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Name ________________________________________ Date __________________ Class__________________
LESSON
4-9
Reteach
Isosceles and Equilateral Triangles
Theorem
Examples
Isosceles Triangle Theorem
If two sides of a triangle are congruent, then the
angles opposite the sides are congruent.
If RT ≅ RS, then �T � �S.
Converse of Isosceles Triangle Theorem
If two angles of a triangle are congruent, then
the sides opposite those angles are congruent.
If �N � �M, then LN ≅ LM.
You can use these theorems to find angle measures in isosceles triangles.
Find m∠E in UDEF.
m∠D = m∠E
5x� = (3x + 14)�
2x = 14
x=7
Isosc. U Thm.
Substitute the given values.
Subtract 3x from both sides.
Divide both sides by 2.
Thus m∠E = 3(7) + 14 = 35�.
Find each angle measure.
1. m∠C = _____________________
2. m∠Q = _____________________
3. m∠H = _____________________
4. m∠M = _____________________
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4-70
Holt McDougal Geometry
Name ________________________________________ Date __________________ Class__________________
LESSON
4-9
Reteach
Isosceles and Equilateral Triangles continued
Equilateral Triangle Corollary
If a triangle is equilateral, then it is equiangular.
(equilateral U → equiangular U)
Equiangular Triangle Corollary
If a triangle is equiangular, then it is equilateral.
(equiangular U → equilateral U)
If �A � �B � �C, then AB ≅ BC ≅ CA .
You can use these theorems to find values in equilateral triangles.
Find x in USTV.
USTV is equiangular.
Equilateral U → equiangular U
(7x + 4)� = 60°
The measure of each ∠ of an
equiangular U is 60°.
7x = 56
x=8
Subtract 4 from both sides.
Divide both sides by 7.
Find each value.
5. n = _____________________
6. x = _____________________
7. VT = _____________________
8. MN = _____________________
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4-71
Holt McDougal Geometry
9. 693 ft
11. 6.3
13. 4
1
yd
2
10. 50°
2. x = 36; y = 72; z = 36
12. 60°
3. 2A
14. 65°
4. Possible answer: The coordinates of Y
are (a, 3a ). The Midpoint Formula
shows that the midpoints of the sides of
⎛1
3 ⎞ ⎛3
3 ⎞
a ⎟ , B ⎜ a,
a⎟ ,
UXYZ are A ⎜ a,
⎜2
⎟
⎜
2 ⎠ ⎝2
2 ⎟⎠
⎝
and C(a, 0). The Distance Formula gives
these distances: AX = AY = AC = AB =
BC = XC = BY = BZ = CZ = a. Thus by
SSS, UABC ≅ UXAC ≅ UYAB ≅ UCBZ,
and the triangles are equilateral.
15. 8
Practice B
1. Possible answer: It is given that HI is
congruent to HJ , so ∠I must be
congruent to ∠J by the Isosceles Triangle
Theorem. ∠IKH and ∠JKH are both right
angles by the definition of perpendicular
lines, and all right angles are congruent.
Thus by AAS, UHKI is congruent to
UHKJ. IK is congruent to KJ by
CPCTC, so HK bisects IJ by the
definition of segment bisector.
2. 58.1 ft
4.
2
Reteach
1. 51°
2. 47°
3. 72°
4. 60°
3. 45°
5. 12
6. 33
5. 36 or 9
7. 18
8. 13
4
3
6. 76°
7.
8. 10
9. 30°
Challenge
1. x = 10; By Isosc. U Thm. and U Sum
Thm., m∠TSV = 54°. UQSV ≅ UTSV by
SSS, so m∠VSQ = 54° by CPCTC.
m∠QSR = 72° by Lin. Pair Thm. and by
∠ Add. Post. By Isosc. U Thm. and U
Sum Thm., m∠SQR = 36°. Solve
(3x + 6)° = 36°. x = 10
10. 89
Practice C
1. Possible answer: UABC is an isosceles
triangle with vertices A(0, b), B(a, 0), and
C(−a, 0). D is the midpoint of BC , so D
has coordinates (0,0). The slope of AD
b−0 b
= , so the slope is undefined. A
is
0−0 0
line with an undefined slope is a vertical
line. The slope of BC is
0−0
0
=
= 0 . A line with a zero
a − ( −a ) 2a
2. 42.2; UFJH is an equilat. U, so HJ = JF
= FH = 15 and x = 3. So GJ = 12.2.
FG ≅ FJ since the sides opp. the ∠s are
≅. So FG = FJ = 15. UEFH ≅ UGFJ by
AAS, so the corr. sides are ≅ by CPCTC.
P = 12.2 + 15 + 15 = 42.2
3. 108°; UFJH is equiangular, so m∠JFH =
m∠FJH = 60°. By Lin. Pair Thm. and ∠
Add. Post., m∠FJG = 66°. By Isosc. U
Thm. and U Sum Thm., m∠GFJ = 48°.
m∠GFH = 48° + 60° = 108°
slope is a horizontal line. Because AD is
vertical and BC is horizontal, AD ⊥ BC .
Problem Solving
1. 14 in.
2. 40°
3. 11 ft; m∠GJH = 72° − 36° = 36°. m∠GHJ
= 36° by Alt. Int. ∠s Thm. By Converse of
Isosceles Triangle Theorem, GJ = GH =
11 ft.
Original content Copyright © by Holt McDougal. Additions and changes to the original content are the responsibility of the instructor.
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Holt McDougal Geometry