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  Transcript
  ```Sides Not Included
  5.6
  Angle-Angle-Side Congruence Theorem
  Learning Goals
  In this lesson, you will:
  Key Term
  • Angle-Angle-Side Congruence Theorem
  • Explore the Angle-Angle-Side Congruence
  Theorem using constructions.
  • Explore the Angle-Angle-Side Congruence
  Theorem on the coordinate plane.
  • Prove the Angle-Angle-Side
  Congruence Theorem.
  S
  ometimes, good things must come to an end, and that can be said for determining
  if triangles are congruent, given certain information.
  You have used many different theorems to prove that two triangles are congruent
  based on different criteria. Specifically,
  • Side-Side-Side Congruence Theorem
  • Side-Angle-Side Congruence Theorem
  • and Angle-Side-Angle Congruence Theorem.
  So, do you think there are any other theorems that can be used to prove that two
  triangles are congruent? Here’s a hint: we have another lesson—so there must be at
  least one more congruence theorem!
  389
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  Problem 1 Using Constructions to Support AAS
  There is another way to determine if two triangles are congruent
  that is different from the congruence theorems you have already
  proven. You will prove the Angle-Angle-Side Congruence Theorem.
  The Angle-Angle-Side Congruence Theorem states: “If two
  angles and a non-included side of one triangle are congruent to the
  corresponding angles and the corresponding non-included side of
  a second triangle, then the triangles are congruent.”
  Recall that
  a non-included side
  is a side that is not
  located between the
  two angles.
  First, you will prove this theorem through construction.
  5
  390 Chapter 5 Congruence Through Transformations
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  ___
  1. Construct triangle ABC given AB ​
  ​  and angles A and C. Then,
  write the steps you performed to construct the triangle.
  A
  A
  It might
  construct angle B first
  and use that to create
  B
  C
  5
  5.6 Angle-Angle-Side Congruence Theorem 451445_Ch05_333-418.indd 391
  391
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  ___
  ___
  2. How does the length of side AB ​
  ​  compare to the length of your classmates’ side AB ​
  ​
  ?
  3. Use a protractor to measure angle A and angle C in triangle ABC. What do you notice
  ?
  4. Thomas claims that his constructed triangle is not congruent because he drew a
  vertical starter line that created a triangle that has side AB being vertical rather than
  horizontal. Denise claims that all the constructed triangles are congruent even though
  Thomas’s triangle looks different. Who’s correct? Why is this student correct?
  5. Is your triangle congruent to your classmates’ triangles? Why or why not?
  5
  Chapter 5 Congruence Through Transformations
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  Problem 2 Using Reflection to Support AAS
  If two angles and the non-included side of a triangle are reflected, is the image of the triangle
  congruent to the pre-image of the triangle?
  1. Graph triangle ABC by plotting the points A (23, 26), B (29, 210), and C (21, 210).
  y
  8
  6
  4
  2
  0
  28 26 24 22
  2
  4
  6
  8
  x
  22
  24
  26
  28
  2. Calculate the length of each side of triangle ABC. Record the exact measurements in
  the table.
  Side of Triangle ABC
  Length of Side
  ___
  ​
  AB
  ​
  ____
  ​
  BC
  ​
  5
  ___
  ​ ​
  AC
  5.6 Angle-Angle-Side Congruence Theorem 451445_Ch05_333-418.indd 393
  393
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  ___
  3. Reflect angle A, angle
  B, and side BC ​
  ​  over the line of reflection y 5 22 to form angle D,
  ___
  angle E, and side EF ​
  ​  . Then, connect points D and E to form triangle DEF. Record the
  image coordinates in the table.
  Coordinates of Triangle ABC
  Coordinates of Triangle DEF
  A (23, 26)
  B (29, 210)
  C (21, 210)
  4. Calculate the length of each side of triangle DEF. Record the exact measurements in
  the table.
  Side of Triangle DEF
  Length of Side
  ___
  DE
  ​
  ​
  ___
  EF
  ​
  ​
  ___
  DF ​
  ​
  5
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  5. Compare the corresponding side lengths of the pre-image and image. What do
  you notice?
  You have shown that the corresponding sides of the image and pre-image are congruent.
  Therefore, the triangles are congruent by the SSS Congruence Theorem. However, you are
  proving the Angle-Angle-Side Congruence Theorem. Therefore, you need to verify that angle
  A and angle C are congruent to the corresponding angles in triangle DEF.
  6. Use a protractor to determine the angle measures of each triangle.
  a. What is the measure of angle A and angle C?
  b. Which angles in triangle DEF correspond to angle A and angle C?
  c. What do you notice about the measures of the corresponding angles in the
  triangles? What can you conclude from this information?
  5
  You have used a protractor to verify that the corresponding angles of the two triangles
  are congruent.
  In conclusion, when the measure of two angles and the length of the non-included side of
  one triangle are equal to the measure of the two corresponding angles and the length of the
  non-included side of another triangle, the two triangles are congruent by the AAS
  Congruence Theorem.
  5.6 Angle-Angle-Side Congruence Theorem 451445_Ch05_333-418.indd 395
  395
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  PROBLEM 3 And Finally the Proof . . .
  Prove the Angle-Angle-Side Congruence Theorem.
  ___
  ___
  Given: /A > /D, /B > /E, BC ​
  ​  > EF ​
  ​
  Prove: nABC > nDEF
  B
  E
  C
  A
  F
  D
  5
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  Problem 4 ASA Congruence or AAS Congruence
  Determine if there is enough information to prove that the two triangles are congruent
  by ASA or AAS. Write the congruence statements to justify your reasoning.
  ?
  1. nABS ˘ nAVF
  A
  F
  K
  B
  S
  V
  ?
  2. nGAB ˘ nSBA
  G
  S
  5
  T
  B
  A
  5.6 Angle-Angle-Side Congruence Theorem 451445_Ch05_333-418.indd 397
  397
  27/05/13 2:47 PM
  ?
  3. nEQD ˘ nDWE
  E
  D
  15o
  15o
  O
  88
  o
  88o
  Q
  W
  ?
  4. nABC ˘ nPQR
  Q
  B
  75o
  A
  32o
  32o
  5 mm R
  C
  75o
  5 mm P
  5
  398 Chapter 5 Congruence Through Transformations
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  Talk the Talk
  This chapter focused on four methods that you
  can use to prove that two triangles are congruent.
  Complete the graphic organizer by providing an
  illustration of each theorem.
  Use markers
  to show congruent
  sides and congruent
  angles.
  Side-Angle-Side
  Congruence Theorem
  Side-Side-Side
  Congruence Theorem
  Triangle
  Congruence
  Theorems
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