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6.4 Prove Triangles Similar by AA

6.4

Bell Thinger
1. In ABC and XZW, m A = m X and m B = m Z.
What can you conclude abo...
6.4
Example 1
6.4
Determine whether the triangles are similar. If they
are, write a similarity statement. Explain your
reasoni...
Example 1
6.4
SOLUTION
Because they are both right angles, ∠D and ∠G are
congruent.
By the Triangle Sum Theorem, 26° + 90°...
Example 2
6.4
Show that the two triangles are similar.

a. ∆ABE and ∆ACD

SOLUTION
a.

You may find it helpful to redraw t...
Example 2
6.4
Show that the two triangles are similar.

b. ∆SVR and ∆UVT
SOLUTION
b. You know SVR
UVT by
the Vertical Angl...
Guided Practice
6.4
Show that the triangles are similar. Write a similarity
statement.
1.

∆FGH and ∆RQS

ANSWER

In each ...
Guided Practice
6.4
Show that the triangles are similar. Write a similarity
statement.

2.

∆CDF and ∆DEF

ANSWER
Since m ...
Guided Practice
6.4
3. REASONING Suppose in Example 2, part (b), SR
Could the triangles still be similar? Explain.

ANSWER...
Example 3
6.4
Example 3
6.4
SOLUTION
The flagpole and the woman form sides of two right
triangles with the ground, as shown below. The s...
Example 3
6.4
You can use a proportion to find the height x. Write 5
feet 4 inches as 64 inches so that you can form two
r...
Guided Practice
6.4
4. WHAT IF? A child who is 58 inches tall is standing
next to the woman that is 5’4” with a shadow 40”...
Guided Practice
6.4
5.

You are standing in your backyard, and you
measure the lengths of the shadows cast by both
you and...
Exit Slip
6.4
Determine if the two triangles are similar. If they are
write a similarity statement.
1.

ANSWER

Yes;

ABE ...
Exit Slip
6.4
Determine if the two triangles are similar. If they are
write a similarity statement.
2.

ANSWER

no
Exit Slip
6.4
3. Find the length of BC.

ANSWER

7.5
Exit Slip
6.4
4.

A tree casts a shadow that is 30 feet long. At the
same time a person standing nearby, who is five
feet ...
Exit Slip
6.4

ANSWER Yes;

XYZ ~

CBA

ANSWER

No
6.4

Homework
Pg 402-405
#9, 13, 16, 17, 32
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6.4 prove triangles similar by aa

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6.4 prove triangles similar by aa

  1. 1. 6.4 Prove Triangles Similar by AA 6.4 Bell Thinger 1. In ABC and XZW, m A = m X and m B = m Z. What can you conclude about m C and m W? They are the same. x 2. Solve = 54 . 9 18 ANSWER ANSWER 3. ABC ~ ANSWER 108 DEF. Find x. 10
  2. 2. 6.4
  3. 3. Example 1 6.4 Determine whether the triangles are similar. If they are, write a similarity statement. Explain your reasoning.
  4. 4. Example 1 6.4 SOLUTION Because they are both right angles, ∠D and ∠G are congruent. By the Triangle Sum Theorem, 26° + 90° + m ∠E = 180°, so m ∠E = 64°. Therefore, ∠E and ∠H are congruent. So, ∆CDE ~ ∆KGH by the AA Similarity Postulate.
  5. 5. Example 2 6.4 Show that the two triangles are similar. a. ∆ABE and ∆ACD SOLUTION a. You may find it helpful to redraw the triangles separately. Because m ∠ABE and m ∠C both equal 52°, ∠ABE ≅ ∠C. By the Reflexive Property, ∠A ≅ ∠A. So, ∆ ABE ~ ∆ ACD by the AA Similarity Postulate.
  6. 6. Example 2 6.4 Show that the two triangles are similar. b. ∆SVR and ∆UVT SOLUTION b. You know SVR UVT by the Vertical Angles Congruence Theorem. The diagram shows RS ||UT so S U by the Alternate Interior Angles Theorem. So, ∆SVR ~ ∆UVT by the AA Similarity Postulate.
  7. 7. Guided Practice 6.4 Show that the triangles are similar. Write a similarity statement. 1. ∆FGH and ∆RQS ANSWER In each triangle all three angles measure 60°, so by the AA similarity postulate, the triangles are similar ∆FGH ~ ∆QRS.
  8. 8. Guided Practice 6.4 Show that the triangles are similar. Write a similarity statement. 2. ∆CDF and ∆DEF ANSWER Since m CDF = 58° by the Triangle Sum Theorem and m DFE = 90° by the Linear Pair Postulate the two triangles are similar by the AA Similarity Postulate; ∆CDF ~ ∆DEF.
  9. 9. Guided Practice 6.4 3. REASONING Suppose in Example 2, part (b), SR Could the triangles still be similar? Explain. ANSWER Yes; if S ≅ T, the triangles are similar by the AA Similarity Postulate. TU.
  10. 10. Example 3 6.4
  11. 11. Example 3 6.4 SOLUTION The flagpole and the woman form sides of two right triangles with the ground, as shown below. The sun’s rays hit the flagpole and the woman at the same angle. You have two pairs of congruent angles, so the triangles are similar by the AA Similarity Postulate.
  12. 12. Example 3 6.4 You can use a proportion to find the height x. Write 5 feet 4 inches as 64 inches so that you can form two ratios of feet to inches. x ft = 50 ft. 40 in. 64 in. 40x = 64(50) x = 80 Write proportion of side lengths. Cross Products Property Solve for x. ANSWER The flagpole is 80 feet tall. The correct answer is C.
  13. 13. Guided Practice 6.4 4. WHAT IF? A child who is 58 inches tall is standing next to the woman that is 5’4” with a shadow 40” long. How long is the child’s shadow? ANSWER 36.25 in.
  14. 14. Guided Practice 6.4 5. You are standing in your backyard, and you measure the lengths of the shadows cast by both you and a tree. Write a proportion showing how you could find the height of the tree. SAMPLE ANSWER tree height your height = length of shadow length of your shadow
  15. 15. Exit Slip 6.4 Determine if the two triangles are similar. If they are write a similarity statement. 1. ANSWER Yes; ABE ~ ACD
  16. 16. Exit Slip 6.4 Determine if the two triangles are similar. If they are write a similarity statement. 2. ANSWER no
  17. 17. Exit Slip 6.4 3. Find the length of BC. ANSWER 7.5
  18. 18. Exit Slip 6.4 4. A tree casts a shadow that is 30 feet long. At the same time a person standing nearby, who is five feet two inches tall, casts a shadow that is 50 inches long. How tall is the tree to the nearest foot? ANSWER 37 ft
  19. 19. Exit Slip 6.4 ANSWER Yes; XYZ ~ CBA ANSWER No
  20. 20. 6.4 Homework Pg 402-405 #9, 13, 16, 17, 32

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