Download 4-5 ISOSCELES AND EQUILATERAL TRIANGLES (p. 210

4-5 ISOSCELES AND EQUILATERAL TRIANGLES (p. 210-216)
The investigation can be skipped.
Recall that an isosceles triangle has at least two congruent sides.
Example: Sketch an isosceles triangle that has exactly two congruent sides. The
congruent sides are called the legs. The other side is called the base. The base is not
always on the bottom, however. Identify the two base angles and the vertex angle.
Theorem 4-3 Isosceles Triangle Theorem
If two sides of a triangle are congruent, then the angles opposite those sides are
congruent.
Example: To prove this theorem, sketch an isosceles triangle and its mirror image next
to each other. Use tick marks to identify the two pairs of congruent sides. By what
method are the two triangles congruent (you have different options here)? Why are the
opposite angles congruent?
Show the symbolic way to remember the Isosceles Triangle Theorem from other
geometry textbooks.
Theorem 4-4 Converse of the Isosceles Triangle Theorem
If two angles of a triangle are congruent, then the sides opposite the angles are
congruent.
Example: To prove this theorem, sketch an isosceles triangle and its mirror image next
to each other. Use tick marks to identify the two pairs of congruent angles. By what
method are the two triangles congruent? Why are the opposite sides congruent?
Show the symbolic way to remember the Converse of the Isosceles Triangle Theorem
from other geometry textbooks.
Theorem 4-5
The bisector of the vertex angle of an isosceles triangle is the perpendicular
bisector of the base.
How do you show that a segment is perpendicular to another segment? How do you
show that a segment is being bisected by another segment?
You will prove Theorem 4-5 in a homework problem. You will prove triangles
congruent and use CPCTC more than once in the same proof.
Example: Sketch an isosceles triangle and the bisector of its vertex angle. Demonstrate
what you will need to do in order to prove Theorem 4-5. Come up with a plan for
proving this theorem by using tick marks on a diagram.
Example 1 and #1 are alternative ways to prove Theorems 4-3 and 4-4. We do not have
to cover this.
Example: In the following diagram, AB
CD and ABC  BDC. Explain why
BCD is isosceles.
G
A
E
B
D
C
Example: In the following diagram, PQ and PS are the legs of isosceles PQS.
PT bisects QPS and
m
Q  58. Find the measures of S, 1, and 2.
P
2
1
Q
T
S
A corollary is a mathematical statement that follows directly from a theorem. Like a
theorem, a corollary can be proved.
Do you recall what it means for a triangle to be equilateral?
Do you recall what it means for a triangle to be equiangular?
Corollary to Theorem 4-3
If a triangle is equilateral, then the triangle is equiangular.
Example: Sketch an equilateral triangle. Describe how you can use the isosceles
triangle theorem twice to get all three angles congruent.
Corollary to Theorem 4-4
If a triangle is equiangular, then the triangle is equilateral.
Example: Sketch an equiangular triangle. Describe how you can use the converse of the
isosceles triangle theorem twice to get all three sides congruent.
Discuss Ex. 4 on p. 212. The rectangles are congruent and the equilateral triangles are
congruent in the path around the regular hexagon. Why is the sum of the measures of the
four angles equal to 360  ? Can you also find the value of x by using the polygon anglesum theorem?
Do 4 on p. 212.
Example: Using the diagram for Ex. 4, create a triangle inside the regular hexagon that
uses two rectangle sides as two sides of the triangle with the third side being a diagonal
opposite the x (see the following sketch).
P
rectangle side
O
x
diagonal
rectangle side
D
How does PO compare to PD? What type of triangle is POD? Find the measure of
O and D in this triangle.
Homework p. 213-216: 1,9-11,15,17,20,23,25,27,28,30,33,40,41,46,48,51,53
41. Prove the triangles congruent by SAS. Use CPCTC twice to get congruent right
angles (to satisfy perpendicular) and congruent segments (to satisfy bisect). To get right
angles, use the theorem that says if two angles are both congruent and supplementary,
then they are right angles.