Question

14 multiple choice 1 point which of the following is true for all parallelograms? diagonals are congruent. opposite angles are congruent. adjacent angles are congruent. adjacent sides are congruent. 15 multiple choice 1 point a quadrilateral must be a parallelogram if one pair of opposite sides is parallel, only parallel and the other pair of opposite sides is congruent congruent and parallel congruent, only

Explanation:

Question 14

Step1: Analyze each option

• Diagonals are congruent: This is true for rectangles (a type of parallelogram) but not for all parallelograms. For example, in a non - rectangle parallelogram (a rhombus that is not a square), the diagonals are not congruent.
• Opposite angles are congruent: By the properties of parallelograms, in a parallelogram \(ABCD\) with \(AB\parallel CD\) and \(AD\parallel BC\), we can use the alternate interior angles and the fact that the sum of adjacent angles is \(180^{\circ}\) to prove that \(\angle A=\angle C\) and \(\angle B = \angle D\).
• Adjacent angles are congruent: In a parallelogram, adjacent angles are supplementary (\(\angle A+\angle B = 180^{\circ}\)). They are congruent only when the parallelogram is a rectangle.
• Adjacent sides are congruent: This is true for rhombuses (a type of parallelogram) but not for all parallelograms. A general parallelogram (e.g., a non - rhombus rectangle) has adjacent sides of different lengths.

Question 15

Step1: Analyze each option

• One pair of opposite sides is parallel, only: A trapezoid has one pair of parallel sides and is not a parallelogram.
• One pair of opposite sides is parallel and the other pair of opposite sides is congruent: Consider an isosceles trapezoid. It has one pair of parallel sides and the non - parallel sides are congruent, but it is not a parallelogram.
• One pair of opposite sides is congruent and parallel: Let \(AB\parallel CD\) and \(AB = CD\). We can use the SAS (side - angle - side) congruence criterion. If we draw a diagonal \(AC\), in \(\triangle ABC\) and \(\triangle CDA\), \(\angle BAC=\angle DCA\) (alternate interior angles since \(AB\parallel CD\)), \(AB = CD\) (given) and \(AC=CA\) (common side). So, \(\triangle ABC\cong\triangle CDA\) by SAS. Then \(AD = BC\) and \(AD\parallel BC\) (since \(\angle CAD=\angle ACB\)), so the quadrilateral is a parallelogram.
• One pair of opposite sides is congruent, only: We can construct a non - parallelogram quadrilateral with one pair of congruent sides.

Answer:

1. Opposite angles are congruent.
2. congruent and parallel.