Question

in exercises 2 and 3, determine which two statements contradict each other. explain your reasoning.② a $\triangle lmn$ is equilateral.b $lm
eq mn$c $angle l = angle m$③a $\triangle abc$ is a right triangle.b $angle a$ is acute.c $angle c$ is obtuse.ccomplete the phrase with the most logical conclusion.1. if there is no right angle in a triangle, then...2. if two lines do not have the same slope, then...3. if a quadrilateral does not have four right angles, then...4. if no two angles of a triangle are congruent, then...5. if the sum of the measures of the interior angles of a polygon is not $180^circ$, then...

Explanation:

For Exercise 2:

An equilateral triangle has all sides congruent ($LM=MN=NL$) and all angles congruent ($\angle L=\angle M=\angle N$). Statement A claims $\triangle LMN$ is equilateral, while Statement B says $LM
eq MN$ (a direct contradiction of equilateral side properties), and Statement C says $\angle L=\angle M$ (consistent with equilateral triangle angle properties).

A right triangle has exactly one right angle (90°), with the other two angles being acute (sum to 90°). Statement A claims $\triangle ABC$ is a right triangle, while Statement C says $\angle C$ is obtuse (greater than 90°). A triangle cannot have both a right angle and an obtuse angle (their sum would exceed 180°, violating the triangle angle sum theorem). Statement B says $\angle A$ is acute, which is consistent with a right triangle.

1. A triangle's angles are either acute, right, or obtuse; no right angle means all angles are between 0° and 90° (acute) or one is obtuse.
2. Lines with different slopes are not parallel, so they intersect at exactly one point.
3. A quadrilateral can have 0, 1, 2, 3, or 4 right angles; not having 4 means it has 0-3 right angles.
4. A triangle with no congruent angles has all sides of different lengths by the converse of the isosceles triangle theorem.
5. The sum of interior angles of a triangle is 180°, so if the sum is not 180°, the shape cannot be a triangle.

Answer:

A. $\triangle LMN$ is equilateral, B. $LM
eq MN$

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For Exercise 3: