Download 4.7 – Isosceles Triangles

4.7 – Isosceles Triangles
Geometry
Ms. Rinaldi
Isosceles Triangles
• Remember that a triangle is
isosceles if it has at least two
congruent sides.
• When an isosceles triangle has
exactly two congruent sides, these
two sides are the legs.
• The angle formed by the legs is the
vertex angle.
• The third side is the base of the
isosceles triangle.
• The two angles adjacent to the base
are called base angles.
Base Angles Theorem
If two sides of a triangle are congruent, then the
angles opposite them are congruent.
If AB  AC , then B  C
Converse of Base Angles
Theorem
If two angles of a triangle are congruent, then
the sides opposite them are congruent.
If B  C , then AB  AC
EXAMPLE 1
In
DEF, DE
Apply the Base Angles Theorem
DF . Name two congruent angles.
SOLUTION
DE
DF , so by the Base Angles Theorem,
E
F.
EXAMPLE 2
Apply the Base Angles Theorem
In PQR , PQ  QR . Name two congruent angles.
P
Q
R
Apply the Base Angles Theorem
EXAMPLE 3
Copy and complete the statement.
1.
If HG
2.
If
KHJ
HK , then
?
KJH, then ?
? .
? .
EXAMPLE 4
Apply the Base Angles Theorem
Find the measures of the angles.
SOLUTION
Q
P
Since a triangle has 180°, 180 – 30 = 150° for
the other two angles.
Since the opposite sides are congruent, angles
Q and P must be congruent.
150/2 = 75° each.
(30)°
R
EXAMPLE 5
Apply the Base Angles Theorem
Find the measures of the angles.
Q
P
(48)°
R
EXAMPLE 6
Apply the Base Angles Theorem
Find the measures of the angles.
Q
P
(62)°
R
EXAMPLE 7
Apply the Base Angles Theorem
Find the value of x. Then find the measure of each angle.
P
SOLUTION
Since there are two congruent sides, the
angles opposite them must be congruent
also. Therefore, 12x + 20 = 20x – 4
20 = 8x – 4
(20x-4)°
Q
R
24 = 8x
3=x
Plugging back in,
mP  12(3)  20  56
mR  20(3)  4  56
And since there must be 180 degrees in
the triangle,
mQ  180  56  56  68
EXAMPLE 8
Apply the Base Angles Theorem
Find the value of x. Then find the measure of each angle.
Q
P
(11x+8)°
(5x+50)°
R
EXAMPLE 9
Apply the Base Angles Theorem
Find the value of x. Then find the length of the labeled sides.
Q
P
(80)°
(80)°
SOLUTION
Since there are two congruent sides, the
angles opposite them must be congruent
also. Therefore, 7x = 3x + 40
4x = 40
x = 10
3x+40
7x
Plugging back in,
QR = 7(10)= 70
PR = 3(10) + 40 = 70
R
EXAMPLE 10
Apply the Base Angles Theorem
Find the value of x. Then find the length of the labeled sides.
P
(50)°
5x+3
(50)°
R
Q
10x – 2