More than one feature can be combined into a triangle. Decide which of the following combinations are possible. If the combination is possible, draw a sketch. If not, explain why not.
a.
a scalene acute triangle
b.
an isosceles acute triangle
c.
an equilateral acute triangle
d.
a scalene obtuse triangle
e.
an isosceles obtuse triangle
f.
an equilateral obtuse triangle
g.
an equilateral triangle that is also isosceles
Problem H2
A certain quadrilateral has one diagonal that is 2 inches long and another diagonal that is 3 inches long. A diagonal is a line segment connecting any two non-adjacent vertices.
a.
Draw two such quadrilaterals. What, if anything, do they have in common? How are they different?
b.
Draw such a quadrilateral where the diagonals bisect each other but are not perpendicular. What does it look like?
c.
Draw such a quadrilateral where the diagonals are perpendicular but do not bisect each other. What does it look like?
d.
Draw such a quadrilateral where the diagonals bisect each other and are perpendicular. What does it look like?
Problem H3
Create a quadrilateral with diagonals that are the same length and bisect each other. What kind of quadrilateral is it? Can you explain why?
Problem H4
Create a quadrilateral with diagonals that are the same length, bisect each other, and are perpendicular. What kind of quadrilateral is it? Can you explain why?
Problem H5
For each part below, draw two different triangles that fit the information given. What do you notice?
a.
One side is 2 inches long; another side is 3 inches long. The angle between them is 45°.
b.
One side is 2 inches long; another side is 3 inches long. The angle between them is 75°.
c.
One side is 2 inches long; another side is 3 inches long. The angle between them is 90°.
Are the triangles different or congruent? If you think they are congruent, try to draw a triangle that fits the description but is not congruent. Close Tip
Problem H6
You have already seen the SSS (side-side-side) congruence test for triangles: If the three sides of one triangle have the same lengths as the three sides of another triangle, then the two triangles are congruent. That is, they have exactly the same size and shape. Describe and name a new congruence test based on your work in Problem H5.
Problem H7
For each part below, draw two different triangles that fit the information given. What do you notice?
a.
The three angle measures are 45°, 45°, and 90°.
b.
The three angle measures are 60°, 60°, and 60°.
c.
The three angle measures are 100°, 30°, and 50°.
Are the triangles different or congruent? If you think they are congruent, try to draw a triangle that fits the description but is not congruent. Close Tip
Problem H8
Is there an angle-angle-angle (AAA) congruence test for triangles? That is, if the three angles of one triangle have the same measures as the three angles of another triangle, are the two triangles necessarily congruent? Explain your answer.
Use your work in Problem H7 to answer this question. Close Tip
