Question

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

Explanation:

Step1: Recall line - symmetry property

A line of symmetry divides a figure into two congruent parts. In a quadrilateral with a line of symmetry, the parts on either side of the line of symmetry are mirror - images of each other.

Step2: Analyze side lengths

Since the two parts are mirror - images, at least two sides (the ones that are mirror images of each other) must have the same length. So, option C is correct and option E is incorrect. A quadrilateral with a line of symmetry doesn't necessarily have all sides equal (e.g., a kite), so option A is incorrect.

Step3: Analyze angle measures

Similarly, at least two angles (the ones that are mirror images of each other) must have the same measure. So, option D is correct and option F is incorrect. A quadrilateral with a line of symmetry doesn't necessarily have all angles equal (e.g., a kite), so option B is incorrect.

Answer:

C. Two sides of the quadrilateral have the same length.
D. Two angles of the quadrilateral have the same measure.