Congruent Triangles in the World By Sierra Smith

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?

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

Theorems 4.1 and 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 - 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 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 <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 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

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 congruent.

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 are congruent.

4.5 Using Congruent Triangles: Denmark 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 the perimeter of ABD is 16 feet ( ). A CD 2 B 1 E “Hey! I used congruent triangles for that!”

4.6 Isosceles, Equilateral and Right Triangles: Greece 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 congruent.

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. A B C The distance formula

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. Engineers have to make sure they design products with congruent triangles so their machines work the way they are invented to work.