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Chapter 3.7 Angle-Side Theorems.

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Works Cited Geometry for Enjoyment and Challenge. New Edition. Evanston, Illinois: McDougal Littell, 1991. Isosceles Triangle Proofs. – PowerPoint PPT presentation

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Title: Chapter 3.7 Angle-Side Theorems.


1
Chapter 3.7Angle-Side Theorems.
  • Erin Sanderson Mod 9.

2
Objective.
  • This section will teach you how to apply theorems
    relating to the angle measure and side lengths of
    triangles.

Triangle D
3
Theorem 20.
  • If two sides of a triangle are congruent, the
    angles opposite the sides are congruent.
  • (If , then .)

4
But Why?
A
Statement. Reason.
A A Given Reflexive Property SAS (1,2,1) CPCTC
B
C
Given
Prove
5
Theorem 21 the Reverse.
  • If two angles of a triangle are congruent, the
    sides opposite the angles are congruent.
  • (if , then .)

6
How Come?
G
Statement. Reason.
Given Reflexive Property ASA (1,2,1) CPCTC
E
M
Given Prove
7
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.

8
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.

9
Basically
  • This means that the longest side is across from
    the largest angle and the shortest side is across
    from the smallest angle.

10
It Would Kind of Look Like...
LARGER
SMALLER
SHORTER
LONGER
That.
11
This means...
  • Equilateral triangles are also equiangular
    because all of the sides are congruent, thus all
    of the angles are congruent.

12
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
13
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
15
E
G
D
F
Given F GE ED Prove EF
bisects GFD
16
Answers.
Statement. Reason.
QR ST UR US QS RT 3 2 QUS TUR Given Addition If sides, then angles SAS (1,2,3)
17
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
18
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.
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