Download 4.7 – Equilateral Triangles

4.7 – Equilateral Triangles
Geometry
Ms. Rinaldi
Equilateral Triangles
Remember that a triangle is equilateral if it
has all congruent sides.
Corollary to the
Base Angles Theorem
If a triangle is equilateral, then it is equiangular.
Corollary to the
Converse of Base Angles Theorem
If a triangle is equiangular, then it is equilateral.
EXAMPLE 1
Find measures in an Equilateral Triangle
Find the measures of angles P, Q,
and R.
P
R
SOLUTION
Q
The diagram shows that the triangle is equilateral.
Therefore, it is also equiangular.
Since there is 180 degrees in a triangle, we need to split
that into 3 equal parts.
180/3 = 60º each.
EXAMPLE 2
Angle Measure in an Equilateral Triangle
In Example 1, you found that the angle measures of
the equilateral triangle were 60º each.
Is it possible for an equilateral triangle to have
different angle measures?
SOLUTION
No.
There must be 180º inside a triangle, and the only way to divide
that into three equal angles is if each angle is 60º
EXAMPLE 3
Find Measures in Equilateral Triangles
Find ST in the triangle at the right.
SOLUTION
STU is equiangular, therefore its is equilateral.
ANSWER
Thus ST = 5
EXAMPLE 4
Use isosceles and equilateral triangles
ALGEBRA
Find the values of x and y in the diagram.
SOLUTION
STEP 1
STEP 2
Find the value of y. Because
KLN is
equiangular, it is also equilateral and KN
Therefore, y = 4.
KL .
Find the value of x. Because LNM
LMN,
LN
LM and
LMN is isosceles. You also
know that LN = 4 because
KLN is equilateral.
(Continued on next slide)
EXAMPLE 4
Use isosceles and equilateral triangles (cont.)
LN = LM
Definition of congruent segments
4=x+1
Substitute 4 for LN and x + 1 for LM.
3=x
Subtract 1 from each side.
EXAMPLE 5
Use Isosceles and Equilateral Triangles
Find the values of x and y in the diagram.
SOLUTION
Since the triangle to the right is equilateral, it is also equiangular. Therefore x
and the other angles are 60º.
We now know that part of the right angle is 60º. The other part, in the
triangle at the left, must be 90 – 60 = 30º.
Since that triangle is isosceles, the base angles are congruent. Find y by
combining those two angles and subtracting them from 180º.
30+30 = 60
180 – 60 = 120º = y