Download Lesson 23: Base Angles of Isosceles Triangles

GEOMETRY
Lesson 23
NYS COMMON CORE MATHEMATICS CURRICULUM
Name ____________________________
M1
Date _______________
Lesson 23: Base Angles of Isosceles Triangles
Today we examine a geometry fact that we already accept to be true. We are going to prove this
known fact by using SAS triangle congruence criteria.
Here is isosceles triangle ABC. We accept that an isosceles triangle, which has (at least) two
congruent sides, also has congruent base angles.
Label the congruent angles (base angles) in figure 1.
Label the vertex angle in figure 1.
The following is a proof which demonstrates that the base angles of an
isosceles triangle are always congruent.
Figure 1
Prove Base Angles of an Isosceles Triangle are Congruent: SAS
Given:
Isosceles △ 𝐴𝐵𝐶, with 𝐴𝐵 = 𝐴𝐶
Prove:
∠𝐵 ≅ ∠𝐶
Statements
Reasons
1. Isosceles
Triangle
1. given
2. AD bisects BAD
2. auxiliary line
3. mBAD  mCAD
3. angle bisectors divide angles
into 2 equal measures
4. AD = AD
4. reflexive property
5. BAD  CAD
5. SAS congruence
6. B  C
6. corresponding angles of
congruent triangle are
congruent
Lesson 23:
Date:
Construction: Draw the
angle bisector AD of
∠𝐴, where 𝐷 is the
intersection of the
̅̅̅̅ . The
bisector and 𝐵𝐶
auxiliary line will be
used for the SAS
criteria.
Base Angles of Isosceles Triangles
11/23/15
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
S.125
This work is licensed under a
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
In proofs, we can state that “base angles of an isosceles triangle are equal in measure” or that
“base angles of an isosceles triangle are congruent” if we already have the isosceles triangle with
the congruent sides stated in the given.
Otherwise, if we do not know that we have an isosceles triangle already then we can state…
“In a triangle, angles opposite equal sides are equal” as well as “In a triangle, sides opposite
equal angles are equal.”
Exercises
1.
2.
3.
4.
Lesson 23:
Date:
Base Angles of Isosceles Triangles
11/23/15
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
S.126
This work is licensed under a
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.
5.
6.
Additional Properties:
*** Two angles that are equal in measure that form a linear pair are right angles.
*** Two angles that are supplementary to the same angle or angles of equal measure
are equal.
Lesson 23:
Date:
Base Angles of Isosceles Triangles
11/23/15
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
S.127
This work is licensed under a
Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License.