1) We know that the sum of the measures of the angles of a triangle equals \(180^{\circ}\) on the Euclidean plane.
Create a new disk (File-New). Draw a triangle on the hyperbolic plane. Use the hyperbolic measure tool to measure the interior angles of the triangle. Does the Triangle Sum Theorem hold on the hyperbolic plane?
Create a new disk (File-New). Draw a triangle on the hyperbolic plane. Use the hyperbolic measure tool to measure the interior angles of the triangle. Does the Triangle Sum Theorem hold on the hyperbolic plane?
2) We know that an exterior angle of a triangle equals the sum of its two non-adjacent interior angles in the Euclidean plane.
Create a new disk (File-New). Draw a triangle on the hyperbolic plane. Extend one of the sides of the triangle. Measure the exterior angle and compare this measure with the measure of the sum of the measure of the two non-adjacent interior angles. Does the exterior angle theorem hold on the hyperbolic plane?
Create a new disk (File-New). Draw a triangle on the hyperbolic plane. Extend one of the sides of the triangle. Measure the exterior angle and compare this measure with the measure of the sum of the measure of the two non-adjacent interior angles. Does the exterior angle theorem hold on the hyperbolic plane?
3) On the Euclidean plane, if two angles of one triangle are congruent to two angles of another triangle, then the third angles are congruent.
Create a new disk (File-New). Draw a triangle on the hyperbolic plane. Measure the angles of the triangle. Create a second triangle with two angles in the second triangle congruent to two angles in the first. Measure the third angle of the triangle. Are the third angles congruent?
Create a new disk (File-New). Draw a triangle on the hyperbolic plane. Measure the angles of the triangle. Create a second triangle with two angles in the second triangle congruent to two angles in the first. Measure the third angle of the triangle. Are the third angles congruent?
4) On Euclidean plane, the base angles of an isosceles triangle are congruent.
Create a new disk (File-New). Draw an isosceles triangle on the hyperbolic plane. Measure the angles at the base of the congruent sides. Are the base angles congruent?
Create a new disk (File-New). Draw an isosceles triangle on the hyperbolic plane. Measure the angles at the base of the congruent sides. Are the base angles congruent?
5) On Euclidean plane, if a triangle has two angles congruent then the sides opposite he congruent angles are congruent.
Create a new disk (File-New). Draw a triangle with two angles congruent on the hyperbolic plane. Measure the sides opposite the congruent. Are those sides congruent?
Create a new disk (File-New). Draw a triangle with two angles congruent on the hyperbolic plane. Measure the sides opposite the congruent. Are those sides congruent?
6) On Euclidean plane, each measure of each angle of an equilateral triangle is \(60^{\circ}\).
Create a new disk (File-New). Draw an equilateral triangle on the hyperbolic plane. Determine the measure of each angle of the equilateral triangle. How do your observations on the hyperbolic plane compare with those on the Euclidean plane?
Create a new disk (File-New). Draw an equilateral triangle on the hyperbolic plane. Determine the measure of each angle of the equilateral triangle. How do your observations on the hyperbolic plane compare with those on the Euclidean plane?
7) In a right triangle on the Euclidean plane, the square of the hypotenuse is equal to the sum of the squares on the legs of the triangle. Does Pythagorean Theorem hold for triangles on the hyperbolic plane?
Construct a number of right triangles on the hyperbolic plane. Use the hyperbolic measure segment option to measure the lengths of the hypotenuse and legs. Use the calculate command under the measure command to discover whether this theorem is valid on the hyperbolic plane.
Construct a number of right triangles on the hyperbolic plane. Use the hyperbolic measure segment option to measure the lengths of the hypotenuse and legs. Use the calculate command under the measure command to discover whether this theorem is valid on the hyperbolic plane.
8) The SAS Congruence Postulate states that if two sides and the included angle of one triangle are congruent respectively to two sides and the included angle of another triangle, then the two triangles are congruent.
Investigate whether this postulate can be accepted on the hyperbolic plane.
Investigate whether this postulate can be accepted on the hyperbolic plane.
9) The SSS Congruence Postulate states that if three sides of one triangle are congruent to three sides of another triangle, then the two triangles are congruent.
Investigate whether this postulate can be accepted on the hyperbolic plane. Explain your findings.
Investigate whether this postulate can be accepted on the hyperbolic plane. Explain your findings.
10) The ASA Congruence Postulate states that if two angles and the included side of one triangle are congruent respectively to two angles and the included side of another triangle, then the two triangles are congruent.
Investigate whether this postulate can be accepted on the hyperbolic plane. Explain your findings.
Investigate whether this postulate can be accepted on the hyperbolic plane. Explain your findings.
11) On the Euclidean plane, if three angles of one triangle are congruent to three angles of another triangle, then the corresponding sides of the triangles are in proportion and the two triangles are similar.
On the hyperbolic plane, if three angles of one triangle are congruent to three angles of another, what can we say about these two triangles?
On the hyperbolic plane, if three angles of one triangle are congruent to three angles of another, what can we say about these two triangles?
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