Here’s my answers from the Wrap-Up Discussion questions from page 528 of the article…
1) Can a triangle have more than one obtuse angle?
No. An obtuse angle equals to more than 90 degrees, and having two of them would equal to more than 180 degrees (not even including the third angle in the triangle). A triangle, by definition, has all the three angles equal to 180 degrees.
2) Can a triangle have three acute angles?
Yes. An acute angle is an angle less than 90 degrees. An example of a triangle with three acute angles could be: 45, 75, and 60 degrees or all three angles equaling 60 degrees (for a total of 180 degrees).
3) Can a triangle or a quadrilateral have exactly two right angles?
A triangle cannot have exactly two right angles because those two angles would equal to 180 degrees, which should be the total of all three angles of a triangle.
A quadrilateral can have exactly two right angles though. For example, this trapezoid:
4) Can a trapezoid have exactly one right angle?
No. A trapezoid can only have two (as pictured in the last answer) or none. Remember, a trapezoid is a quadrilateral with one pair of parallel sides.
5) Can a quadrilateral have two obtuse angles?
Yes. Examples include a rhombus, parallelogram, and a kite.
6) Can a quadrilateral have exactly three right angles?
No. By definition, the sum of the angles of a quadrilateral must equal to 360 degrees. Therefore, if it had three right angles, that would equal to 270 degrees, leaving 90 degrees for the last angle- which in essence would give the quadrilateral four (not exactly three) right angles. A quadrilateral can either have exactly two or exactly four right angles.
7) Can all four sides of a quadrilateral have different lengths?
Yes. A quadrilateral is any four-sided figure.
8) Can a five-sided shape, a pentagon, have five acute angles?
No. The sum of the interior angles of any polygon is (N-2) * 180 degrees, where N is the number of sides (and interior angles). Therefore, the sum of the interior angles of a pentagon would equal to (5-2) * 180 = 540 degrees. An acute angle must be equal to less than 90 degrees. If we had all five acute angles (even at their highest possible whole degree), that would give us five angles of 89 degrees. 89 * 5 = 445, which is (540-445) 95 degrees less than what we need to make this shape a pentagon. The most acute angles a pentagon could have would be four, with the fifth angle measuring more than 180 degrees (being called a reflex angle).
9) Can a six-shaded shape, a hexagon, have four obtuse angles?
Yes. For an example, take a regular hexagon that has 6 obtuse angles and make two of the side angles acute… overall, making the hexagon look a lot flatter.
10) Can an eight-sided shape, an octagon, have eight right angles?
No. Following the same formula I included in my answer for #8 above: the sum of the interior angles in an octagon will always be 1080 degrees, (8-2) * 180. If all of the angles were right angles, that would give us a total of 720 degrees, 8*90.
11) Describe what you have learned about each kind of polygon.
Basically, this exercise helped me to remember the definition of several different geometric terms. Here’s a few to keep in mind:
A trapezoid is a quadrilateral with only one pair of parallel sides.
A rhombus is a parallelogram with opposite equal acute and obtuse angles and four equal sides.
A parallelogram is a four-sided plane rectilinear figure with opposite sides parallel.
A kite, or deltoid, is a quadrilateral with two disjoint pairs of congruent adjacent sides. (In contrast, a parallelogram also has two pairs of equal-length sides, but they are opposite each other rather than adjacent.)
The interior angles of a triangle must equal to 180 degrees.
The sum of the interior angles of any polygon is (N-2) * 180 degrees, where N is the number of sides (and interior angles).
Here’s a chart of the different names of the polygons:
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