Question

1. according to the alternate interior angles conjecture, if parallel lines ( l_1 ) and ( l_2 ) are cut by a transversal ( t ), and ( angle a = 75^{circ} ), what is the measure of the alternate interior angle ( angle t )? a. ( 70^{circ} ) b. ( 75^{circ} ) c. ( 85^{circ} ) d. ( 80^{circ} ) 26. the diagonals of a certain quadrilateral intersect at 90 degrees. this quadrilateral could be a a. trapezoid b. rhombus c. rectangle d. kite 27. why are squares and other regular polygons often used in cultural mosaic artworks? a. they are symmetrical b. they can tell stories or highlight cultural significance through patterns c. they are resistant to wear d. they are easy to draw

Explanation:

Step1: Recall alternate - interior angles property

When two parallel lines are cut by a transversal, alternate - interior angles are congruent. Given $\angle4 = 75^{\circ}$, the alternate - interior angle $\angle T$ has the same measure. So for question 25, $\angle T=75^{\circ}$.

Step2: Recall properties of quadrilaterals

The diagonals of a trapezoid are not perpendicular in general. The diagonals of a rectangle are equal in length but not perpendicular (except for a square which is a special rectangle). The diagonals of a kite are perpendicular. The diagonals of a rhombus are perpendicular bisectors of each other. So for question 26, both a rhombus and a kite have diagonals that intersect at 90 degrees.

Step3: Analyze the use of regular polygons in mosaics

Regular polygons like squares are often used in cultural mosaic artworks because they can create patterns that can tell stories or highlight cultural significance. While they are symmetrical, the main reason in the context of cultural art is their pattern - forming ability to convey meaning. So for question 27, the best answer is that they can tell stories or highlight cultural significance through patterns.

Answer:

1. B. 75°
2. B. rhombus, d. kite
3. B. They can tell stories or highlight cultural significance through patterns