Question 1: An isosceles triangle has two equal sides of length 7 cm. The base angles are equal. If one of the base angles measures 50°, find the vertex angle.

Answer: 80

Question 2: In an isosceles triangle, the vertex angle is 40°. Find the measure of each of the base angles.

Answer: 70

Question 3: Construct an isosceles triangle where the equal sides are 5 cm each, and the vertex angle is 60°. Use the grid to sketch your triangle.

Answer: Student's drawn triangle on grid

Question 4: Calculate the base angles of an isosceles triangle if the vertex angle is 30°.

Answer: 75

Question 5: A triangle with sides of lengths 8 cm, 8 cm, and 5 cm is isosceles. Which sides are equal? Justify your answer.

Answer: Sides of 8 cm are equal; the triangle is isosceles with equal sides of 8 cm.

Question 6: In an isosceles triangle, if the equal sides are each 10 cm and the base angles are 45°, find the length of the base.

Answer: 14.1

Question 7: Explain why the base angles in an isosceles triangle are equal. Provide a brief reasoning.

Answer: Because the sides opposite the equal angles are equal, the angles themselves are equal by the Isosceles Triangle Theorem.

Question 8: Identify the error in the following statement: 'In an isosceles triangle, the base angles are always 60°.'

Answer: The statement is incorrect; base angles are equal but not necessarily 60°.

Question 9: A student claims that in an isosceles triangle, if one base angle is 80°, then the vertex angle must be 20°. Is this correct? Explain your reasoning.

Answer: No, because the sum of all angles in a triangle is 180°, so vertex angle = 180° - 2×80° = 20°. The claim is correct.

Question 10: Challenge: An isosceles triangle has a perimeter of 24 cm. The two equal sides are each 7 cm. Find the length of the base.

Answer: 10