Question

isosceles and equilateral triangles

1. which of the following statements is incorrect?

a ( fe = 58 )
b ( fe = 40 )
c ( x = 8 )
d ( df = 40 )
(figure: triangle dfe with ( df = (8x - 24) ), ( de = 40 ), ( fe = (6x + 10) ), angles at f and e marked as equal)

1. which of the following statements is incorrect?

a a triangle with three congruent sides is equiangular.
b the isosceles triangle theorem can be applied to equilateral triangles.
c the measure of each angle of an equilateral triangle is 120.
d a triangle with three congruent angles is equilateral.

1. in the figure shown, what is ( mangle a )? explain.

a 57; ( \triangle abc ) is an isosceles triangle with base angles a and c. ( mangle a = mangle c ).
b 66; ( \triangle abc ) is an isosceles triangle with base angles b and c. ( mangle b = mangle c = 57 ), and ( mangle a + mangle b + mangle c = 180 ).
c 57; ( \triangle abc ) is an equilateral triangle.
d there is not enough information to find ( mangle a ).
(figure: triangle abc with ab and ac marked as equal, ( angle c = 57^circ ))

1. an equiangular triangle has one side of length six inches. what is the perimeter of the triangle, in inches?

(blank box for answer)

1. what are the measures of the angles in the triangle?

a 60; 60; 60
b 78; 51; 51
c 34; 112; 112
d 112; 34; 34
(figure: isosceles triangle with angles ( (x + 17)^circ ), ( (4x - 34)^circ ), ( (6x + 10)^circ ), two sides marked as equal)

Explanation:

Question 1

Step1: Identify triangle type

Triangle \( DEF \) has \( \angle F \cong \angle E \), so it's isosceles with \( DF = DE \). Given \( DE = 40 \), so \( DF = 40 \) (so D is correct). Thus, \( 8x - 24 = 40 \).

Step2: Solve for \( x \)

\( 8x - 24 = 40 \)
\( 8x = 40 + 24 = 64 \)
\( x = \frac{64}{8} = 8 \) (so C is correct).

Step3: Find \( FE \)

\( FE = 6x + 10 = 6(8) + 10 = 48 + 10 = 58 \). So \( FE = 58 \) (A correct), \( FE
eq 40 \) (B incorrect).

• A: Equilateral triangles (3 congruent sides) are equiangular (3 congruent angles) – correct.
• B: Isosceles Triangle Theorem (base angles congruent) applies to equilateral (all sides congruent, so any two sides form base) – correct.
• C: Sum of angles in triangle is \( 180^\circ \). Equilateral triangle angles are \( \frac{180}{3} = 60^\circ \), not \( 120^\circ \) – incorrect.
• D: 3 congruent angles imply 3 congruent sides (equilateral) – correct.

Triangle \( ABC \) has \( AB \cong AC \) (marked), so it's isosceles with base \( BC \). Thus, base angles \( \angle B \cong \angle C \), so \( m\angle B = m\angle C = 57^\circ \). Using angle sum: \( m\angle A + 57 + 57 = 180 \) → \( m\angle A = 180 - 114 = 66^\circ \).

• A: Incorrect (base angles are \( B \) and \( C \), not \( A \) and \( C \)).
• B: Correct (matches calculation).
• C: Not equilateral (only two sides marked congruent).
• D: Enough info (isosceles + angle sum).

Answer:

B. \( FE = 40 \)

Question 2