 to conclude the two
  sides opposite the angles were congruent.
• Anna sees she could also create two right
  triangles with congruent hypotenuses. Since they
  share the same base, or leg, she knows from the
  reflexive property that the leg is congruent to
  itself. Using the Hypotenuse-Leg Congruence
  Theorem 4.8, Anna knows that those two triangles
  are congruent.

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4.7 Triangles and Coordinate Proof Im Going Home
As Anna was on the plane ride home, she pulled
out her map and began looking at three of the
earlier countries she traveled to, picking out
Turkey A, Egypt B and China C. She noticed ltABC
was indeed a right angle, and pulling out her
ruler again, measured the lengths of the legs. AB
was .5 inches, BC was 3 inches, and just as she
was about to measure CA, Anna dropped her ruler.
No worries! She said, unable to find it. Ill
just graph this!
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The distance formula
A
C
B
Anna decided to make every four units measure 1
inch by her map. Using the distance formula, she
calculated the measurement of the hypotenuse,
CA. CA 3.04 inches by her map.
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• Careers Using Congruent Triangles
• Architects have to use congruent triangles in
  order to keep their design measurements the same.
• Designers use many congruent triangles in modern
  art and decorating rooms. They like to keep unity
  around the room and congruent figures, triangles
  especially, pull the room together.
• Airplane Pilots use triangles in coordinate grids
  to figure out the distance of their routes.
• Artists use congruent triangles in their
  compositions.
• Construction Workers use triangles in coordinate
  grids to figure out where everything should be
  built and placed and the measurements.
• The people who map constellations use triangles
  in coordinate grids to lay out where the stars
  are and the distances between the stars in
  constellations.
• Carpenters have to make sure the triangles they
  use are congruent so that the pieces of wood will
  fit the way they are meant to.
• Painters have to make sure they paint congruent
  triangles in their patterns.
• Sailors have to map their routes, sometimes
  triangles, in coordinate planes and find the
  distances between each point to make sure they
  have enough supplies to last.