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
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?
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.
m<A + m<B + m<C = 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.
m<D = m<A + m<B
- Corollary to the Triangles Sum Theorem-
The acute angles of a right triangle are complementary.
m<A + m<B = 90 ̊
4.2 Congruence and Triangles: the
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 <1 is
congruent to <2, and she knows that triangle A is congruent to
triangle B. She also knows that m<4 = m<3. Using the Third
Angles Theorem, theorem 4.3, she concludes that the third
angles of the triangles are congruent.
What’s the other theorem in lesson 4.2? You guessed it! It’s 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 1
triangle DEF is congruent to triangle ABC. 2
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.
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 it’s 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 what’s 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
4.4 Proving Triangles are Congruent:
ASA and AAS: Egypt
Anna’s 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.
She’d 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
4.5 Using Congruent Triangles:
Anna’s 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 A B
the perimeter of ABD is 16 feet (5 + 5 + 6). 1 2
C D E
“Hey! I used congruent
triangles for that!”
4.6 Isosceles, Equilateral and Right
Anna’s 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
4.7 Triangles and Coordinate Proof: I’m 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 <ABC 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. “I’ll just graph this!”
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.
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
•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.
•Engineers have to make sure they design products with congruent triangles so their machines work the way they are invented