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Isosceles Triangles
Bill Zahner
Lori Jordan
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Printed: October 7, 2012
AUTHORS
Bill Zahner
Lori Jordan
www.ck12.org
C ONCEPT
Concept 1. Isosceles Triangles
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Isosceles Triangles
Here you’ll learn the definition of an isosceles triangle as well as two theorems about isosceles triangles.
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
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CK-12 Foundation: Chapter4IsoscelesTrianglesA
Watch the first part of this video.
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James Sousa:HowTo Construct AnIsosceles Triangle
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James Sousa:Proof of the Isosceles TriangleTheorem
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James Sousa:Using the Properties ofIsosceles Trianglesto Determine Values
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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 and the angles between the base and the congruent sides are
called base angles. The angle made by the two legs of the isosceles triangle is called the vertex angle.
Investigation: Isosceles Triangle Construction
Tools Needed: pencil, paper, compass, ruler, protractor
1. Using your compass and ruler, draw an isosceles triangle with sides of 3 in, 5 in and 5 in. Draw the 3 in side
(the base) horizontally 6 inches from the top of the page.
2. Now that you have an isosceles triangle, use your protractor to measure the base angles and the vertex angle.
The base angles should each be
We can generalize this investigation into the Base Angles Theorem.
Base Angles Theorem: The base angles of an isosceles triangle are congruent.
To prove the Base Angles Theorem, we will construct the angle bisector through the vertex angle of an isosceles
triangle.
Given: Isosceles triangle 4DEF with DE ∼
= EF
Prove: 6 D ∼
=6 F
TABLE 1.1:
Statement
1. Isosceles triangle 4DEF with DE ∼
= EF
6
2. Construct angle bisector EG for E
3. 6 DEG ∼
= 6 FEG
4. EG ∼
= EG
5. 4DEG ∼
= 4FEG
6. 6 D ∼
=6 F
Reason
Given
Every angle has one angle bisector
Definition of an angle bisector
Reflexive PoC
SAS
CPCTC
By constructing the angle bisector, EG, we designed two congruent triangles and then used CPCTC to show that the
base angles are congruent. Now that we have proven the Base Angles Theorem, you do not have to construct the
angle bisector every time. It can now be assumed that base angles of any isosceles triangle are always equal. Let’s
further analyze the picture from step 2 of our proof.
Because 4DEG ∼
= 4FEG, we know that 6 EGD ∼
= 6 EGF by CPCTC. Thes two angles are also a linear pair, so
◦
they are congruent supplements, or 90 each. Therefore, EG⊥DF. Additionally, DG ∼
= GF by CPCTC, so G is the
midpoint of DF. This means that EG is the perpendicular bisector of DF, in addition to being the angle bisector
of 6 DEF.
Isosceles Triangle Theorem: The angle bisector of the vertex angle in an isosceles triangle is also the perpendicular
bisector to the base.
The converses of the Base Angles Theorem and the Isosceles Triangle Theorem are both true.
Base Angles Theorem Converse: If two angles in a triangle are congruent, then the opposite sides are also
congruent.
So, for a triangle 4ABC, if 6 A ∼
= 6 B, then CB ∼
= CA. 6 C would be the vertex angle.
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Concept 1. Isosceles Triangles
Isosceles Triangle Theorem Converse: The perpendicular bisector of the base of an isosceles triangle is also the
angle bisector of the vertex angle.
∼ DB, then 6 CAD =
∼ 6 BAD.
In other words, if 4ABC is isosceles, AD⊥CB and CD =
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.
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. Recall that the base angles are equal.
116◦ + b + b = 180◦
2b = 64◦
b = 32◦
Watch this video for help with the Examples above.
MEDIA
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CK-12 Foundation: Chapter4IsoscelesTrianglesB
Vocabulary
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.
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Guided Practice
1. Find the value of x and 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. Set the angles equal to each other and solve for x.
(4x + 12)◦ = (5x − 3)◦
15◦ = x
If x = 15◦ , then 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.
Complete 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
11. Given: Isosceles 4ICS with 6 C and 6 SIO is the perpendicular bisector of CSProve: IO is the angle bisector
of 6 CIS
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Concept 1. Isosceles Triangles
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.
(-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)
5