Question 1: In an isosceles triangle, the two equal angles measure 35° each. Calculate the measure of the third angle.

Answer: 110

Question 2: Construct an isosceles triangle where the base angles are each 40°. Use the grid to draw the triangle accurately.

Answer: Student draws with base angles 40°, vertex angle 100°

Question 3: An isosceles triangle has a vertex angle of 60°. Find the measure of each of the equal side angles.

Answer: 60

Question 4: Plot a triangle on the grid where two sides are equal, and the vertex angle is 50°. Label all angles clearly.

Answer: Student creates an isosceles triangle with vertex angle 50°, base angles 65° each

Question 5: In an isosceles triangle, if the two equal sides measure 8 cm each and the base angles are 45°, find the height of the triangle from the vertex to the base.

Answer: 5.66

Question 6: A real-world scenario: A triangular traffic sign is isosceles with a vertex angle of 70°. The two equal sides are 15 meters long. How far apart are the two base points?

Answer: Approximately 16.6 meters

Question 7: Identify and correct the mistake: An isosceles triangle has base angles 50° each, and the vertex angle is 80°. Is this possible? Explain.

Answer: No, because 50° + 50° + 80° = 180°, which is valid; but as base angles are 50°, vertex angle should be 80°, which matches. The mistake is in thinking it's impossible; actually, it's valid.

Question 8: Derive a formula for the measure of the vertex angle in an isosceles triangle if the equal sides are of length 'a' and the base is of length 'b'.

Answer: Using Law of Cosines: angle = arccos((a^2 + a^2 - b^2)/(2a^2))

Question 9: Challenge: Construct an isosceles triangle with a vertex angle of 30° and equal sides of length 10 cm. Calculate the base length using the grid.

Answer: Approximately 17.32 cm

Question 10: Compare two isosceles triangles: one with base angles 40° and the other with base angles 60°. Which one has a larger area assuming the same side lengths? Explain your reasoning.

Answer: The triangle with 60° base angles has a larger area because its height is greater given the same side length.

Question 11: Identify and fix the error in this statement: 'In an isosceles triangle, the sum of the two equal angles is always 120°.'

Answer: The sum of the two equal angles depends on the third angle; they are not necessarily 120° together.