4.7 – isosceles triangles geometry ms. rinaldi. isosceles triangles remember that a triangle is...
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4.7 – Isosceles Triangles
GeometryMs. 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 , thenACAB CB
Converse of Base Angles Theorem
If two angles of a triangle are congruent, then the sides opposite them are congruent.
If , thenCB ACAB
EXAMPLE 1 Apply the Base Angles Theorem
SOLUTION
In DEF, DE DF . Name two congruent angles.
DE DF , so by the Base Angles Theorem, E F.
EXAMPLE 2 Apply the Base Angles Theorem
In . Name two congruent angles.QRPQPQR ,
P
RQ
EXAMPLE 3 Apply the Base Angles Theorem
Copy and complete the statement.
1. If HG HK , then ? ? .
If KHJ KJH, then ? ? .If KHJ KJH, then ? ? .2. 2.
EXAMPLE 4 Apply the Base Angles Theorem
P
R
Q
(30)°
Find the measures of the angles.
SOLUTION
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.
EXAMPLE 5 Apply the Base Angles Theorem
P
R
Q
(48)°
Find the measures of the angles.
EXAMPLE 6 Apply the Base Angles Theorem
P
R
Q(62)°
Find the measures of the angles.
EXAMPLE 7 Apply the Base Angles Theorem
Find the value of x. Then find the measure of each angle.
P
RQ(20x-4)°
(12x+20)°
SOLUTION
Since there are two congruent sides, the angles opposite them must be congruent also. Therefore, 12x + 20 = 20x – 4
20 = 8x – 4
24 = 8x
3 = xPlugging back in,
And since there must be 180 degrees in the triangle,
564)3(20
5620)3(12
Rm
Pm
685656180Qm
EXAMPLE 8 Apply the Base Angles Theorem
Find the value of x. Then find the measure of each angle.
P
R
Q(11x+8)° (5x+50)°
EXAMPLE 9 Apply the Base Angles Theorem
Find the value of x. Then find the length of the labeled sides.
P
R
Q(80)° (80)°
SOLUTION
Since there are two congruent sides, the angles opposite them must be congruent also. Therefore, 7x = 3x + 40
4x = 40
x = 107x 3x+40
Plugging back in,
QR = 7(10)= 70PR = 3(10) + 40 = 70
EXAMPLE 10 Apply the Base Angles Theorem
Find the value of x. Then find the length of the labeled sides.
P
RQ
(50)°
(50)°
10x – 2
5x+3