Question

isosceles triangles
key terms
isosceles triangle (eye•sahs•uh•leez) a triangle with at least two sides congruent
equiangular triangle (oh•kwee•an•gyu•lur) a triangle in which all three angles are congruent
equilateral triangle (ee•kwuh•lat•uh•rul) a triangle in which all sides are congruent
reading the lesson

1. determine whether each statement is always, sometimes, or never true. if the statement is not always true, explain why.

a. if a triangle has three congruent sides, then it has three congruent angles.
b. if a triangle is isosceles, then it is equilateral.
c. if two sides of a triangle are congruent, then the angles opposite those sides are congruent.
d. if an isosceles triangle has three acute angles, then it is equilateral.
e. if a triangle is equiangular, then it has three obtuse angles.

1. suppose an isosceles triangle has vertex angle ( v ) and other vertices labeled ( d ) and ( e ). describe the relationship between the median from ( v ), the angle bisector of ( angle v ), the altitude from ( v ), and the perpendicular bisector of ( overline{de} ).
2. a right triangle is also isosceles. what is the measure of each congruent angle of the triangle? explain how you found the measure.

helping you remember

1. write theorem 6–5 as two different if - then statements. one statement will be the converse of the other statement.

Explanation:

Step1: Address 1a (Always/Sometimes/Never)

An equilateral triangle (3 congruent sides) is always equiangular, so this is always true by the definition of equilateral/equiangular triangles.

Step2: Address 1b (Always/Sometimes/Never)

Isosceles triangles only need 2 congruent sides; equilateral needs 3. So this is sometimes true (only when the isosceles triangle has all 3 sides congruent).

Step3: Address 1c (Always/Sometimes/Never)

This is the Isosceles Triangle Theorem, a fundamental geometric rule, so this is always true.

Step4: Address 1d (Always/Sometimes/Never)

An isosceles triangle can have 3 acute angles without being equilateral (e.g., angles 70°, 70°, 40°). So this is sometimes true.

Step5: Address 1e (Always/Sometimes/Never)

An equiangular triangle has three 60° angles (acute, not obtuse). Obtuse angles are >90°, so this is never true.

Step6: Address Question 2

In an isosceles triangle, the median, angle bisector, altitude from the vertex angle, and perpendicular bisector of the base are the same line segment (they coincide).

Step7: Address Question 3

Sum of triangle angles is $180^\circ$. Right angle = $90^\circ$, so remaining two congruent angles sum to $180^\circ - 90^\circ = 90^\circ$. Each angle is $\frac{90^\circ}{2} = 45^\circ$.

Step8: Address Question 4

Theorem 6-5 refers to the Isosceles Triangle Theorem and its converse:

1. If two sides of a triangle are congruent, then the angles opposite those sides are congruent.
2. If two angles of a triangle are congruent, then the sides opposite those angles are congruent.

Answer:

1.
a. Always true: An equilateral triangle (3 congruent sides) is always equiangular.
b. Sometimes true: An isosceles triangle only needs 2 congruent sides, so it is only equilateral if all 3 sides are congruent.
c. Always true: This is the Isosceles Triangle Theorem.
d. Sometimes true: An isosceles triangle can have 3 acute angles (e.g., 70°, 70°, 40°) without being equilateral.
e. Never true: An equiangular triangle has three 60° acute angles, not obtuse angles.

1. The median from $V$, the angle bisector of $\angle V$, the altitude from $V$, and the perpendicular bisector of $\overline{DE}$ are all the same line segment (they coincide).

1. $45^\circ$. The right angle is $90^\circ$, so the remaining two congruent angles sum to $180^\circ - 90^\circ = 90^\circ$. Dividing this by 2 gives $45^\circ$ for each congruent angle.

4.

1. If two sides of a triangle are congruent, then the angles opposite those sides are congruent.
2. If two angles of a triangle are congruent, then the sides opposite those angles are congruent.