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Isosceles Triangles
Andrew Gloag
Bill Zahner
Dan Greenberg
Jim Sconyers
Lori Jordan
Victor Cifarelli
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Printed: October 30, 2012
AUTHORS
Andrew Gloag
Bill Zahner
Dan Greenberg
Jim Sconyers
Lori Jordan
Victor Cifarelli
EDITOR
Annamaria Farbizio
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C ONCEPT
Concept 1. Isosceles Triangles
1
Isosceles Triangles
Here you’ll learn the definition of an isosceles triangle as well as two theorems about isosceles triangles: 1) The
angle bisector of the vertex is the perpendicular bisector of the base; and 2) The base angles are congruent.
What if you were presented with an isoceles triangle and told that its base angles measure x◦ and y◦ ? What could you
conclude about x and y? After completing this Concept, you’ll be able to apply important properties about isoceles
triangles to help you solve problems like this one.
Watch This
MEDIA
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CK-12
Watch the first part of this video.
MEDIA
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James Sousa:HowTo Construct AnIsosceles Triangle
Then watch this video.
MEDIA
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James Sousa:Proof of the Isosceles TriangleTheorem
Finally, watch this video.
MEDIA
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James Sousa:Using the Properties ofIsosceles Trianglesto Determine Values
1
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Guidance
An isosceles triangle is a triangle that has at least two congruent sides. The congruent sides of the isosceles triangle
are called the legs. The other side is called the base. The angles between the base and the legs are called base
angles. The angle made by the two legs is called the vertex angle. One of the important properties of isosceles
triangles is that their base angles are always congruent. This is called the Base Angles Theorem.
For 4DEF, if DE ∼
= EF, then 6 D ∼
= 6 F.
Another important property of isosceles triangles is that the angle bisector of the vertex angle is also the perpendicular bisector of the base. This is called the Isosceles Triangle Theorem. (Note this is ONLY true of the vertex angle.)
The converses of the Base Angles Theorem and the Isosceles Triangle Theorem are both true as well.
Base Angles Theorem Converse: If two angles in a triangle are congruent, then the sides opposite those angles are
also congruent. So for 4DEF, if 6 D ∼
= 6 F, then DE ∼
= EF.
Isosceles Triangle Theorem Converse: The perpendicular bisector of the base of an isosceles triangle is also the
angle bisector of the vertex angle. So for isosceles 4DEF, if EG ⊥ DF and DG ∼
= GF, then 6 DEG ∼
= 6 FEG.
Example A
Which two angles are congruent?
This is an isosceles triangle. The congruent angles are opposite the congruent sides. From the arrows we see that
6 S∼
= 6 U.
Example B
If an isosceles triangle has base angles with measures of 47◦ , what is the measure of the vertex angle?
Draw a picture and set up an equation to solve for the vertex angle, v. Remember that the three angles in a triangle
always add up to 180◦ .
47◦ + 47◦ + v = 180◦
v = 180◦ − 47◦ − 47◦
v = 86◦
Example C
If an isosceles triangle has a vertex angle with a measure of 116◦ , what is the measure of each base angle?
Draw a picture and set up and equation to solve for the base angles, b.
116◦ + b + b = 180◦
2b = 64◦
b = 32◦
2
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Concept 1. Isosceles Triangles
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CK-12 Isosceles Triangles
Guided Practice
1. Find the value of xand the measure of each angle.
2. Find the measure of x.
3. True or false: Base angles of an isosceles triangle can be right angles.
Answers:
1. The two angles are equal, so set them equal to each other and solve for x.
(4x + 12)◦ = (5x − 3)◦
15 = x
Substitute x = 15; the base angles are [4(15) + 12]◦ , or 72◦ . The vertex angle is 180◦ − 72◦ − 72◦ = 36◦ .
2. The two sides are equal, so set them equal to each other and solve for x.
2x − 9 = x + 5
x = 14
3. This statement is false. Because the base angles of an isosceles triangle are congruent, if one base angle is a right
angle then both base angles must be right angles. It is impossible to have a triangle with two right (90◦ ) angles. The
Triangle Sum Theorem states that the sum of the three angles in a triangle is 180◦ . If two of the angles in a triangle
are right angles, then the third angle must be 0◦ and the shape is no longer a triangle.
Practice
Find the measures of x and/or y.
1.
2.
3.
4.
5.
Determine if the following statements are true or false.
6.
7.
8.
9.
Base angles of an isosceles triangle are congruent.
Base angles of an isosceles triangle are complementary.
Base angles of an isosceles triangle can be equal to the vertex angle.
Base angles of an isosceles triangle are acute.
3
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Fill in the proofs below.
10. Given: Isosceles 4CIS, with base angles 6 C and 6 SIO is the angle bisector of 6 CISProve: IO is the perpendicular bisector of CS
TABLE 1.1:
Statement
1.
2.
3. 6 CIO ∼
= 6 SIO
4.
5. 4CIO ∼
= 4SIO
∼
6. CO = OS
7.
8. 6 IOC and 6 IOS are supplementary
9.
10. IO is the perpendicular bisector of CS
Reason
1. Given
2. Base Angles Theorem
3.
4. Reflexive PoC
5.
6.
7. CPCTC
8.
9. Congruent Supplements Theorem
10.
11. Given: Isosceles 4ICS with 6 C and 6 SIO is the perpendicular bisector of CSProve: IO is the angle bisector
of 6 CIS
TABLE 1.2:
Statement
1.
2. 6 C ∼
=6 S
3. CO ∼
= OS
6
4. m IOC = m6 IOS = 90◦
5.
6.
7. IO is the angle bisector of 6 CIS
Reason
1.
2.
3.
4.
5.
6. CPCTC
7.
On the x − y plane, plot the coordinates and determine if the given three points make a scalene or isosceles triangle.
12.
13.
14.
15.
16.
4
(-2, 1), (1, -2), (-5, -2)
(-2, 5), (2, 4), (0, -1)
(6, 9), (12, 3), (3, -6)
(-10, -5), (-8, 5), (2, 3)
(-1, 2), (7, 2), (3, 9)