Download TOPIC 7-1: Isosceles Triangles and Special Segments

TOPIC 7-1: Isosceles Triangles and Special Segments in
Triangles
Term
Definition
Sketch
Isosceles
Triangle
A triangle with 2 congruent
sides.
A
EXAMPLE 1:
legs _____________________
base ___________________
base angles ________________
vertex angle _______________
C
B
If two sides of a triangle are congruent, then the angles opposite them
are also congruent.
If two angles of a triangle are congruent, then the sides opposite them
A
are also congruent.
If AB ≅ AC, then ________________.
If ∠C ≅ ∠B, then ________________.
B
EXAMPLES: Find the value of ‘x’ in each.
2.
3.
30°°
x°°
x°°
40°°
C
4.
5.
2x + 2
41
42
2x - 4
x+5
62°°
56°°
x
Term
Definition
Sketch
angle bisector A ray that cuts an angle into 2
congruent angles.
altitude
Perpendicular segment from
any vertex to its opposite side.
Median
Line segment drawn from any
vertex to the midpoint of
opposite side.
Perpendicular Segment that is perpendicular
bisector
to a side at its midpoint.
Every triangle has 3 of each of these.
EXAMPLES:
6. BG is a median. Find the value of ‘x’ if BG = 4x + 10,
A
AG = 6x + 4, CG = 7x – 5.
G
B
C
7. Given AF is an altitude, find the value of x if m∠
∠1 = (8x+18)°.
B
A
1
F 2
C
An isosceles triangle is a special case. In the picture below, the
segment AD is an angle bisector, altitude, median, and
perpendicular bisector.
A
m∠
∠__________ = m∠
∠__________
________ ⊥ ________
________ ≅ ________
B
D
C
As long as it is drawn from the vertex angle, a segment that is one of
the special segments is all four of the special segments.
EXAMPLE 8: In isosceles ∆ABC below, BD is an angle bisector.
Find the values of ‘x’, ‘y’, and ‘z’ if m∠
∠1 = (6x + 7)°,
m∠
∠2 = (3x + 16)°, m ∠3 = (3y - 1)°, AD = 2z + 1, and DC = 5z – 8.
B
A
D
C