Question

a triangle has a line of symmetry. select all conclusions that must be true. a all sides of the triangle have the same length. b all angles of the triangle have the same measure. c no sides of the triangle have the same length. d no angles of the triangle have the same measure. e two sides of the triangle have the same length. f two angles of the triangle have the same measure.

Explanation:

Step1: Recall the definition of a line - symmetric triangle

A triangle with a line of symmetry is an isosceles triangle (at least two sides equal) or an equilateral triangle (all sides equal).

Step2: Analyze each option

• Option A: An equilateral triangle has all sides equal, but an isosceles triangle (which also has a line of symmetry) does not. So this is not always true.
• Option B: An equilateral triangle has all angles equal (\(60^{\circ}\)), but an isosceles triangle (with a line of symmetry) has two angles equal and one different. So this is not always true.
• Option C: If a triangle has a line of symmetry, it has at least two sides equal. So this is false.
• Option D: If a triangle has a line of symmetry, it has at least two angles equal. So this is false.
• Option E: By the definition of a triangle with a line of symmetry (isosceles or equilateral), it has at least two sides equal.
• Option F: By the property of an isosceles or equilateral triangle (which has a line of symmetry), it has at least two angles equal.

Answer:

E. Two sides of the triangle have the same length; F. Two angles of the triangle have the same measure.