13-2: Isosceles Triangles

13-2: Isosceles Triangles
Objectives:
1. To develop and
prove theorems
about Isosceles
Triangles
Assignment:
• P. 188: 18-21
• P. 190: 12-20
• Challenge Problems
Objective 1
You will be able to
develop and prove
theorems about
Isosceles Triangles
Isosceles Triangle
A triangle is an isosceles triangle if and only if
it has at least two congruent sides.
Exercise 1
If the lengths of two sides of an isosceles
triangle are 3 and 7, what is the perimeter
of the triangle?
Exercise 2
Use your
straightedge to
draw an acute
angle. How could
you use your
compass to turn
this angle into an
isosceles
triangle?
Investigation 1
Use the following investigation
to complete the Base Angles
Theorem. Half of the class
should work with isosceles
triangles with acute vertex
angles, and the other
half should work with
isosceles triangles
with obtuse vertex
angles.
Investigation 1
1. Draw an angle and
label the vertex 𝐶.
This will be your
vertex angle.
2. Using point 𝐶 as
center, swing an arc
that intersects both
sides of ∠𝐶.
Investigation 1
3. Label the points of
intersection 𝐴 and 𝐵.
Construct side 𝐴𝐵.
You have constructed
isosceles Δ𝐴𝐵𝐶 with
base 𝐴𝐵.
4. Use your protractor to
measure the base
angles of isosceles
Δ𝐴𝐵𝐶.
Investigation 1
Compare your results with the rest of the
class. What relationship do you notice
about the base angles of each isosceles
triangle?
Base Angles Theorem
If two sides of a
triangle are
congruent, then the
angles opposite them
are congruent.
Exercise 3
C
A
Median
M
B
Exercise 4
Exercise 5
Find the value of x.
2x+5
30
Exercise 6: SAT
If AB = BC, what is y in terms of x?
A
3x - 2
B
2y
C
Investigation 2
Use the following to
investigate the
Converse of the
Base Angles
Theorem.
Investigation 2
1. Draw a line
segment 𝐴𝐵.
2. Draw an acute
angle at 𝐴 with
your protractor.
Investigation 2
3. Duplicate ∠𝐴 at
point 𝐵 with your
protractor. Label
the point of
intersection 𝐶.
4. Use your compass
or ruler to
compare 𝐴𝐶 and
𝐵𝐶.
Converse of the Base Angles Theorem
If two angles of a
triangle are congruent,
then the sides
opposite them are
congruent.
Proof of Converse
This proof is similar to the one for The Base
Angles Theorem, but you draw an altitude
instead of a median. Of course, the other
proof would have worked with an altitude…
C
A
Altitude
L
B
Investigation 3
In this Investigation,
we will use some
more compass and
straightedge
wizardry to
construct a perfect
equilateral triangle.
Investigation 3
1. Draw AB.
A
B
Investigation 3
2. Using A as center and AB as the radius,
draw a quarter-circle arc through B.
A
B
Investigation 3
3. Using B as center and AB as the radius,
draw a quarter-circle arc through A.
A
B
Investigation 3
4. Label the point of intersection C.
C
A
B
Investigation 3
5. Draw AC and BC.
C
A
B
Investigation 3
In ΔABC, measure each angle with the
protractor. What do you notice about the
angle measurements?
Use deductive reasoning to show that these
measurements must be true based on the
Base Angles Theorem.
Equilateral Triangle Theorem
A triangle is equilateral if and only if it is
equiangular.

Exercise 7
Find the values of x and y in the diagram.
Exercise 8: SAT
Find the value of x.
x
18
60
24
60
Exercise 9: SAT
O
70
x
In the figure shown, if
O is the center of
the circle, what is
the value of x?
Exercise 10: SAT
In the figure shown, if
AB = AC and
AC||BD, what is the
value of x?
130
B
A
C
x
D
13-2: Isosceles Triangles
Objectives:
1. To develop and
prove theorems
about Isosceles
Triangles
Assignment:
• P. 188: 18-21
• P. 190: 12-20
• Challenge Problems