• Isosceles, Equilateral, and Right Triangles

4.6 Isosceles, Equilateral, and
Right Triangles
Goals p Use properties of isosceles and equilateral triangles.
p Use properties of right triangles.
VOCABULARY
Base angles Base angles are the two angles adjacent to the
base of a triangle.
Vertex angle The vertex angle is the angle opposite the base
of a triangle.
THEOREM 4.6: BASE ANGLES THEOREM
If two sides of a triangle are congruent, B
then the angles opposite them are congruent. A
If AB c AC , then aB c a C .
& & C
THEOREM 4.7: CONVERSE OF THE BASE ANGLES THEOREM
If two angles of a triangle are congruent,
then the sides opposite them are congruent. B
If aB c aC, then AB c AC .
& & A
C
COROLLARY TO THEOREM 4.6
If a triangle is equilateral, then it is equiangular.
COROLLARY TO THEOREM 4.7
If a triangle is equiangular, then it is equilateral.
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 Example 1 Using Isosceles Triangles
Find the value of y.
70 y
Solution
Notice that y represents the measure of a base angle of an
isosceles triangle. From the Base Angles Theorem, the other
base angle has the same measure. The vertex angle forms a linear
pair with a 70 angle, so its measure is 110 .
110 2y 180 Apply the Triangle Sum Theorem.
y 35 Solve for y.
Checkpoint Solve for x and y.
1. 2.
70
75 5x y
x
3y
x 15, y 10 x 40, y 65
THEOREM 4.8: HYPOTENUSE-LEG (HL) CONGRUENCE
THEOREM
If the hypotenuse and a leg of a A D
right triangle are congruent to the
hypotenuse and a leg of a second
right triangle, then the two triangles
are congruent. B C E F
If BC c EF and AC c DF , then TABC c T DEF .
& & & &*
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 Example 2 Proving Right Triangles Congruent
The pole holding up one end of a
volleyball net is perpendicular to the V
plane containing the points W, X, Y,
and Z. Each of the lines running from
the top of the pole to X, Y, and Z uses
the same length of rope. Prove that
X W Z
TVWX, TVWY, and TVWZ are
congruent. Y
Given: VW ∏ WX, VW ∏ WY , VW ∏ WZ , VX c VY c VZ
*& *& *& *& *& *& & & &
Prove: TVWX c TVWY c TVWZ
Paragraph Proof You are given that VW ∏ WX and VW ∏ WY ,
*& *& *& *&
which implies that aVWX and aVWY are right angles. By
definition, TVWX and TVWY are right triangles. You are given
& &
that the hypotenuses of these two triangles, VX and VY , are
congruent. Also, VW is a leg for both triangles, and VW c VW
*& & & &&
by the Reflexive Property of Congruence. Thus, by the
Hypotenuse-Leg Congruence Theorem, TVWX c TVWY .
Similar reasoning can be used to prove that TVWY c TVWZ .
So, by the Transitive Property of Congruent Triangles,
TVWX c TVWY c TVWZ .
Checkpoint Complete the following exercise.
3. Given: RV c ST ; aRTV and aSVT are right angles
& &
Prove: TRTV c TSVT
R S Statements (Reasons)
1. RV c ST (Given)
*& *&
2. aRTV and aSVT are right angles (Given)
3. aRTV c aSVT (Right angles are
congruent.)
4. TV c TV (Reflexive Property of
*& *&
T V
Congruence)
5. TRTV c TSVT (HL Congruence Theorem)
Copyright © McDougal Littell/Houghton Mifflin Company All rights reserved. Chapter 4 • Geometry Notetaking Guide 90