Download Geometry Notes TC – 5: Isosceles Triangle Theorem Angle

Geometry Notes TC – 5: Isosceles Triangle Theorem
Angle Bisectors (review)
Definition: An angle bisector
Postulate: Every angle has
Isosceles Triangle Theorem
We wish to prove: If two sides of a triangle are congruent, the angles opposite those sides are also congruent.
Complete the Given and Prove below and draw a suitable diagram.
Given:
Prove:
The plan is to bisect the vertex angle of the given triangle
and then prove the two new triangles are congruent. Fill in
missing statements and reasons (including the blanks in
statement 2) to complete the proof below.
Statement
Reason
1. Given
1.
2. Let point
be on base
such that
2.
is the angle bisector of vertex angle
3.
3.
4.
4.
5.
5.
6.
6.
Ex: Draw a suitable diagram, then give an appropriate conclusion and reason:
Given: In RAT, RA  AT
Conclusion:
Reason:
Ex: CAT, shown at right, has perimeter 52.
A
a. Find the value of x.
2y + 3z
2
x
5x
z
C
5y + 30
T
2x + 8
b. Find the values of y and z.
The converse of the Isosceles Triangle Theorem is also true. (Proof is on a later HW).
Write the converse:
Ex: Determine which two sides of HUG are congruent.
U
2x + 42
H
3x + 12
5x + 6
G
Geometry HW: Triangle Congruence - 5
1. In ABC, if AB  BC and mB = 80 find mC.
2. Each of the congruent angles of an isosceles triangle measures 9 less than 4 times the vertex angle. Find
the measures of all three angles of the triangle.
Find the value of x in each of the diagrams below.
3.
A
130
D
50
D
C
A
x
C
AD  BD , BDC
mADC = 130
AD  DE , AB  CB ,
ACD and BCE
(2x)
(5x)
l1
7.
y
l1 || l1
y
l2
(2x)
55
50
C
E
Find the values of x and y in each of the diagrams below.
6.
B
5.
x = mABD
x
B
B
4.
A
(3x)
D
8. In ABC, A  C, AB = 5x + 6, BC = 3x + 14, and AC = 6x – 1. Find the lengths of all three sides of the
triangle.
9. Prove the following: In an isosceles triangle, the median from the vertex, the altitude from the vertex and
the angle bisector of the vertex are all the same.
Given: Isosceles triangle ABC with AC  BC , CP bisects ACB
a. Prove (in statement – reason format) that CP is a median.
b. Explain (in paragraph format) why CP is an altitude.
10. Given: Isosceles ABC with AC  BC , M is the midpoint of AB , AME  BMF
Prove: CE  CF
C
E
A
11. In the diagram at right, MS is both an angle bisector and
an altitude of BMW. Find the values of x, y and z.
F
B
M
M
(2x + 14) 7 x + 8 
3
B
(5x)
y
z –7
2
S
3z + 3
W