Chapter 3.7 Angle-Side Theorems.

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Chapter 3.7Angle-Side Theorems.

• Erin Sanderson Mod 9.

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Objective.

• This section will teach you how to apply theorems
  relating to the angle measure and side lengths of
  triangles.

Triangle D
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Theorem 20.

• If two sides of a triangle are congruent, the
  angles opposite the sides are congruent.
• (If , then .)

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But Why?
A
Statement. Reason.
A A Given Reflexive Property SAS (1,2,1) CPCTC
B
C
Given
Prove
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Theorem 21 the Reverse.

• If two angles of a triangle are congruent, the
  sides opposite the angles are congruent.
• (if , then .)

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How Come?
G
Statement. Reason.
Given Reflexive Property ASA (1,2,1) CPCTC
E
M
Given Prove
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How Do I know if a is Isosceles?

1. If at least two sides of a triangle are
   congruent, the triangle is isosceles.
2. If at least two angles of a triangle are
   congruent, the triangle is isosceles.

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The Inverses Also Work...

• If two sides of a triangle are not congruent,
  then the angles opposite them are not congruent,
  and the larger angle is opposite the longer side.
• If two angles of a triangle are not congruent,
  then the sides opposite them are not congruent,
  and the longer side is opposite the larger angle.

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Basically

• This means that the longest side is across from
  the largest angle and the shortest side is across
  from the smallest angle.

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It Would Kind of Look Like...
LARGER
SMALLER
SHORTER
LONGER
That.
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This means...

• Equilateral triangles are also equiangular
  because all of the sides are congruent, thus all
  of the angles are congruent.

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Sample Problems.
Statement. Reason.
ACDE is a square. B bisects Given. Given All sides of a square are cong. If a line is bisected, it is divided into 2 cong. lines All angles of a square are cong. SAS (3,4,5) CPCTC If sides, then angles
A
B
C
D
E
Given ACDE is a square.B bisects .
Prove
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2
B
C
9x-72
x40
Given Angle measures as shown ABC is
isosceles. Find The measure of angle A.
Since you know that B C, you can
say that x409x-728x112x14 Then, you
can substitute 14 in for the x in
A.6(14)-12The answer is 72.
A
6x-12
14
Now, do some on your own.
U
1
2
3
4
T
Q
S
R
Given QR ST UR US Prove QUS
TUR
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E
G
D
F
Given F GE ED Prove EF
bisects GFD
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Answers.
Statement. Reason.
QR ST UR US QS RT 3 2 QUS TUR Given Addition If sides, then angles SAS (1,2,3)
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And another
Statement. Reason
F GE ED GF FD EGF EDF EGF EDF GFE DFE EF bisects GFD Given Radii of a circle are congruent. If sides, then angles. SAS (1,2,3) CPCTC If a ray divides an angle into 2 congruent angles, the ray bisects the angle
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Works Cited

• Geometry for Enjoyment and Challenge. New
  Edition. Evanston, Illinois McDougal Littell,
  1991.
• Isosceles Triangle Proofs. Math Warehouse. 29
  May 2008. lthttp//www.mathwarehouse.com/geometry/c
  ongruent_triangles/isosceles-triangle-theorems-pro
  ofs.phpgt.