Download Honors Geometry: 2.4b: Isosceles and Equilateral Triangles

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Date:
Honors Geometry: 2.4b: Isosceles and Equilateral Triangles
(Section 3.6 in your book)
BACKGROUND/INTRODUCTION: We have spent some amount of time discussing what is required to guarantee that
two different triangles are congruent. Today, we use congruent triangles to make formal arguments about the
properties of isosceles triangles.
1.
An isosceles triangle is a triangle with AT LEAST two equal sides. Using this definition, fill in the
blanks with the word always, sometimes, or never.
(a) Isosceles triangles are _______________________ equilateral.
(b) Equilateral triangles are _______________________ isosceles.
2. In the drawings below, there are several isosceles triangles. In each case, the legs and the base are all
labeled. Using these drawings, come up with definitions for the terms: legs of an isosceles triangle and
base of an isosceles triangle.
leg
leg
leg
base
leg
leg
base
leg
base
The legs of an isosceles triangle are:
The base of an isosceles triangle is:
3. The following statement is a known fact in Geometry: In an isosceles triangle, the angles opposite the
congruent sides are also congruent. Using the drawings below, show what is meant by the “angles opposite
the congruent sides.”
leg
leg
leg
leg
leg
leg
4. The hypotenuse of a right triangle is always the segment of a right triangle, which is opposite the right
angle. In the drawings below, show what is meant by the phrase opposite the right angle.
5. In the drawings below, there are several isosceles triangles. In each case, the vertex angle is labeled.
Using these drawings, come up with a definition for the term: vertex angle of an isosceles triangle.
Vertex Angle
Vertex Angle
Vertex Angle
The vertex angle of an isosceles triangle is:
6. In class we proved the statement: If two sides of a triangle are congruent, then the angles opposite
those sides are congruent.
(a) What is the converse of this statement?
(b) Are we allowed to assume the converse is true?
(c) Do you think the converse of this statement has to be true as well? Just write yes or no.
7. See if the converse can be proven using any of these three methods:
A
E
B
H
C
F
G
By adding a segment from
By adding a segment
point A to the midpoint of BC
from point E ⊥ to FG
8. Find x in the problems below:
I
By adding an ∠
bisector of∠IHJ
J
9. In each figure, O is the center of the circle. Solve for x.
10) AB & XY are the same length. Are they the same distance from the center of the circle? Be able
to prove your answer.
ANSWERS:
1. A) sometimes
b) always
2. The legs are the two equal sides. The base is the third side.
3. The angle “opposite” a side is the angle whose sides intersect the end points of that side. . . for example,
the hypotenuse of a right triangle is always OPPOSITE the 90° angle.
4.
5. The vertex angle of an isosceles triangle is the one formed by the two congruent sides.
6.
a. If the two base angles of a triangle are congruent, then the two sides opposite those angles are
also congruent.
b. No
c. .yes
7.
8. a) 36
b) 20
c) 45
d) 45
9.
a. 36
b. 69
c. 30
d. 36
e. 18
f. 40
g. 36
h. 60
i. 50
j. 120
k. 20
l. 25
10. Yes. . . my suspicion is that the proof involves either congruent triangles or isosceles triangles . . . or both.